Extracting Ordering Exponents

An accurate description of the magnetic ordering can be obtained through the col- lection of hysteresis loops at discrete temperatures. This has an advantage over a continuous measurement of the magnetisation while increasing the temperature as it eliminates domain e↵ects. It does, however, allow fewer temperature values to be probed in a given time. Element specific hysteresis loops are collected using XRMS at a fixed scattering vector, chosen to produce the maximum F.R. atHsat,

by recording the measured F.R. as a function of applied field. The maximum F.R. is often at the point of lowest scattered intensity (see figure 3.8), causing the required counting times to increase. In order to reduce noise in the data,M(H) can be fitted to

M(H) =A.arctan(H±Hc

where A,B,C, ⌫ and Hc are all parameters to be fitted. With the exception of

Hc, the values of the fitting parameters do not relate to physical properties and

should be considered arbitrary. The shape parameter, ⌫, governs the roundness of the loop. The loop fit then comprises two arctan functions, one each for the increasing and decreasing field directions, which are identical except for a translation of_±Hc. Furthermore, the invariance of the shape parameter with field forces the fit

to have two-fold rotational symmetry. The true loop shape often di↵ers subtly from this simplification. As the parameter of interest, the remanent magnetisation, is unlikely to be a↵ected by this di↵erence, the simplification is justified. Similarly, B, the gradient atHsat, should be zero if the sample is truly fully saturated. As some

additional paramagnetic response may remain, it is fitted to prevent any adverse influence of this discrepancy on the remanent moment. Furthermore, B is allowed to di↵er between temperatures, again ensuring the best possible fit to the remanent moment.

The fitted hysteresis loops allows the both the moment and susceptibility as a function of field to be extracted. The susceptibility, =dM/dH, can be calculated directly from equation 4.1 yielding:

(H) = A ⌫ ✓ 1 +⇣(H±HC) ⌫ ⌘2◆ +B. (4.2)

The susceptibility is again optimised around the zero-field region due to the fitting compromises faced by equation 4.1. Analysis of the moment and susceptibility as a function of field allows the information contained within the shape of the loop to be extracted and will be described in more detail in chapter 7.

As an example of how the ordering exponent can be extracted from hysteresis loop data, loops from a 0.7 ML Pd/Fe/Pd trilayer, at the Pd edge, can be examined. The value ofmatH= 0 obtained from the fit is the desired remanent magnetisation. Examples of typical hysteresis loops can be seen in the inset to figure 4.7. Using the remanent magnetisation from the hysteresis loop fitting, shown in the main body of figure 4.7, the magnetisation as a function of temperature can be plotted, wheremis described by the power lawm= (TC T) ef f. The scaling exponent, ef f, describes

the dimensionality of the ordering which is governed by the spin and spacial degrees of freedom available to the magnetic lattice.

Extracting the ordering exponent from these data can be accomplished through log-log fitting. Plotting log(m) against log(TC T) yields a linear trend with gra-

dient ef f. Small field induced tails in the data nearTC, visible in figure 4.7, make

Figure 4.7: Magnetic ordering behaviour of a 1.1 ML FePd trilayer film deduced from arctan fits to hysteresis loops collected as a function of temperature (select loops shown in inset with half point density for clarity). Blue line shows the results of log-log fitting.

inflection in the data, where the tail begins, and is then adjusted to maximise the extent of the linear region over which the linear trend holds. This is the basis of the method outlined by Durr et al. [98].

The magnetic ordering data typically split into three regions governed by di↵erent power law dependencies. Below approximately 0.3 TC, a Bloch law tem-

perature dependence dominates. This region contains no additional information about the nature of the magnetic ordering dimensionality, so is often ignored. Loops collected in this region, where available, are used to improve the accuracy of nor- malisation, where m(T = 0) = 1 is used. For approximately 0.3 < TC/T < 0.95,

critical behaviour dominates. This region is used to determine the magnetic or- dering dimensionality through log-log fitting. Above approximately 0.95 TC, field

dependent tails may appear due to inhomogeneities in the applied field or material composition. The upper limit beyond which critical behaviour is lost varies from sample to sample; if the determinedTC remains close to the point of inflection, this

Figure 4.8: Log-log analysis of the Fe edge magnetic ordering data from a 1.1 ML FePd trilayer film showing the influence ofTC on the data linearity. The maximum

range over which the linear trend hold occurs at 178.8 K providing a value of 0.11(1) from the gradient of the linear fit (blue line). Near TC data points for 170

and 175 K showed significant deviation from the linear trend and are outside of the plotted range.

the low temperature data where critical behaviour is not expected. The low point density close to the determined TC prevents the appearance of the field induced

region as the field a↵ected data all appear aboveTC.

The remanent magnetisation is obtained from the zero field loop height of the arctan fit to the data. Quantification of an error using this method is not straightforward, as the extracted value is not a fitting parameter. A reasonable approximation is made by examining the standard deviation from the fitted trend of the five data points closest to zero field. If more data points are included in this analysis, the chance of an outlier adversely a↵ecting the data is minimised. However, due to the rapid fallo↵ in magnetisation approaching TC, the larger a field range

used, the greater the a↵ect of uncertainties in TC. The resultant error bars, though

too small to be seen in figure 4.7, become more critical when analysing the log-log data; the propagated errors in figure 4.8 are much more substantial. Though these

error bars appropriately weight the linear fit used to deduce the ordering exponent,

ef f, the uncertainty in ef f is dominated by the uncertainty inTC. To quantify the

uncertainty in ef f, the range of TC values over which the linearity of the log-log

data remains indistinguishable is manually determined. The concomitant change observed in ef f over the limits of this range defines the uncertainty.

It should also be noted that it is also possible to directly fit a power law be- haviour to the data. An additional convolution with a Gaussian broadening function can simulate the field induced tails. The scaling exponent, ef f, and the broadening

function are, however, strongly coupled to TC. As there is not a reliable and con-

sistent way to define TC during this fitting, the results obtained using this method

are more erroneous. All data presented in this work will therefore be analysed using the log-log method.