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.
Keywords
- soft X-ray
- magnetism
- thin films
- scattering
- diffraction
- reflectivity
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 (
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
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
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].
Here
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.
If we assume that
Where
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.
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
where
where the
3.1 Commensurate antiferromagnet
The system
The antiferromagnetic phase in
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
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.
In this equation we have assumed a complex atomic form factor
To first order gives.
By writing
Here the
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
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].
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
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.
An example of such measurements using small angle scattering is shown in Figure 7 where skyrmions were measured on thin samples of
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
It can immediately be seen from Figure 8 that both hysteresis loops are exchange biased. However, the
By tuning to the
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.
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
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
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
With this we can work out the scattered intensity. Here we work out a general expression with the applied magnetic field along any direction.
We note here that the
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
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
This will give the general result for
Here we have simplified the equation since
For
Note that Eq. (22) now has a linear and a quadratic term in the
(which looks similar to Eq. (21)) and moments perpendicular to the scattering plane.
Results from the disordered
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
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
Here
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
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
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