X-ray investigation of long-range antiferromagnetic ordering in FeRh

The meta-magnetic phase transition of equiatomic FeRh has been used as an archetype for investigations of the three-temperature model of phase transitions^1,2 as it undergoes (i) an antiferromagnetic (AF) to ferromagnetic (FM) transition,^1,3 (ii) a lattice expansion,⁴ and (iii) a change in electronic structure⁵ on increasing the temperature. The coupled nature of this transition provides a wide range of parameters with which one can alter the boundary between the two phases.⁶ As the ground states of the AF and FM phases are close in energy,⁷ perturbation of the electronic, crystal or spin structure can promote the stability of one phase and change the transition temperature.⁶ For example, this can be achieved by changing the strain at an interface, either with a capping layer⁸ or via a lattice mismatch to the substrate,⁹ or by changing the contribution of strain by tailoring the sample thickness.¹⁰ Interfacial strain has also been shown to stabilise the FM phase.¹¹ The crystal structure and atomic disorder in the sample are also important and, by doping the material, the properties can be changed.¹² The properties can also be modified by artificially increasing the disorder in the sample using ion irradiation.^7,13,14 Patterning films for potential applications can change the magnetic properties, either as a by-product of the lithography process¹⁵ or as a result of magnetic interactions between the patterned elements.¹⁶ 

FeRh thin films are of interest in a wide range of fields, the most developed of which is magnetic data storage. Two examples of applications are (i) as a component in an exchange-spring magnet^17–19 for heat-assisted magnetic recording (HAMR); or (ii) as an AF magneto-resistor acting as a write/read bit.²⁰ In both cases the bit is read whilst in the AF state,²¹ while the write action takes advantage of high saturation moment and (relatively) low coercivity of the FM phase.¹⁹ In order to optimise FeRh thin films for applications a knowledge of the exact mechanism of how the spins evolve with temperature is a prerequisite. Theoretical work has suggested that the transition is instigated by canting of the AF spins²² which helps promote the transition over the small energy barrier from the AF to FM phase.⁶ This is consistent with the strong magnetic field dependence of the phase transition which is 9 K T⁻¹, since the field induced canting of the Fe spins reduces the thermal energy required to promote the transition.²³ A simultaneous probe of the lattice and the AF order could verify such a model.²⁴ However, AF order has previously only been investigated with static techniques based on neutron diffraction²⁵ or correlation lengths extracted from x-ray magnetic linear dichroism (XMLD).²⁶ In addition, it is known that the lattice expansion occurs within 30 ps following laser excitation,²⁷ but it is not clear if this is commensurate with a loss in AF order.

In this article, we describe a method to probe the long-range order of the Fe spins in the AF phase. Shown in Fig.1 is the distribution of spins on the Fe sites with anti-parallel ordering on a cubic structure which is defined as a G-type anti-ferromagnet. Such regular order allows us to extend the Bragg condition for diffraction to the spins, which gives a weak coherent scattering of x-ray photons.²⁸ Within the family of $(121212)$ lattice planes, the spins align parallel and therefore can scatter x-rays coherently with a phase shift due to the transfer of angular momentum.²⁹ When viewed in k-space, the reciprocal lattice points are non-coincident with the integer Bragg peaks that scatter from the charge centres. As such, they can be probed independently. This allows us to monitor the evolution of AF order and compare it to the lattice. It is possible, in principle, to determine the spin and orbital contributions to the moment in the AF state,³⁰ which is information that is not available in neutron diffraction experiments. However, this analysis is best applied to materials that have a large relative orbital moment,³⁰ which is not the case for FeRh. Based on the FM moment and ab initio calculations, it is expected that the Fe centres will have moments that are dominated by the spin.^22,31,32 As such the ratio m_l/m_s is close to zero and we expect a negligible contribution from the orbital angular momentum to the scattered intensity.

The samples investigated were nominally 500 nm thick Fe₅₀Rh₅₀ films magnetron sputtered on MgO substrates from an equiatomic FeRh alloy target. The films were deposited using an AJA ATC-2200 sputter system and, following the deposition at 650°C, the samples were annealed for 2 hours at 750°C.³³ After cooling to room temperature, a Pt cap (3 nm) was deposited to inhibit surface oxidation. The annealing step is important to achieve the necessary B2 crystal structure for the required meta-magnetic transition to occur.³³ Further details of the deposition can be found in previously published work.^11,34

The crystal structures of the FeRh films were confirmed by XRD analysis. The film is found to align in-plane at 45° to the cubic MgO substrate (Fig.1). This fulfils the lattice matching condition with a mismatch of <2%³⁵ for planes FeRh[001](011)‖ MgO[001](001). The lattice expansion of the FeRh through the meta-magnetic phase transition was measured to be 0.75% when heated from 303 K to 493 K, with the expansion in the MgO lattice being 0.01%. A factor of 70 difference implies that interfacial strain from the substrate is not driving factor of the transition. Vibrating sample magnetometry (VSM) confirmed the meta-magnetic transition occurs at 358 K, with a maximum saturation Ms of 920 kA m⁻¹ (@ 413 K).

X-ray scattering experiments were carried out on 5 mm x 5 mm films at the I16 beamline of Diamond Light Source, and the MS beamline of the Swiss Light Source using x-ray energies of 4.998 keV and 6.408 keV. These energies do not coincide with any resonant Fe absorption edges (Fe K-edge = 7.112 keV). During the I16 measurements, the temperature was controlled with the available 6K-800K ARS GM cooler.

For our experiment, we focused on measuring the magnetic Bragg peaks of a Fe₅₀Rh₅₀ film originating from the long-range AF order. In order to establish the efficiency of the x-ray magnetic scattering, the ‘charge’ peaks - those corresponding to the nuclear x-ray scattering - were used as a reference. The noise was reduced by defining a region of interest (ROI) for each peak on the Pilatus 3-100K 2-d detector and normalising against a background ROI. To confirm that the signal of the $(121212)$ peaks were indeed of magnetic origin, a Pyrolithic graphite polarisation crystal was initially used (with the x-ray energy tuned to 5.22 keV). The absorption and re-emission of x-rays from magnetic scattering require a transfer in angular momentum that rotates the polarisation relative to the incident x-rays. Therefore, it is expected that significant intensity is measured in the rotated polarisation channel (so called σ-π’ scattering^30,36). In the vertical experimental configuration (see Fig.2), with a grazing incident angle α=2°, we obtained count rates of the order of 40 ct/s for the $(321212)$ reflection. We lowered the energy of the x-rays to 4.998 keV to establish more favourable conditions for measuring AF peaks. At this energy the $(321212)$ magnetic peak occurs for a horizontal scattering angle of 90°, where nuclear Bragg peaks are suppressed. It is therefore possible to measure without a polarisation analyser, as only the magnetic signal is present at this scattering angle. Indeed, we found a count rate of 2500 ct/s for the $(321212)$ magnetic peak (α=2°). In order to confirm these efficiencies a higher x-ray energy (6.408 keV) was used to check the $(521212)$ Bragg peak. In this case we obtained a count rate of 1000 ct/s (α = 5°). Finally, we attempted to observe resonant enhancement of the magnetic scattering in the vicinity of the Fe K-edge. However, above the edge (@ 7.6 keV^37,38), the signal is dominated by the background due to the Fe fluorescence. This was despite the near perpendicular scattering angle (90.42°).³⁹ 

We initially observed the incidence angle dependence shown in Fig.3 at an energy of 4.998 keV. The incidence angle dependence was confirmed for higher x-ray energy (6.408 keV), and for the charge peak (2 0 1). The angular dependence in all cases shows a steep drop off in signal for angles below 1°. In the case of magnetic scattering, no discernible peak is present for angles less than 0.5°. The angular dependence can be explained by considering the effect of grazing incidence geometry on the spot size of the x-ray beam. The cross-section of the beam profile was fitted to a Gaussian function and found to have a FWHM of 80 μm. When this beam is incident at near grazing angles, the larger footprint results in an increased volume of the thin film sample being illumined. For a perfectly flat film, we estimate the footprint to be several mm when angles of less than 1° are employed. Below such angles, total external reflection of the x-ray beam is expected, with θ_critical estimated to be 0.2 - 0.5° from the refractive index (n = 1 – 4.85x10⁻⁵) of FeRh at x-ray energies.^40–42 The vanishing signal around 0.5° is attributed to such reflections.

The efficiency of the magnetic scattering can be estimated by comparing the intensity of the magnetic and charge peaks. The ratio of these intensities is estimated to be I_charge/I_magnetic = 1 × 10⁻⁷ when comparing similar incidence angles and correcting for the attenuation of the x-rays. Based on the theoretical work of Blume and Gibbs,²⁸ this is close to the expected ratio. From the relative overlap with the x-ray photon wavefunction, it is predicted that the intensity of the magnetic peaks will be 10⁵ – 10⁶ times weaker than for the charge peaks.⁴³ 

In order to confirm that the signal from the $(121212)$ family of peaks of FeRh has no contribution from the charge centres, we measured the intensity of the peaks as a function of temperature through the phase transition. The results are presented in Fig.4 where the intensities are obtained from fitting measured data to Voigt functions. The intensity was found to steadily decrease for increasing temperature as expected due to increasing thermal energy and therefore increased non-coherent phonon scattering in the sample. At temperatures close to the phase transition determined from the VSM data, the intensity drops significantly on increasing the temperature in the interval 360-390 K (Fig.4 a)). Based on the measured XRD data of the (001) and (002) peaks of the FeRh we expect no significant change in the form factor. The charge Bragg peaks measured at 303 K and 493 K show no change in the integrated intensity. These measurements demonstrate that the observed $(321212)$ peak is due to magnetic scattering from the AF lattice.

As the AF-FM transition occurs over a range of temperatures, the AF order does not evolve as a single first-order phase transition where the properties would be expected to change over 0.1-1 K. Instead, previous experiments into the microscopic structure of FeRh have shown that the sub-micron regions have an independent transition temperature,⁴⁴ each with a first order transition. Averaging across these regions yields a Gaussian distribution of transitions for macroscopic films.³⁴ By fitting the first derivative of the intensity vs. temperature with a Gaussian function in Fig.4, we can compare the transition as monitored by VSM (T_VSM = 356K) to the change in AF order (T_x-ray = 377K). From the signal, it appears that the FM moment begins to emerge before the AF order is completely lost. The mixed phase of the material still shows some long-range AF ordering in the Fe spins. Even accounting for an uncertainty in the calibration temperatures for the two different systems (± 5K), the difference in the two transition temperatures is significant. This is in agreement with previous work where the nucleation and growth of FM domains through the phase transition was investigated.^45,46 The domains have been seen to first nucleate and then grow by absorbing neighbouring AF domains. If we assume that the AF domains are much smaller than the FM domains, it is possible to still have coherence within the AF domains while a significant portion of the material has transitioned to the FM state. Therefore, we suggest that this is further evidence for the previously reported dynamics of the phase transition.²⁶ It will also be useful to have such long-range order in the mixed phase when measuring the lattice dynamics of the meta-magnetic transition.⁴⁷ By being able to record signal from the AF ordering it should be possible to examine with fine detail how quickly the film transitions with respect to the spin flop on the Fe sites.

The magnetic and charge peaks of Fig.3 were fitted with Gaussian functions to estimate the coherence length of the AF ordering and crystal structure with full width at half maxima (FWHM) of 0.38° and 0.39° respectively. Being at the same energy and employing the same experimental set-up, it can be assumed that any differences in the FWHM are not due to either spectral or instrumental broadening.⁴⁸ We do not observe any increase in the FWHM for the half-integer peaks, implying that the ordering of the Fe spins in the AF phase are of a similar long-range order, comparable to the crystalline coherence length. This remains consistent across small angles with increasing certainty at higher angles. Averaging values obtained from a Scherrer analysis of these fits yields a minimum mean coherence length of 40 ± 3 nm, consistent with our assumption that the AF domains are much smaller than the FM domains. There is a slight peak broadening at lower incidence angles suggesting that we have increased incoherent scattering. This is consistent with the assumption that grazing incidence scans will have greater diffuse reflection due to the surface defects.⁴¹ 

A previously reported experiment looking at the correlation length of XMLD signal could not definitively resolve the size of the AF domains.²⁶ They suggested that the spatial resolution of the technique limited this measurement and that the domains are likely to be 30-60 nm. This would yield AF domain structure smaller or close to the crystallite size of sputtered FeRh films, consistent with the assumption that such domains are limited by crystallographic defects. This has been previously seen in AF thin film samples, with domain sizes of less than 50 nm observed in NiO and LaFeO₃.^49,50

We have demonstrated an x-ray diffraction technique to probe long-range AF ordering of Fe spins in FeRh films at RT. The temperature dependence of the peaks was shown to correlate with the emergence of FM moment in this material. The non-resonant x-ray scattering of the magnetic order shows similar peak widths to the integer Bragg peaks suggesting the coherence length of the AF order is limited by the grain boundaries of the sputtered thin film. The expected domain size of AF domains has previously shown to be limited by crystallographic defects in agreement with our measurements. We have further demonstrated the incidence angle and energy dependence of these peaks which will be important factors to consider in the future investigation of the AF order in FeRh. This technique could be adapted to the time-dependent regime as a simultaneous probe of the lattice and spin dynamics of FeRh.

The authors wish to gratefully acknowledge the contribution of the x-ray characterisation facilities of the Henry Royce Institute through EPSRC Grant Nos. EP/S019367/1 and EP/P025021/1. We acknowledge Diamond Light Source for time on Beamline I16 under Proposal MM21966-1, and the Swiss Light Source by allocation P202011A. M.G. acknowledges funding from the Department of Computer Science, University of Manchester, UK; and the Paul Scherrer Institut, Switzerland. V.S. and N.G. acknowledge funding by the Swiss National Science Foundation (200021-162863). H.U. acknowledges the National Centers of Competence in Research in Molecular Ultrafast Science and Technology (NCCR MUST-No. 51NF40-183615) from the Swiss National Science Foundation and from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 801459 - FP-RESOMUS. The authors would like to acknowledge E. Pomjakushina for her contributions to the sample preparation.

The authors have no conflicts to disclose.

The data that support the findings of this study are available within the article. Raw data were generated at the Diamond Light Source (MM21966-1), and Swiss Light Source (P202011A) large scale facilities. Derived data supporting the findings of this study are available from the corresponding author upon reasonable request.