Fourier transformation based analysis routine for intermixed longitudinal and transversal hysteretic data for the example of a magnetic topological insulator

We present a symmetrization routine that optimizes and eases the analysis of imperfect, experimental data featuring the anomalous Hall hysteresis. This technique can be transferred to any hysteresis with (point-)symmetric behavior. The implementation of the method is demonstrated exemplarily using intermixed longitudinal and transversal data obtained from a chromium-doped ternary topological insulator revealing a clear hysteresis. Furthermore, by introducing a mathematical description of the anomalous Hall hysteresis based on the error function precise values of the height and coercive field are determined.

A common problem when dealing with experimentally gained magnetotransport data from Hall bars is the intermixing of longitudinal (R xx ) and transversal (R xy ) resistance data [21][22][23][24][25].Even when excluding errors, there are internal origins for an overlay of the signals that cannot be solved experimentally.Possible reasons are sketched in Fig. 1 on transversal contact pairs of a Hall bar.On the left contact pair inhomogeneous potential fluctuations due to charge puddles are shown [26][27][28].In the middle a geometrical displacement (∆S) of the contacts is sketched that gains importance when approaching smaller structures dependent on the fabrication technique [2].In this case, a longitudinal signal in the order of ∆S/L • R xx would contribute to the Hall signal R xy .On the right contact pair possible grain boundaries causing inhomogeneous potential drops are depicted.The consequence of the presented mechanisms is comparable to the one of a diagonal measurement over the Hall bar [29].Despite these irregularities, the data could hold valuable information about the sample.In the following, the data processing of intermixed conventional and anomalous Hall data with respect to their symmetries is explained.Subsequently, the analysis routine is demonstrated using the data of the MTI Hall bar.Here, the focus lies on the symmetrization of the Hall data, the longitudinal data can be treated analogously.

Symmetries of Magnetic Topological Insulators
When analyzing measurement data that shows an intermixing of longitudinal and Hall signal, symmetrizing the data (see supplementary material) is a common method to separate the signals from each other, as the longitudinal data is expected to show axial symmetry with respect to the ordinate while the transversal data is point symmetric.Thus, a fast Fourier transformation (FFT) can help separating the respective contributions.In order to do so, the signal is Fourier transformed to the frequency space giving complex values in general.For the longitudinal data all odd contributions given by the imaginary part are filtered out while for the transversal data all even ones given by the real part are omitted.Then, the remaining quantities are transformed back giving the unperturbed signal using inverse FFT.
For MTIs showing the anomalous Hall effect or even the quantum anomalous Hall effect there are major differences.Figure 2 shows such signatures in the longitudinal (a) and Hall resistance (b) for two magnetic field sweeps: The blue curve shows data of the sweep from negative magnetic field to positive and the orange curve shows the data for the sweep in opposite direction [30].One can see that, compared to e.g.nonmagnetic TIs, the sweeps differ around zero magnetic field.The origin are the magnetic moments M introduced by the doping material that align in a ferromagnetic order with the external magnetic field B, so an internal magnetization inside the MTI is created [31], similar to diluted magnetic semiconductors [32].At the coercive magnetic field B c , the external magnetic field is strong enough to switch the orientation of the opposingly aligned internal magnetic moments.In the longitudinal signal around the switching point a shifted peak for each sweep direction is visible.In the Hall signal a hysteretic behaviour is seen that is identified by the coercive magnetic field B c and the height in resistance R AH .Due to the hysteretic behaviour, it becomes apparent, that the symmetrization procedure mentioned before is not directly transferable.Intuitively, one could think of shifting the data by ±B c and do the same procedure, but especially when considering the longitudinal data one can see that this would only fit well for the peak position but not for larger magnetic fields.Indeed, one would also loose information about the individual curvature of the anomalous Hall hysteresis.A better but more complex solution is the reallocation of the data points in the data set to again find axial and point symmetry.
Figure 3 illustrates, how the data needs to be restructured in order to have data sets that show a certain symmetry.In Fig. 3 a) the longitudinal data is shown.Colored in blue and orange, respective data set pairs are marked that are symmetric with respect to the ordinate shown with a black mirror line.The Hall hysteresis is depicted in Fig. 3 b).Compared to the longitudinal signal that shows an axial symmetry with respect to the ordinate, the Hall data is point symmetric to the middle of the hysteresis at the coordinate origin as indicated by a black dot.When combining the orange and blue symmetries, it becomes apparent that the two sweep directions are not symmetric in itself but with respect to each other.Thus, for many systems merging both sweep directions to one data set is an equivalent alternative with the same symmetries.Having this new symmetry in mind, the Fourier symmetrization discussed above can be done.

Mathematical Description of the Hall Hysteresis
The parameters B c and R AH (cf.Fig. 2) are taken as a suitable measure for the hysteresis.Thus, their accurate determination is crucial.Hence, after the symmetrization process described before, the cleared data set is resorted in the manner of the process depicted in Fig. 2 b) and a fit is performed.Even after symmetrization the ideal data still consists of the hysteresis arising from the anomalous Hall effect superimposed by the classical Hall slope.As the slope may be non-negligible the data is corrected by the point symmetric conventional Hall slope determined far away from the hysteresis.Now, just the anomalous hysteresis is remaining.
As one can see in the transversal signal, the switching of the magnetic moments does not result in a steplike behaviour.Instead, the tails are slightly curved.The reason is that not all the magnetic moments are bound exactly the same way but are assumed to follow approximately a Gaussian distribution, for instance due to defects or due to the inhomogeneity of the energy at the edges of the sample.Therefore, e.g. a Heaviside step function as basic model would neglect these factors.Instead, we employed an error function as a mathematical model that describes the normal-ized integration of a Gaussian function.The complete model that describes the development of each branch of the hysteresis results in Here, R AH scales the height of the normalized error function, the term B ± B c takes the shift of the curve by the coercive magnetic field B c with respect to zero magnetic field into account and the parameter A is a measure of the switching curvature.Further discussion regarding the model can be found in the supplementary material.Figure 4 shows the data from Fig. 2 b).One least square fit for each side using equation 1 is performed and shown with a dotted line.
One can see that the fits describe the data quite accurately.Moreover, three error functions with varying parameter A are plotted in the insert to illustrate how the curvature of the function develops.In the following the method is demonstrated exemplarily using the intermixed anomalous Hall data of a chromium-based MTI.

EXEMPLARY ANALYSIS
The following data is obtained from a 6 µm wide and 300 µm long MTI Hall bar with a stoichiometry of Cr 0.15 Bi 0.35 Sb 1.5 Te 3 .Further information on the fabrication and measurement technique is provided in the supplementary material.The sample produced slightly asymmetric data.Using the exact same setup, similar samples have been found to have no intermixing of the signal.Thus, the origin for the intermixing seen in this sample is attributed to an internal issue.The analysis is divided into two parts.First, the classical Hall effect is analyzed where high magnetic fields are beneficial for a destingued determination of the slope.After that, a precise measurement around the hysteresis feature is used for the investigation of the AHE.In the transversal signal the intermixing of longitudinal data is even more pronounced due to the ratio of their magnitudes.In the R xy signal a dip around zero magnetic field, the typical (inverse) longitudinal curvature and an offset to negative values on the ordinate are observed as disruptive factors.To conclude, the signal is highly intermixed and especially the shape of the Hall signal does not correspond to the expectations.

Classical Hall Analysis
The aim of this part of the analysis is to get an accu-rate estimation of the classical Hall slope.Therefore, the Hall data around zero magnetic field is excluded.This allows to handle the symmetrization of the data similar to the one of conventional TIs without need of restructuring the data sets for symmetry reasons.After removing the even contributions in the frequency space the resulting transversal signal is free from longitudinal contributions.With this data, a charge carrier concentration of n 2D = 1.86•10 13 cm −2 and a mobility of µ = 143 cm 2 /Vs are derived from the slope at base temperature using classical Hall analysis and Drude theory.As a check, the value of the charge carrier concentration is also calculated to n 2D = 1.85 • 10 13 cm −2 from the non-linear raw data.The values do not really differ, as the exclusion of the symmetric data arising from an intermixture of the longitudinal data does not effect the point symmetric, linear slope.

Anomalous Hall Analysis
The inserts of Fig. 5 a) and b) show precise measurements around zero magnetic field of the longitudinal and transversal data.Only the data up to 30 K is taken into account, as for higher temperatures no hysteretic behavior is observed.First, the slope determined in the previous large magnetic field measurement is subtracted, as this is caused by the classical Hall effect and the purpose of the precise measurement around zero magnetic field is the determination of the anomalous Hall effect properties.As the slope is a point symmetric feature, it has to be excluded separately.In order to remove the contributions from R xx in the Hall signal, the data of both sweep directions is split at B = 0 T and restructured following the technique shown in Fig. 3.The redistribution of the values is indicated in Fig. 6 a) for base temperature.Next, the data is interpolated to ensure an equidistant spacing of the data points for the Fourier transformation.Making use of the point symmetry of the restructured R xy data, a Fourier analysis is performed that removes all axial symmetric contributions.
The result of the symmetrization process is shown in Fig. 6 b) for multiple temperatures.One can see that the signal is cleared from intermixed perturbations.The symmetric data that is excluded during the symmetrization process is plotted in Fig. S2 in the supplementary material.A fitting of the point symmetric data using equation 1 is made for both branches of each hysteresis.The fits are plotted in Fig. 6 c) together with the data marked with a dashed box in Fig. 6 b).The fit describes the data well, but it can also be seen that a remaining curvature at small magnetic fields of the data measured at higher tem-peratures causes a slight difference between fit and data that results in an uncertainty in the value of A for elevated temperatures.The corresponding fit parameters are shown in Fig. 6 d) -f).For the determination of the values the average of both fits for each temperature is taken.As the curves are similar, also due to the symmetrization, only differences below 0.01 % between the parameters of both fits are observed.For base temperature values of R AH = 463 Ω, B c = 134 mT and A = 59.5 T −1 are obtained.One can see that all parameters decrease with increasing temperatures.The decrease of R AH and B c indicate a decrease of the AHE towards the Curie temperature T c .For T = 20 K the widths of the hysteresis is already close to zero so that for the measurement at T = 30 K no real value for B c could be determined.As the parameter A scales inversely with the width of the transition, the transition region is broadened with increasing temperature.For T ≥ 20 K a larger decrease is observed as the bending of the curve also influences the parameter.

CONCLUSION
In this article the analysis of intermixed conventional and anomalous Hall data was discussed.First, different reasons for intermixed data were listed.Then, a possible symmetrization process of conventional data using a FFT is shown.Thereafter, the existing axial symmetry of the longitudinal and the point symmetry of the transversal anomalous Hall data were discussed.A method for the restructuring of the data by splitting and recombining it at zero magnetic field is suggested in order to maintain the symmetries.This was followed by the symmetrization process using FFT.Furthermore, a mathematical description based on the error function is introduced in order to describe and fit the hysteresis.
Next, the data of an MTI sample was analyzed that showed intermixed anomalous Hall data.The method is carried out exemplarily for the transversal data of the MTI Hall bar.The result is a clear, symmetric anomalous Hall hysteresis where the height and width are precisely determined using the model based on the error function.Slight deviations in curvature from the fitting model are found for elevated temperatures as in addition to the approximately Gaussian distributed binding of the magnetic moments another temperature dependent component that is expected to be Fermi-Dirac distributed contributes.
Besides clearing the hysteretic data from perturbation, the presented approach offers a precise determination of the width and the height of the hysteresis that are comparable to the ones estimated from the raw data.This approach cannot only be used for magnetic field dependent MTI measurements but may be transferred easily to any perturbed hysteretic behaviour that is based on symmetries.
Finally, it is pointed out that the method needs to be handled with care as it may mimic symmetries also for data where no symmetry is expected.Thus, a close comparison between resulting data, raw data and underlying concepts always needs to be made.

FIG. 1 .
FIG. 1. Imperfect Hall bar as origin of intermixed longitudinal and transversal magnetotransport data.The orientation of the external magnetic field B is indicated with an arrow.The examples of charge puddles (left), misaligned contacts (middle) or potential drops at grain boundaries (right) are sketched at opposing contact pairs.

FIG. 2 .
FIG. 2. Anomalous Hall effect.a) The longitudinal data Rxx of two magnetic field sweeps is shown in arbitrary units (a.u.), where the blue data corresponds to a sweep from negative to positive magnetic field and the orange curve vice versa.The peak indicating zero total magnetic field is shifted in B by the coercive magnetic field with respect to the zero position.b) The corresponding Hall data Rxy shows a hysteresis with height RAH and width Bc.

FIG. 3 .
FIG. 3. Symmetries of MTIs.Compared to Fig. 2, here, the colors mark the parts of the data sets that are supposed to be symmetric to each other.a) The longitudinal data of two magnetic field sweeps is shown in arbitrary units (a.u.).A symmetry axis at B = 0 T is shown.b) The corresponding Hall hysteresis shows a point symmetric behaviour.The symmetry point is indicated by a black dot at coordinate origin in the middle of the hysteresis.

FIG. 4 .
FIG. 4. Model for the Hall hysteresis.Using the least square method, the error function is fitted to the data taken from Fig. 2 b).The results are shown with dotted lines.Error functions with different A parameters are sketched in the insert.

FIG. 5 .
FIG. 5. Perturbed magnetoresistance signal of an MTI.The legend is shown in b).a) The raw longitudinal data recorded for different temperatures shows a small influence from the Hall signal.b) The corresponding raw Hall data is shown.The inserts show zoom-ins of precise measurements around zero magnetic field.

Figure 5 a
Figure 5 a) and b) show the longitudinal data and the raw Hall data, respectively.The peaks in the R xx signal are slightly affected by the Hall hysteresis.Furthermore, an offset in resistance between the values for high negative and high positive magnetic field is seen.In the transversal signal the intermixing of longitudinal data is even more pronounced due to the ratio of their magnitudes.In the R xy signal a dip around zero magnetic field, the typical (inverse) longitudinal curvature and an offset to negative values on the ordinate are observed as disruptive factors.To conclude, the signal is highly intermixed and especially the shape of the Hall signal does not correspond to the expectations.The aim of this part of the analysis is to get an accu-

FIG. 6 .
FIG. 6. Symmetrization and analysis of the AHE.The legend is shown in b).a) The redistribution for the symmetrization process of the signal at a base temperature of 1.3 K is shown.In b) the symmetrized data is shown.c) Fits of equation 1 are performed for all temperatures and the temperature dependences of the averaged fit parameters RAH, Bc and A are plotted in d) -f), respectively.