Magnetic Scattering with Polarised Soft X-rays

Abstract

Soft X-ray scattering is a powerful technique for measuring magnetic materials. By highlighting some examples using diffraction, small angle scattering and reflectivity the element sensitivity and strong dependence of the polarisation on both the size and direction of the magnetic moments in both single crystals and thin films will be demonstrated.

1. Introduction

The interaction of light with magnetism was first discovered by Michael Faraday in 1845 when he observed that magnetised heavy glass would rotate the plane of polarised light as it was transmitted [1]. A few decades later John Kerr discovered the same magneto-optical effect but in a reflection geometry [2]. This proved the link between optics and magnetism, theoretically explained by James Clark Maxwell [3]. Whilst these first experiments were done using optical wavelengths [4, 5, 6, 7, 8] the first results using X-rays were not measured until 1972 (de Bergevin and Brunel) [9]. In this experiment, which was built on a previous idea (Platzmann and Tzoar 1970) [9] a laboratory X-ray source was used to measure the antiferromagnetic order in NiO. Several days were needed to collect the weak signal from the ($\frac{1}{4}$$\frac{1}{4}$$\frac{1}{4}$) peak due to the antiferromagnetic ordering between the main structural (charge) Bragg peaks.

This experiment was one of the first to prove that X-rays could be used to measure magnetism and that magnetic diffraction did not have to only rely on neutron diffraction. Indeed de Bergevin and Brunel neatly demonstrated that the interaction of both the electric and magnetic parts of X-ray. Unfortunately the interaction with the spin compared to the charge, is scaled by a relativistic factor of $E_1/{\mathit{mc}}_2$, where $E_1$ is the energy of the incident photon and $m$ is the rest mass of the electron. This means that the magnetic scattering is reduced by a factor of 100 for each electron. Since not all electrons contribute to the magnetic signal the total magnetic signal is very weak compared to the scattering from structural(charge) Bragg peaks.

With investment in synchrotron radiation sources in the early 1980s, such as the SRS, Daresbury, UK or NSLS, New York, USA, the ability to separate the weak magnetic scattering from the noise was increased by several orders of magnitude. With the high intensity of the synchrotron radiation and the well-defined polarisation meant that the effects discovered by de Bergevin and Brunel, which were weak and heavily polarisation dependent could be exploited. Magnetic scattering was now becoming a viable contender for measuring magnetism along with neutron scattering. The two techniques are actually very complimentary. The more bulk sensitive neutron scattering technique compares with a relative surface sensitive X-ray technique. An advantage of X-ray is the ability to be able to separate the spin and orbital parts of the electron angular momentum. This advantage is made possible through the different polarisation dependences of the scattering which had clearly been enhanced by synchrotron radiation.

A big breakthrough came at the end of the 1980s when Hannon et al. [10] discovered that magnetic scattering was enhanced at certain atomic resonances, in particular those from the dipolar transitions. Similarly to non-resonant scattering the spin and orbital parts of the electron could be separated. However, now the technique is element sensitive. The ability to access many energies on beamlines at synchrotrons enabled difference resonances or absorption edges to be accessed. In addition the dipole resonances enhancement are very strong at soft X-ray energies, which cover the $L_{2.3}$ resonances of the transition metals, the $M_{2,3}$ resonances of the rare earths and the $N$ edges of the actinides.

In this chapter we will first discuss some theoretical preliminaries for resonant scattering, then soft X-ray diffraction followed by, small angle scattering, soft X-ray reflectivity and element specific hysteresis curves.

2. Theoretical preliminaries

X-ray magnetic scattering can be measured on or off an atomic resonance. The non-resonant scattering is stronger as the energy of the incident photon increases due to the relativistic factor mentioned previously. It is possible to measure X-ray magnetic scattering for energies above a keV (wavelengths of Angstroms) [11]. However, this is very weak at soft X-ray energies which are defined as energy between 100 and 2000eV (6.2 to 124 Angstroms). At both soft and hard X-ray energies magnetic scattering is enhanced by going to a resonance where a core electron with a well-defined spin (spin up or spin down) is transferred to the unoccupied states in the outer electron levels (same as the Fermi energy in metals). The well-defined spin then becomes a very sensitive probe of its environment which is short lived as it decays back to its core level emitting a photon of equal energy to the incident one (elastic scattering). In the dipole approximation the spin does not flip so spin is preserved throughout this process which make it very sensitive to the magnetic moment of the atom, since the outer electron levels is the magnetic environment. In addition the magnetic order breaks the symmetry of the lattice, since this is a vector quantity thus any experiment involving this resonant process i.e. X-ray absorption or X-ray scattering has strong polarisation dependence.

We will not discuss non resonant scattering but there are reviews in the literature [12, 13, 14] as well as some of the first work by De Bergevin, Brunel, Gibbs and Blume to name but a few [9, 12].

The amplitude of electric dipole transitions can be written as [10, 16].

$$f=\left({\mathbf{e}}_{\mathbf{f}}\cdot {\mathbf{e}}_{\mathbf{i}}\right)F^{\left(0\right)}-i\left({\mathbf{e}}_{\mathbf{f}}\times {\mathbf{e}}_{\mathbf{i}}\right)\cdot \mathbf{M}{F}^{\left(1\right)}+\left({\mathbf{e}}_{\mathbf{f}}\cdot \mathbf{M}\right)\left({\mathbf{e}}_{\mathbf{i}}\cdot \mathbf{M}\right)F^{\left(2\right)}$$E1

Here e${}_i$and e${}_f$ are directional vectors representing the incident and scattered polarisation respectively, M is the magnetic moment and the coefficients F${}^{\left(0\right)}$, F${}^{\left(1\right)}$and F${}^{\left(2\right)}$ depend on the matrix elements involved in the resonant process. The discussion of these coefficients are out of the scope of this work and not necessary for this chapter or our conclusions but some comments will be necessary. The first term is the resonant charge scattering and has the same form as the non-resonant charge scattering here the dot product between the polarisation vectors is simply due to experimental geometry i.e. the position of the detector relative to the incident beam. As the detector angle increases the dot products of the polarisation vectors components in the scattering plane has a cosine dependence and therefore gets weaker. The second term is a first order in magnetic moment. It involves a triple product with the cross product between the two polarisation vectors dotted with the magnetic moment. This term is related to circular dichroism in absorption. The third term is the second order in magnetic moment and depends on the projection of each polarisation vector with the magnetic moment. It gives raise to the linear dichroism in absorption. When referring to the above we will refer to the reference frame shown in Figure 1. Here we define the polarisation vectors ${\pi }_i$ and ${\pi }_f$ in the scattering plane and ${\sigma }_i$ and ${\sigma }_f$ perpendicular to the scattering plane for incoming and outgoing polarisations respectively. With this in mind and defining ${\theta }_i$ and ${\theta }_f$ as the incoming and outgoing grazing angles we can define the following vectors.

Figure 1.

The frame of reference used for the calculations of polarisation dependent scattering. The Greek symbols $\pi$ and $\sigma$ refer to polarisation that are parallel or perpendicular to the scattering plane (plane defined by incoming and outgoing beam) respectively. The suffixes i and f refer to the incident and scattered polarisation. The incident and outgoing angles are represented by ${\theta }_i$ and ${\theta }_f$ respectively. A right handed set with unit vectors i, j and k is shown.

$${\mathit{\pi }}_{\mathbf{i}}={\pi }_{i}sin\left({\theta }_i\right)\mathbf{i}+{\pi }_{i}cos\left({\theta }_i\right)\mathbf{k}$$E2

$${\mathit{\sigma }}_{\mathbf{i}}={\sigma }_i\mathbf{j}$$E3

$${\mathit{\pi }}_{\mathbf{f}}=-{\pi }_{f}sin\left({\theta }_f\right)\mathbf{i}+{\pi }_{f}cos\left({\theta }_f\right)\mathbf{k}$$E4

$${\mathit{\sigma }}_{\mathbf{f}}={\sigma }_f\mathbf{j}$$E5

With this frame of reference we would like to construct the following matrix equation where each element represents a well-defined initial and final polarisation state.

$$f=\left(\begin{array}{cc}{\sigma }_i\to {\sigma }_f& {\pi }_i\to {\sigma }_f\\ {\sigma }_i\to {\pi }_f& {\pi }_i\to {\pi }_f\end{array}\right)F^{\left(0\right)}-i\left(\begin{array}{cc}{\sigma }_i\to {\sigma }_f& {\pi }_i\to {\sigma }_f\\ {\sigma }_i\to {\pi }_f& {\pi }_i\to {\pi }_f\end{array}\right)F^{\left(1\right)}+\left(\begin{array}{cc}{\sigma }_i\to {\sigma }_f& {\pi }_i\to {\sigma }_f\\ {\sigma }_i\to {\pi }_f& {\pi }_i\to {\pi }_f\end{array}\right)F^{\left(2\right)}$$E6

If we assume that ${\theta }_i$ and ${\theta }_f$ are equal to $\theta$ as in specular reflectivity and use Eqs. (2)–(5) one can rewrite the Eq. (1)

$$\begin{array}{c}f=\left(\begin{array}{cc}1& 0\\ 0& cos2\theta \end{array}\right)F^{\left(0\right)}-i\left(\begin{array}{cc}0& m_{i}cos\theta -m_{k}sin\theta \\ -m_{i}cos\theta -m_{k}sin\theta & m_{j}sin2\theta \end{array}\right)F^{\left(1\right)}\\ +\left(\begin{array}{cc}{m}_k^2& m_k\left(m_{i}sin\theta +m_{k}sin\theta \right)\\ -m_k\left(m_{i}sin\theta -m_{k}sin\theta \right)& -{cos}^2\theta \left(m_i^2{tan}^2\theta +m_k^2\right)\end{array}\right)F^{\left(2\right)}\end{array}$$E7

Where

$$\mathbf{M}=m_i\widehat{\mathbf{i}}+m_j\widehat{\mathbf{j}}+m_k\widehat{\mathbf{k}}$$E8

Although this is just another version of Eq. (1) in a particular frame of reference, it makes it easier to see that the off-diagonal components within the first order term only depend on the magnetic moment within the scattering plane i.e. $m_i$ and $m_k$ and the diagonal term only depends on the magnetic moment out of the scattering plane $m_j$. The second order term in magnetic moment is more complicated and allows magnetic scattering in the ${\sigma }_i\to {\sigma }_f$ channel of the matrix. The second order terms matrix tend to be small so we will ignore this for most of the chapter but some comments will be made on this when we discuss diffraction.

We will now apply these equations in three different situations. In the next section we will briefly examine the subject of diffraction, then small angle scattering and followed by a section dedicated to reflectivity measurements.

3. Diffraction

There are many exciting materials with large enough unit cells to enable the Bragg condition to be satisfied at soft X-ray wavelengths. In addition since magnetism lowers the symmetry of the crystal lattice, it is possible that extra diffraction peaks will occur in between the main Bragg peaks, due to the larger magnetic unit cell. This can help enormously with soft X-ray scattering since even if it is not possible to reach one of the main Bragg peaks it may be possible to reach a magnetic diffraction peak.

In kinematical theory we sum up the diffraction amplitudes as follows

$$A\left(\mathbf{Q}\right)=\sum _{n=o}^{N-1}{f}_{n}exp\left(i\mathbf{Q}.{\mathbf{r}}_n\right)$$E9

where $f_n$ is the form factor of a particular element, $\mathbf{q}$ is the reciprocal lattice vector and ${\mathbf{r}}_n$ is the real space position of the atoms in the lattice. In the case where the scattering ion is at resonance we need to replace the form factor $f_n$ with the anomalous corrections as shown in Eq. (10).

$$f\to f+f^{\prime }+i{f^{\prime }}^{\prime }-{\mathit{\text{if}}}_m$$E10

where the $f$ is the non-dispersive atomic form factor, $f^{\prime }$ and ${f^{\prime }}^{\prime }$ are the real and imaginary parts of the dispersion corrections to the charge scattering and $f_m$ is the contribution from the magnetic scattering.

3.1 Commensurate antiferromagnet

The system ${\mathit{Co}}_x{\mathit{Mn}}_{1-x}{\mathit{WO}}_4$ [17] which has many different phases one of which is an interesting multiferroic phase at low temperatures, another is a commensurate antiferromagnetic structure. The magnetic moments on the Mn atoms in this phase align antiparallel along the a direction. The lowering of the crystal symmetry means that the unit cell is doubled compared to the chemical unit cell. This means that in between the chemical Bragg peaks there are peaks at “half order” positions that are purely magnetic. This is demonstrated with a simple schematic in Figure 2.

Figure 2.

A schematic demonstrating how doubling the size of the unit cell, shown on left, results in half order peaks in reciprocal space, shown on right.

The antiferromagnetic phase in ${\mathit{Co}}_x{\mathit{Mn}}_{1-x}{\mathit{WO}}_4$, known as the AF4 phase, exists in samples with $x$ = 0.15 below 18 K. So by going to the Co or Mn $L_{2,3}$ resonance and then putting the sample and detector at the correct point in reciprocal space we should be able to measure the Brag diffraction due to the AF4 antiferromagnetic order. This is shown in 2 where we see the resonance enhancement appears very clearly at the Co $L_3$ edge at the ($\frac{1}{2}$ 0 0) position. One of the advantages of X-ray resonant scattering is the ability to tune and distinguish between different elements. This is nicely demonstrated in this sample where it was possible to tune to the Co $L_3$ and Mn $L_2$ edges and follow their evolution with temperature. Both peaks behave similarly and decay as expected from previous work. Whilst this demonstrates the power of soft X-ray scattering, and X-ray scattering in general, particularly with element specificity it is worth noting the disadvantages. Firstly having the half order peak was necessary so that there was a peak existing in reciprocal space that could be measured i.e. where the Bragg condition could be satisfied. Also since the Mn resonances occur at lower energy right at the limit of where the Bragg condition could be satisfied. From this particular sample, it was only possible to measure the ($\frac{1}{2}$ 0 0) peak at the Mn $L_2$ edge since the Mn $L_3$ edge, occurring at 638.7 eV, was too low in energy.

Another tool one can use in soft X-ray scattering is polarisation analysis. By looking at the form of Eq. (1). In particular the first order in magnetic moment, the triple product $\left({\mathbf{e}}_{\mathbf{f}}\times {\mathbf{e}}_{\mathbf{i}}\right)\cdot \mathbf{M}$ involves the incoming polarisation ${\mathbf{e}}_{\mathbf{i}}$, the outgoing polarisation ${\mathbf{e}}_{\mathbf{f}}$ along with the magnetic moment $\mathbf{M}$. This vector nature of this process means that if one is able to define the incoming polarisation, measure the outgoing polarisation along with its intensity then it should be possible to measure the direction of the magnetic moment. Since there are an infinite number of solutions if the measurement is only done using one incident polarisation, a technique known as full polarisation analysis is used. Here several incoming polarisations are used and then the outgoing polarisation is measured for each one. The outgoing polarisation is measured by taking the scattered beam and diffracting it at 90 ${}^{\circ }$ using a special multilayer analyser. With a specially designed detector mount the scattered beam polarisation is measured by rotating the detector and analyser around the scattered beam. The results of doing this at the Co $L_3$ edge are shown in Figure 3. By fitting the $\left({\mathbf{e}}_{\mathbf{f}}\times {\mathbf{e}}_{\mathbf{i}}\right)\cdot \mathbf{M}$ for the outgoing polarisations for several incoming polarisations it was possible to determine the direction of the magnetic moments. The resonant nature of this scattering also meant that it was possible to ascertain that the Mn and Co moments are non collinear, a measurement that would not have been possible with other techniques. The non collinearity, a result of competition between the ${\mathit{Co}}^{2+}$ and ${\mathit{Mn}}^{2+}$ single ion anisotropies furthers the understanding of the complex magnetic phase diagram of this material (Figure 4) [18].

Figure 3.

The ($\frac{1}{2}$ 0 0) peak of ${\mathit{Co}}_{0.15}{\mathit{Mn}}_{0.85}{\mathit{WO}}_4$. On the left is shown the intensity (symbols) of the peak as one changes the energy through the Co $L_{2,3}$ resonances. Also shown are the fluorescence signals (line) as a function of energy in two different crystalline directions. On the right is the intensity at the Co $L_3$ and Mn $L_2$ edge as a function of temperature.

Figure 4.

On the left is shown the full polarisation analysis measurement at the Co edge. For each angle of incident linear polarisation given by $\chi$ the analysis of the polarisation was performed by rotating the detector analyser around the scattered beam (rotation of detector analyser angle $\eta$). A value of zero is defined as perpendicular to the scattering plane. By doing this at both the Co $L_3$ and Mn $L_2$ edges it was possible to measure both moments. The results are depicted in the picture on the right where it was established that the Co and Mn moments were not parallel.

3.2 Incommensurate structures

In addition to commensurate magnetic lattices there are examples of magnetic lattices that are incommensurate with the chemical structure. Such structures still provide diffraction peaks as can easily be shown in the following example. If we take Eq. (9) for a one dimensional lattice and add in an incommensurate modulation in the magnetic moments similar to an example shown in [19] (see section 4.4.5) but adapted to magnetism.

$$A\left(\mathbf{Q}\right)=\sum _{n=o}^{N-1}\left(f+f^{\prime }+i{f^{\prime }}^{\prime }\right)exp\left(\mathit{iQ}.r_n\right)-{\mathit{\text{if}}}^{m}exp\left(\mathit{iQ}.r_n\right)exp\left({\mathit{iq}}_m.r_{n}ucos\left(q.r_n\right)\right)$$E11

In this equation we have assumed a complex atomic form factor $f+f^{\prime }+i{f^{\prime }}^{\prime }-{\mathit{\text{if}}}_m$ where $f$ is the non-dispersive form factor with the real and imaginary terms of the dispersive form factors $f^{\prime }$ and ${f^{\prime }}^{\prime }$ respectively. There is also an additional part $f_m$ due to the magnetic moment which includes all the magnetic terms in 1. The second order term, however, will be assumed to be negligible. The term $u$ is the amplitude of a wave which forms the incommensurate structure with a periodicity given by $q_m=2\pi /{\lambda }_m$. We can expand the exponential containing the modulation such that:

$$A\left(\mathbf{Q}\right)=\sum _{n=o}^{N-1}\left(f+f^{\prime }+i{f^{\prime }}^{\prime }\right)exp\left(\mathit{iQ}.r_n\right)-{\mathit{\text{if}}}^{m}exp\left(\mathit{iQ}.r_n\right)\left(1+\mathit{iQu}cos\left(q_m.r_n\right)+\dots \right)$$E12

To first order gives.

$$A\left(\mathbf{Q}\right)=\sum _{n=o}^{N-1}\left(f+f^{\prime }+i{f^{\prime }}^{\prime }-{\mathit{\text{if}}}_m\right)exp\left(\mathit{iQ}.r_n\right)+f^m\left(\frac{\mathit{Qu}}{2}\right)\left[exp\left(i\left(Q+q_m\right).r_n\right)+exp\left(i\left(Q-q_m\right).r_n\right)\right]$$E13

By writing $r_n=\mathit{\text{an}}$, where a is the lattice parameter of the one dimensional lattice we can work out the summations. The modulus squared then gives us the intensity which, in the limit of large $N$, yields the following.

$$I\left(\mathbf{Q}\right)=\begin{array}{l}N\left(\frac{2\pi }{a}\right)\left[{\left(f+f^{\prime }\right)}^2+{\left({f^{\prime }}^{\prime }-f_m\right)}^2\right]\delta \left(Q-G\right)\\ +N{\left(\frac{\mathit{Qu}}{2}\right)}^2\left(\frac{2\pi }{a}\right){f_m}^2\left[\delta \left(Q+q_m-G\right)+\delta \left(Q-q_m-G\right)\right]\end{array}$$E14

Here the $\delta$ are the Dirac delta functions and the $G$ are reciprocal lattice vectors along the one dimensional lattice. This means that as well as the structural Bragg peak $Q=G$ the modulated magnetic structure gives magnetic peaks around the Bragg peak at $Q=G\pm q_m$. In this example there are only first order peaks but that is because we only took the expansion in Eq. (12) to first order. Note that magnetic scattering is also presence at the Bragg peak and not just on the peaks around it. This can be seen by the $f_m$ in the factor of the Dirac delta function for the main Bragg peak.

There are many fascinating example of incommensurate magnetic structures. Hexaferrites, an interesting materials with multiferroic properties offer interesting properties to study with soft X-rays [20, 21]. The large unit cells of the M, Y and Z type hexaferrites enable the Bragg condition to be satisfied even at soft X-ray energies (particularly at the Fe and Co $L_{2,3}$ edges). Incommensurate spin structures result in easily reachable magnetic diffraction peaks which can be studied at different temperatures, magnetic and electric fields.

4. Small angle scattering

Another possibility to measure magnetic structures is to perform experiments in transmission enabling the measurement of small angle scattering. Due to the strong absorption of soft X-rays the samples have to be about a few hundred nanometres thick or thinner. The complexity of producing the samples is a contrast to the much simpler experimental set-up. Since the energies are quite low there is the opportunity to study large structures such as magnetic domains. A very good example of this is the study of the domains in FeRh with both circular and linear polarisation [22]. In this work the domains and their evolution over time across the interesting antiferromagnetic to ferromagnetic transition was examined. Another area that has made extensive use of small angle scattering involves magnetic skyrmions. Magnetic skyrmions can best be described as textures of magnetic swirls. They are caused by a balance of magnetic anisotropy, applied field, fluctuating temperature and the Dzyaloshinskii-Moriya interaction. The latter, caused by the electronic spins sensitivity to non-centro-symmetric symmetry via the spin-orbit interaction causes the magnetic spins to spiral in two dimensions (see Figure 5). The topological nature of the spin structure means that they are robust magnetic entities which could potentially be used in magnetic memory applications [23].

Figure 5.

The different phases that exist in skyrmion hosting materials. On the left is shown a schematic of a phase diagram along with pictures of the helical, conical and skyrmion phases. The corresponding diffraction patterns are shown below for each phase.

A typical phase diagram of magnetic states in a skyrmion hosting material is shown in Figure 5. In general in the absent of magnetic field there is a helical arrangement of spins. If a field is applied the spins start to rotate towards the applied field. At certain values of applied field and temperature the skyrmion phase occurs. The exact values of temperature and magnetic field that this phases occur depends on the material and more specifically on the exchange interaction, Dzyaloshinskii-Moriya interaction, spin-orbit interaction and crystalline anistropy. Also shown are the typical in Figure 5 are the diffraction patterns due to the scattering from skyrmions and the competing helical and conical phases.

The large magnetic periodicity of the skyrmion lattices, which can vary from tens to hundreds of nanometres makes them ideal for soft X-ray diffraction. Many experiments have been done on ${\mathit{Cu}}_2\mathit{\text{OSeO}}{3}_3$ which, unlike some of the other B20 materials, is insulating. Here the skyrmion lattice causes a six fold diffraction pattern around the (0 0 1) diffraction peak. This occurs in a similar way to the commensurate antiferromagnet mentioned previously but the magnetic lattice is now many times larger than the chemical unit cell of the ${\mathit{Cu}}_2\mathit{\text{OSeO}}{3}_3$ (see for example [24, 25, 26]).

Another way of measuring skyrmions is to grow very thin samples and measure the small angle scattering in transmission. The technique of small angle neutron scattering (SANS) has already been used extensively for measuring skyrmions (e.g [27, 28]). The hexagonal structure of the skyrmion lattice will produce a hexagonal diffraction pattern around the (0 0 0) incident beam direction. A schematic is shown in Figure 6.

Figure 6.

The setup for small angle scattering. The X-rays pass through an aperture and are transmitted through an ultrathin sample. The small angle scattering is measured using a CCD camera. A schematic of the hexagonal diffraction pattern from a skyrmion lattice is shown on the right.

An example of such measurements using small angle scattering is shown in Figure 7 where skyrmions were measured on thin samples of ${\mathit{Cu}}_2\mathit{\text{OSeO}}{3}_3$ [29]. Here it was demonstrated that by field cooling (in a field of 44 mT) the skyrmion phase existed all the way down to 23.5K. Moreover there is no anomaly at the measured phase boundary (40K) as can be seen in the evolution of the reciprocal lattice vector and the intensity of the diffraction peaks. The intensity was fitted with a power law and gave a critical exponent of 0.73 which is expected for a three dimensional Heisenberg system agreeing with previous work [30, 31].

Figure 7.

The production of metastable skyrmions. The blue line in the phase diagram (a) results in a metastable skyrmion phase (b). (c) and (d) show the evolution of the reciprocal lattice vector Q and the intensity of the skyrmion peaks. The solid line in d is a fit to a power law behaviour yielding a critical exponent of 0.73.

5. Reflectivity

To avoid ambiguity reflectivity in this chapter will refer to the case of specular reflectivity i.e. where the incoming angle is equal to the outgoing angle. A reflectivity scan is generally performed by increasing the detector angle at twice the rate of the sample angle although some commercial diffractometers allow the symmetric increasing of the incoming and outgoing angle by increasing the detector and X-ray angles but keeping the sample constant.

Although it is a scattering technique like diffraction, reflectivity is different. Whilst diffraction refers to scattering from planes of atoms, reflectivity refers to scattering from a surface or interface or a combination of both. In many cases diffraction can be described by kinematical theory where amplitudes can be summed up. Reflectivity is often best described by optical theories using the Fresnel coefficients for reflection from each interface. In soft X-ray reflectivity this usually works well since the wavelengths are large enough to assume that the material is a continuum and not discrete planes atoms (as in diffraction). However, if a Bragg condition is satisfied during a reflectivity measurement (which would be quite common in a hard X-ray measurement) then the optical theory will no longer adequately describe the scattering and more complicated dynamical theories are needed [32].

An example of an optical theory that works well with soft X-ray scattering involves that of Zak et al [32, 33, 34, 35]. It involves calculating the wave properties as it propagates through a multilayer system. Two matrices are formulated: one that calculates the electromagnetic waves due to the reflection and refraction at each interfaces and a second one calculates the phase of the wave. The details are included in the references. Although it is based on optical theory, for calculating the Kerr and Faraday rotations it works well for soft X-rays as long as there is not have any Bragg diffraction i.e. that we can model the films as continuous media. It is a classical equivalent of the theory represented by Eq. (1) to first order in magnetisation.

Soft X-ray reflectivity is a very powerful technique for studying thin films and multilayers and therefore very relevant for device applications. A good example is exchange bias. Exchange bias occurs when a ferromagnetic is grown next to an antiferromagnetic material. The coupling at the interface causes a unidirectional anistropy; a hysteresis loop of the ferromagnetic material is not centred at zero applied field but offset by a quantity known as the exchange field H_ex. Discovered in 1956 by Meiklejohn and Bean [36] the effect is still not properly understood despite being used in read heads in the latest hard drives. IrMn₃ is an antiferromagnet and the most commonly used in spin valves in hard drives. A layer of Fe grown on top of IrMn₃ is exchange biased. The nature of the reversal process and the exchange bias field H_ex depends strongly on the structure of the antiferromagnet and therefore on its method of preparation. IrMn₃ can be prepared by molecular beam epitaxy or the more industrially relevant sputtering. It is well known that when IrMn₃ is grown at room temperature the structure is a $\gamma$ phase where the atoms are arranged randomly in a face centred cubic structure. When it is grown with the substrate at 400°C the material orders in the L1₂ structure. Here the structure can be described as having the Mn atoms ordered in the centre of the faces giving a simple cubic structure. In addition an Fe film grown on top of this will have a very different magnetic reversal behaviour. The hysteresis loops are shown in Figure 8.

Figure 8.

The hysteresis curves from IrMn₃. The loop for the $\gamma$ disordered phase is shown on the left and that corresponding L1₂ phase to the right.

It can immediately be seen from Figure 8 that both hysteresis loops are exchange biased. However, the $\gamma$ phase has a sharp loop with a H_ex of 150 Oe, the L1₂ sample is much smoother and with a much higher H_ex. Understanding the mechanisms for this is vital for understanding exchange bias and improving spintronics devices.

By tuning to the $L_{2,3}$ resonances of Mn and Fe it is possible to separate out the magnetism from the ferromagnetic and the antiferromagnetic layer. By using circular polarisation the technique also benefits from the significant linear component of magnetisation in the scattering cross section. This is shown in Figure 9. On the left is shown two reflectivities; the top one is measured at the Fe L₃ resonance (sensitive to the ferromagnet) and the bottom one is measured at the Mn L₃ edge (sensitive to the antiferromagnet). In each reflectivity opposite helicities of circular polarisation were measured. These are designated CP and CN in the plots. At the Fe $L_3$ edge there is a clear difference between the two reflectivities measured at opposite helicities which is not so apparent in the reflectivities measured at the Mn $L_3$ edge. This is represented in the plot of the difference on the right. Here the difference, often called the dichroism, is measured. This is not to be confused with the magnetic circular dichroism of X-ray absorption although it is strongly related. It is noteworthy here that whilst there is a clear difference at the Fe edge, hardly surprising for a ferromagnetic material, there is also a small but significant difference at the Mn edge.

Figure 9.

Reflectivity measure for the disordered $\gamma$ phase of IrMn₃. On the right are shown the reflectivities measured at the Fe $L_3$ edge (top) and Mn $L_3$ edge at the bottom. The reflectivities were measured with incident circularly polarised light at both helicities indicated by CN and CP in the plots. A schematic of the measurement geometry is shown at the top right. The plot on the right is the difference between the two helicities (dichroism). A schematic of the thin film with the thicknesses of the Fe and IrMn₃ layers is shown at the bottom right.

To examine the magnetic signal more we could fix the sample and detector angles at a convenient point in reciprocal space and measure the intensity as the sample goes through a hysteresis cycle. The result of this measurement is shown in Figure 10. Here it can clearly be seen that the signal follows the hysteresis much like that produced by a vibrating sample magnetometer. With X-rays we have the added advantage of being element specific which is nicely demonstrated here; by tuning to the Fe resonance we are measuring the ferromagnetic behaviour and at the Mn resonance we are measuring the antiferromagnet.

Figure 10.

Magnetic reflectivity measured during a hysteresis cycle. This was achieved by measuring at the Fe and Mn $L_3$ edges (Fe in black and Mn in red). On the left are shown the results from disordered $\gamma$ phase of IrMn₃. The corresponding ones from the $L{1}_2$ ordered phase are shown on the right.

It is hardly surprising that we can measure the ferromagnet. The terms in the first order (in magnetic moment) part of Eq. (7) show that magnetic scattering measures the magnetic moment in several directions depending on the magnetic moment.

To show this we can write out Eq. (7) in the following way

$$f=\left(\begin{array}{cc}{\sigma }_{\mathit{CR}11}& 0\\ 0& {\pi }_{\mathit{CR}22}\end{array}\right)+i\left(\begin{array}{cc}{\sigma }_{\mathit{CI}11}& 0\\ 0& {\pi }_{\mathit{CI}22}\end{array}\right)-i\left(\begin{array}{cc}0& {\sigma }_{M12}\\ {\pi }_{M21}& {\pi }_{M22}\end{array}\right)$$E15

Here we have added an imaginary term for the charge scattering to allow for the phase change during the resonant process. The imaginary term is assumed to have the same polarisation dependence as the real term. The magnetic term is assumed only to have an imaginary part. We have also ignored the second order part of the equation which we assume to be negligible. For circular polarisation we need to construct the polarisation as two orthogonal components with a $\pi$ phase difference i.e.

$$\begin{array}{c}{P}_i^{+}={\sigma }_i+i{\pi }_i\\ P_i^{-}={\sigma }_i-i{\pi }_i\end{array}$$E16

For both helicities respectively. Here the + and - refer to the different helicities of the circular polarisation. Including now the phase factors the structure factors for both helicities become

$$\begin{array}{c}{f}^{+}=\left(\begin{array}{cc}{\sigma }_{\mathit{CR}11}& 0\\ 0& i{\pi }_{\mathit{CR}22}\end{array}\right)+i\left(\begin{array}{cc}{\sigma }_{\mathit{CI}11}& 0\\ 0& i{\pi }_{\mathit{CI}22}\end{array}\right)-i\left(\begin{array}{cc}0& i{\sigma }_{M12}\\ {\pi }_{M21}& i{\pi }_{M22}\end{array}\right)\\ f^{-}=\left(\begin{array}{cc}{\sigma }_{\mathit{CR}11}& 0\\ 0& -i{\pi }_{\mathit{CR}22}\end{array}\right)+i\left(\begin{array}{cc}{\sigma }_{\mathit{CI}11}& 0\\ 0& -i{\pi }_{\mathit{CI}22}\end{array}\right)-i\left(\begin{array}{cc}0& -i{\sigma }_{M12}\\ {\pi }_{M21}& -i{\pi }_{M22}\end{array}\right)\end{array}$$E17

With this we can work out the scattered intensity. Here we work out a general expression with the applied magnetic field along any direction.

$$\begin{array}{l}{I}^{+}=F_{\mathit{\text{Total}}}{F}_{\mathit{\text{Total}}}^{\ast }=F_C{F}_C^{\ast }+F_C{F}_M^{\ast }+F_M{F}_C^{\ast }+F_M{F}_M^{\ast }\\ ={\sigma }_{\mathit{CR}11}^2+{\pi }_{\mathit{CR}22}^2+{\sigma }_{\mathit{CI}11}^2+{\pi }_{\mathit{CI}22}^2+{\sigma }_{M12}^2+{\pi }_{M21}^2+2\left({\sigma }_{\mathit{CR}11}{\sigma }_{M12}-{\pi }_{\mathit{CR}22}{\pi }_{M21}\right)\\ +{\pi }_{M22}^2-2{\pi }_{\mathit{CI}22}{\pi }_{M22}\\ \\ I^{-}=F_{\mathit{\text{Total}}}{F}_{\mathit{\text{Total}}}^{\ast }=F_C{F}_C^{\ast }+F_C{F}_M^{\ast }+F_M{F}_C^{\ast }+F_M{F}_M^{\ast }\\ ={\sigma }_{\mathit{CR}11}^2+{\pi }_{\mathit{CR}22}^2+{\sigma }_{\mathit{CI}11}^2+{\pi }_{\mathit{CI}22}^2+{\sigma }_{M12}^2+{\pi }_{M21}^2-2\left({\sigma }_{\mathit{CR}11}{\sigma }_{M12}-{\pi }_{\mathit{CR}22}{\pi }_{M21}\right)\\ +{\pi }_{M22}^2-2{\pi }_{\mathit{CI}22}{\pi }_{M22}\end{array}$$E18

We note here that the $\pi$ phase difference in front of the magnetic terms means that it is possible to have interference between the charge and magnetic scattering, interference terms which are linear in ${\sigma }_{M12}$, ${\pi }_{M21}$ and ${\pi }_{M22}$. Note also that these terms change sign with helicity. However there are also quadratic terms in ${\sigma }_{M12}$, ${\pi }_{M21}$ and ${\pi }_{M22}$. As the field is being applied in the scattering plane the ${\pi }_{M22}$ terms, which only depend on moments out of the scattering plane are small. This will work reasonably for the disordered sample where the moments switch abruptly. If we were to ignore the pure charge scattering terms then the two terms above can be simplified to the following.

$$\begin{array}{l}{I}^{+}={\sigma }_{M12}^2+{\pi }_{M21}^2+2\left({\sigma }_{\mathit{CR}11}{\sigma }_{M12}-{\pi }_{\mathit{CR}22}{\pi }_{M21}\right)\\ \\ I^{-}={\sigma }_{M12}^2+{\pi }_{M21}^2-2\left({\sigma }_{\mathit{CR}11}{\sigma }_{M12}-{\pi }_{\mathit{CR}22}{\pi }_{M21}\right)\end{array}$$E19

We can now see that the effect of changing the helicity during a scattering measurement of a hysteresis would resulted in the reverse the loop. However, the quadratic terms cannot always be ignored. Since the quadratic terms obviously do not reverse with helicity a simple way of removing this uncertainty is to measure the scattering during hysteresis with opposite helicities and subtract one from the other i.e. take the dichroism of the measured hysteresis. The important result from Eq. (19) is that there is a linear dependence on magnetic moment which reverses with helicity explaining why we see the hysteresis curves in 10.

The hysteresis curves measured at the Fe $L_3$ edge in Figure 10 are hardly surprising. Fe is a ferromagnetic material such that the resonant scattering process measures a net moment. In this example the behaviour of the reflectivity nicely follows that of the magnetometry shown in Figure 8. It also demonstrates in this case that, the second order terms in Eq. (19) are actually quite negligible. This should not be assumed to be a general case though. More surprising is the ability to be able to measure the magnetic moments in the antiferromagnetic layer i.e. the ${\mathit{\text{IrMn}}}_3$. One possibility is the second order term in Eq. (1). Here the square dependence on the magnetic moment would make it possible to measure the antiferromagnetism. It is exactly this term, in absorption, that is responsible for the magnetic linear dichroism that is often used to measure antiferromagnetic materials. However, in our case this is definitely not the explanation. It would be impossible for the squared dependence to give a hysteresis loop such as those in 10 as the loop would need to have equal reflectivities at both negative and positive saturation. More likely it is due to uncompensated moments near the interface. This could be caused by uncompensated moments domain walls. However, more fundamentally this could just be the effect of measuring the moments near an interface where the moments, even in an antiferromagnetic material do not cancel out.

Measurements of the hysteresis can also be done with linear polarisation. For this we need to work out the equivalent to Eqs. (17) and (18) for linear light. The general result is written out in Eq. (20) for both linear out of the scattering plane $\sigma$ and linear parallel to the scattering plane $\pi$.

$$f=\left(\begin{array}{cc}{\sigma }_{\mathit{CR}11}& 0\\ 0& {\pi }_{\mathit{CR}22}\end{array}\right)+i\left(\begin{array}{cc}{\sigma }_{\mathit{CI}11}& 0\\ 0& {\pi }_{\mathit{CI}22}\end{array}\right)-i\left(\begin{array}{cc}0& {\sigma }_{M12}\\ {\pi }_{M21}& {\pi }_{M22}\end{array}\right)$$E20

This will give the general result for $\sigma$ polarisation

$$\begin{array}{l}I=F_C{F}_C^{\ast }+F_C{F}_M^{\ast }+F_M{F}_C^{\ast }+F_M{F}_M^{\ast }\\ I={\sigma }_{\mathit{CR}11}^2+{\sigma }_{\mathit{CI}11}^2+{\pi }_{M21}^2\end{array}$$E21

Here we have simplified the equation since $\sigma$ polarisation is insensitive to moments out of the scattering plane (we are not taking into account the second order term in 1) and also that all the magnetic scattered x-ray polarisation has flipped $\pi$ polarisation in agreement with the triple product in the first order term of 1.

For $\pi$ polarisation Eq. (20) gives us something more complicated.

$$\begin{array}{l}I=F_{\mathit{\text{Total}}}{F}_{\mathit{\text{Total}}}^{\ast }=F_C{F}_C^{\ast }+F_C{F}_M^{\ast }+F_M{F}_C^{\ast }+F_M{F}_M^{\ast }\\ ={\pi }_{\mathit{CR}22}^2+{\pi }_{\mathit{CI}22}^2-2{\pi }_{\mathit{CI}22}{\pi }_{M22}+{\pi }_{M22}^2+{\sigma }_{M12}^2\end{array}$$E22

Note that Eq. (22) now has a linear and a quadratic term in the $\pi$ channel as well as a quadratic one in the $\sigma$ channel. The scattering that rotates into the $\sigma$ channel is only sensitive to moments in the scattering plane (c.f. Eq. 21). The scattering into the $\pi$ channel is sensitive to moments perpendicular to the scattering plane. To make this more readable we can split the equation into two parts: for magnetic moments in the scattering plane

$$I={\pi }_{\mathit{CR}22}^2+{\pi }_{\mathit{CI}22}^2+{\sigma }_{M12}^2$$E23

(which looks similar to Eq. (21)) and moments perpendicular to the scattering plane.

$$I={\pi }_{\mathit{CR}22}^2+{\pi }_{\mathit{CI}22}^2-2{\pi }_{\mathit{CI}22}{\pi }_{M22}+{\pi }_{M22}^2$$E24

Results from the disordered $\gamma$ phase sample using both incident linear polarisations $\sigma$ and $\pi$ are shown in Figure 11. At the top is shown linear horizontal $\sigma$ and the bottom is shown $\pi$ incident polarisation. The schematics next to the graphs represent the scattering processes with the moments saturated in the four principal directions. Firstly for $\sigma$ polarisation, which is only sensitive to moments in the plane, in this case as the moments reduce, so does the reflectivity. It reaches a minimum at the coercive field, and then increases again. Since the dependence on magnetisation is quadratic (see Eq. 21) the reflectivity should be equal for both negative and positive saturation. The small difference is due to locked moments in the ferromagnetic film [37]. Whereas the $\sigma$ at coercive field shows a minimum, the incident $\pi$ polarised beam shows a maximum. This is most likely due to the increase of moments and increase in magnetic disorder out of the scattering plane which maximises the expression represented by Eq. (23). Again there is a slight offset possibly caused by some locked moments in the ferromagnetic film.

Figure 11.

Magnetic reflectivity hysteresis loops taken at the $L_3$ edges of Fe on the disordered $\gamma$ phase IrMn₃. At the top is shown linear horizontal and at the bottom linear vertical.

6. Note on second order term

In the equation describing the magnetic atomic form factor (Eq. (1)) there are three terms. The last term representing the second order in magnetic moment has been ignored up until now. It is often ignored in most studies due to the assumption that it is small. To measure this in an experiment, particularly with the uncertainty of the coefficients $F^{\left(0\right)}$, $F^{\left(1\right)}$ and $F^{\left(2\right)}$, requires careful exploitation of the polarisation and vector nature of the magnetisation. For example this could be done using polarisation analysis and then measuring magnetic dependence in the $\sigma$-$\sigma$ channel where the polarisation (defined as out of the scattering plane) does not rotate. Since the first order term is zero for this channel any change in the scattering due to manipulation of the moments must be from this second order term. Unfortunately, the inefficient detection in polarisation analysis (less than 1 %) will make this very difficult. A good estimate could also be made without polarisation analysis. Low scattering angles could be used, where all the channels apart from the angle independent $\sigma$-$\sigma$ channel would be weak. Again this could be combined with applied magnetic field dependence so that charge and magnetic scattering can be separated.

Whereas the first order term will provide the first order diffraction peaks from a magnetic lattice the second order term will in addition produce second order satellites. This can easily be demonstrated by inserting a phase factor, for a one dimensional commensurate structure into Eq. (1). In the following we assume that the charge scattering is a real number. Although this is incorrect it simplifies the mathematics and does not influence the main conclusion. If we assume that both the charge and magnetic lattice has a lattice parameter $a$ and insert the corresponding phase factors $exp\mathit{iq}.\mathit{na}$. Following this, and exactly analogous to Eq. (14), we work out the intensity for a large number of planes $N$.

$$\begin{array}{l}I=N\left({\left(\left({\mathbf{e}}_{\mathbf{f}}\cdot {\mathbf{e}}_{\mathbf{i}}\right)F^{\left(0\right)}\right)}^2+{\left(\left({\mathbf{e}}_{\mathbf{f}}\times {\mathbf{e}}_{\mathbf{i}}\right)\cdot \mathbf{M}{F}^{\left(1\right)}\right)}^2\right)\left(\frac{2\pi }{a}\right)\delta \left(\mathbf{Q}-\mathbf{G}\right)\\ +N{\left(\left({\mathbf{e}}_{\mathbf{f}}\cdot \mathbf{M}\right)\left({\mathbf{e}}_{\mathbf{i}}\cdot \mathbf{M}\right)F^{\left(2\right)}\right)}^2\left(\frac{2\pi }{a}\right)\delta \left(2\mathbf{Q}-\mathbf{G}\right)\end{array}$$E25

Here $\mathbf{G}$ is a reciprocal lattice vector and $\delta$ is the Dirac delta function. Eq. (25) shows, in the case of a commensurate structure, that the first two terms of Eq. (1) will give peaks at the the reciprocal lattice vector $\mathbf{Q}$ but the second order term in magnetic moment gives peaks at $2\mathbf{Q}$.

7. Conclusions

This chapter has summarised some of the main techniques in polarised soft X-ray scattering: diffraction, small angle scattering and reflectivity. It has been demonstrated that by tuning to a suitable dipole electric resonance e.g. the $L_{2,3}$ edges of the transition metals or the $M_{4,5}$ edges of the rare earths, which both occur at soft X-ray energies, scattering experiments at these energies are very sensitive to the arrangements of magnetic moments in a material. In diffraction it was demonstrated, for both commensurate and incommensurate magnetic structures, that in addition to the powerful enhancement given by the resonance, the different symmetries of the charge and magnetic lattices allow one to measure purely magnetic signals e.g. in the case of a commensurate antiferromagnet incommensurate helical/spiral spin structures. The relatively new technique of small angle scattering in measuring domains and magnetic skyrmions has been made possible by fabricating ultrathin samples making unprecedented advances in measuring magnetism with a relatively simple experimental setup. The measurement of reflectivity has also been reported on. Here it was shown to be a useful probe for complicated magnetic reversal processes in an exchange biased system, enabling not just the measurement of the ferromagnetic layer but also the antiferromagnet. Measurements with both circular and linear light enabled the probing of the magnetic moments, particularly near the ferromagnetic/antiferromagnetic interface. Circular light is shown to cause interference between charge and magnetic scattering giving a very strong linear magnetic component. Whilst circular light is only sensitive to moments in the scattering plane linearly polarised light is sensitive to moments both parallel and perpendicular to the scattering plane. The sensitive dependence on polarisation means that it is possible to use X-ray scattering as an element specific depth sensitive vector magnetometer.

Acknowledgments

The authors acknowledge J. Herrero-Martin for discussion and the use of figures in section 3.1 on the commensurate antiferromagnetic (AF4) phase in ${\mathit{Mn}}_{0.85}{\mathit{Co}}_{0.15}{\mathit{\text{MnWO}}}_4$ The authors also wish to thank P.D. Hatton and M.T. Birch for discussions about the small angle scattering and measuring magnetic skyrmions in section 6.

Conflict of interest

The authors declare no conflict of interest