Unveiling the Mechanism for the Split Hysteresis Loop in Epitaxial Co2Fe1-xMnxAl Full-Heusler Alloy Films

Utilizing epitaxial Co2Fe1-xMnxAl full-Heusler alloy films on GaAs (001), we address the controversy over the analysis for the split hysteresis loop which is commonly found in systems consisting of both uniaxial and fourfold anisotropies. Quantitative comparisons are carried out on the values of the twofold and fourfold anisotropy fields obtained with ferromagnetic resonance and vibrating sample magnetometer measurements. The most suitable model for describing the split hysteresis loop is identified. In combination with the component resolved magnetization measurements, these results provide compelling evidences that the switching is caused by the domain wall nucleation and movements with the switching fields centered at the point where the energy landscape shows equal minima for magnetization orienting near the easy axis and the field supported hard axis.

, where M s is the saturation magnetization. Dumm et al. 4 , recognized that K 2 and K 4 depend on both k and H s . Meanwhile, Oepen et al. 6 , analyzed the split loops in the framework of the field-driven SRT. Their results showed that the switching fields in the split loops are directly related to the SRT and thus K 2 and K 4 are obtained by the switching fields and k. The different models have been adopted by many groups 11,[15][16][17][18][19][20][21][22][23][24] . However, they have not been cross-checked with additional magnetic anisotropy sensitive techniques such as ferromagnetic resonance (FMR) and magnetic torque measurements.
In this paper, we identified the model which best describes the system by comparing the uniaxial anisotropy field = / H K M 2 obtained with FMR and vibrating sample magnetometer (VSM) measurements on the Co 2 Fe 1-x Mn x Al films with different Mn concentrations. Co 2 Fe 1-x Mn x Al is a kind of Co-based full-Heusler alloys, which possess high spin polarization and have great potential in spintronics application [25][26][27][28][29][30][31][32][33][34] . Interestingly, electronic structure calculations have revealed that Co 2 MnAl can retain half-metallic Scientific RepoRts | 6:18615 | DOI: 10.1038/srep18615 property for different levels of Fe doping 35,36 . The magnetic anisotropy however shows strong variations on concentration which mainly due to a competition between uniaxial and four-fold anisotropies 37,38 . Thus, it provides an interesting system to crosscheck the different models used in the split hysteresis analysis besides the fundamental interest in the exploration of the concentration dependent properties. We first measured the hysteresis loops with VSM and calculated the anisotropy fields from the split loops using different models. The results are quantitatively compared with the measured values utilizing FMR which allows us to identify the most suitable model used in the split loop analysis. The underlying physics is discussed. And the concentration dependent important material parameters such as effective magnetization, anisotropy fields, and damping constants are also given.

Results
Controversy over the analysis for the split hysteresis loop. Figure 1 shows a typical split hysteresis loop of a 45 nm Co 2 FeAl film measured by VSM with the magnetic field being applied within the sample plane and along the [110] direction. A discontinuity appears around H s . This discontinuity is assumed to be the consequence of the superposition of the uniaxial and the fourfold anisotropy in the case that the hard axis of the uniaxial anisotropy coinciding with an easy axis of the fourfold component 1,4,6 . With the measured split loops, we calculated H 2 and H 4 utilizing the three models mentioned above. The results are listed in Table 1 for Co 2 FeAl and Co 2 Fe 0.7 Mn 0.3 Al films. Both films have the same thickness of 45 nm. We can find that the results obtained with different models are significantly different. This raises an interesting question which model describes the magnetic anisotropies of the system best.
FMR measurements and analysis. To obtain independent results for crosschecking the different models, we performed angular dependent measurements utilizing FMR. The samples were positioned on a coplanar waveguide (CPW) fixture with the [110] direction of the film parallel to the x-direction as sketched in Fig. 2. The CPW is connected with a Vector Network Analyzer (VNA), which generates microwave with tunable frequencies.
The VNA also detects the rf signal via the change of the forward transmission coefficient in scattering parameters, S 21 . The external magnetic field H is applied within the film plane. The orientation of magnetic field is controlled by a servo motor with high accuracy of positioning (error margin < 0.15°). We note that all the graphs are plotted with the raw data without any mathematical smoothing. The magnitude of the magnetic field is first set to be 1548 Oe, which is larger than the saturation field. The microwave frequency is swept to find the resonance condition where the maximum absorption occurs. Rotating the magnet yields an angular-dependent resonance frequency for Co 2 FeAl which is plotted in Fig. 3(a). From the plot of the angular dependence one can easily identify twofold and fourfold symmetries with the maximum frequency located at 0° and 180°. Since the resonance frequency is proportional to the effective magnetic field, the maximum (minimum) value is found when the field is parallel to the easy (hard) axis. The angular dependence of the resonance frequency immediately shows that the easy axis Weber et al. 1 Dumm et al. 4 . In our measurements, the magnetic field is applied within the film plane, i.e., θ θ = =°90

M H
. Therefore, it can be simplified as: is the effective magnetization of the sample. According to equation (2), we can fit the angular dependent measurements. The fitted result is plotted as the red line in Fig. 3(a). The plot reproduces the data very well. The best fitting parameters are listed in Table 2 Oe for Co 2 FeAl. The obtained effective magnetization and g are ± 15300 40 Oe and 2.0, respectively. On account of the complexity of multi-parameters fitting, we repeat the same measurements at a field with different amplitude, i.e., 1166 Oe to double-check the fitted results. The experimental data and the fitted curve are shown in Fig. 3(b). The fitting yields the same results within the error margin of experiments (see Table 2). The quantitative agreement evidences that the method used in the angular dependent FMR measurements and analysis is accurate and with high reproducibility.

Comparison of FMR and VSM measurements along different directions.
To further check the accuracy of the fitted results, we also performed field dependent FMR measurements along different directions. In the left column of Fig. 4, we show the two dimensional (2D) gray scale mapping of the S 21 signal of the microwave absorption measured via VNA as a function of the microwave frequency f and magnetic field H with the field applied along the [110], [100] and [110], respectively. The right column displays the corresponding hysteresis loops obtained via VSM. In the FMR measurements we swept the magnetic field from ~ +1400 Oe to zero. The brightness of the grayscale shows the amplitude of the absorption and the brightest spots represent the position of FMR. Since the anisotropy fields cause additional contributions to the torque on the magnetization, FMR spectrum exhibits different behavior in these three directions. The easy axis of magnetization is along the [110] direction, as can be recognized from the rectangular shape of the hysteresis loop in Fig. 4(d). The data for the field along the [110] direction [ Fig. 4(a)] show a typical FMR spectrum. It can be easily fitted with Kittel formula (red curve). In this case, the magnetization precesses around the direction of external magnetic field. The curve along the [100] direction, Fig. 4(b), is however more complicated. With the increase of the magnetic field, the resonance frequency first decreases then increases after an inflection field at ~450 Oe. Consequently, two resonance modes at different fields can be obtained for a given frequency. For instance, a 7 GHz microwave field causes resonance at either 200 Oe or 700 Oe. From Fig. 4(e), it is obvious that [100] is the hard axis of the sample and it saturates at the field higher than 500 Oe. The remanence is about 0.7 of the saturation value which can be easily understood as the field is applied 45° with respect to the easy axis, i.e., the [110] direction and the magnetization is expected to align close to the easy axis at low external field. With the increasing of the magnetic field, the magnetization will align along the field direction eventually. In such case, ϕ M is no longer constant, but is determined by the total energy minimum of equation (1). In fact, it changes from 45° to 0°. From equation (2), we see that the resonance frequency is proportional to (  Hence the resonance frequency is determined by the competition between anisotropy fields and magnetic field. As listed in Table 2, both H 2 and H 4 are comparable with H at low magnetic field which can lead to a decrease of resonance frequency f with increasing field in a certain field range, as shown in Fig. 4(b). Taking the same parameters in the angular dependent FMR measurements as in Fig. 3 and using equations (1) and (2), we have computed the field dependent f (red curve) and found that it agrees well with the experimental data. Similarly, we can understand the field dependence of f for the field applied along the [ 110] direction [ Fig. 4(c)] which is also not an easy axis. The resonance frequency decreases with increasing the field up to 80 Oe. After a sharp drop, the spectrum increases with field as expected. The  direction. At zero field M is oriented along the easy axis and perpendicular to H. Depending on either increase or decrease of the field, the magnetization curve shows two discontinuities at two different switching fields around H s . We further performed the field dependent FMR measurements with ascending field and found that the jumping field in the FMR occurs at exactly the same field as the switching field in the field ascending branch of hysteresis loop. The one-to-one correspondence identifies that the origin of the jump in the field dependent FMR measurements shown in Fig. 4(c) is the magnetization switching. We emphasize that all the three resonance curves shown in Fig. 3(a-c) can be well reproduced with the parameters in  Table 2. Right column (d-f) are the corresponding longitudinal hysteresis loops of Co 2 FeAl film measured by VSM at RT.  (1) and (2) [red lines in Fig. 3(a-c)] once again, proving the validity of the method used for anisotropy field analysis in the FMR measurements.

Underlying mechanisms for the split hysteresis loop analysis. Comparing the results obtained via
FMR with the values derived from the VSM measurements utilizing different models (  6 , the field driven reorientation for a strong uniaxial behavior is discussed. The assumption of the model is that without field the only minimum of the energy landscape appears for magnetization parallel to one axis only while the field applied along the hard axis drives the system through a state of coexisting phases (or metastability [40][41][42] ). The latter scenario can be found only for − < − − <− 6 1 2 . We note here that, in the original paper, the free energy is expressed as ϕ ϕ = + E a b sin s in 2 4 and a translation of = + K a b 2 and K 4 = b is therefore required for comparison. The main assumption of the model is that in descending field the magnetization switches back into the easy axis when the local minimum created by the field is erased. With values of the anisotropy obtained via FMR it is evident that − − >− 1 2 and thus the prerequisite not fulfilled.
In the system studied here, however, the starting point (zero field) is situated within the range of metastability (see Fig. 4b in ref. 41 . In lowest order approximation the field applied along the harder axis weakens the K 2 contribution. Trespassing the range of metastability means to change the depth of the two coexisting minima. Dumm et al. 4 , considered the switching to appear around the point where the magnetic field causes an energy degeneracy of the two states. Apparently the latter is the appropriate assumption to describe the situation in the system investigated here. The key issue to the understanding is the fact that the anisotropies as well as the potential well are very small. As the saturation magnetization is high even the minor values of the in-plane shape anisotropy can have an impact. At the edges the demagnetizing fields can cause a local reorientation of the magnetization which causes an instantaneous reversal of magnetization via domain wall movement on further decrease of field strength. This is confirmed by the simultaneously obtained magnetization along different directions (not shown). The normalized value of the magnetization calculated from the individual components along different direction shows a strong decrease around the switching fields (Fig. 5) evidencing that the switching is caused by the domain wall nucleation and domain wall movements 43 .
In ref. 4 as well as in ref. 6 a reversible rotation near the zero field in the potential that is determined by the sum of two-and fourfold anisotropy contribution is assumed. In case of strong shape anisotropy the magnetization reversal appears fully within the sample plane, the total energy of the system [equation (1)] can be simplified by taking θ =°90 Oe (FMR results) we obtain three local energy minima around 0°, 90° and 180° values (see Fig. 6). The minima have the same value at 80 Oe while for fields that are either smaller (76 Oe) or larger (84 Oe), the local minimum at 90° is higher or lower than the other two minima. Hence the FMR data verify the assumption of equal energy states at the switching field. We note that H s = 77 Oe, the small variance with 80 Oe may origin from the error bar of the measurements. Dumm et al. 4   Oe, are consistent with the fitted data from the FMR measurements.
Confirmation with different Mn concentration. Above, we have investigated the magnetic properties of Co 2 FeAl film via FMR and VSM measurements. In the following, we continue to discuss the property variation with different Mn doping, i.e., Co 2 Fe 0.7 Mn 0. 3 Al and Co 2 Fe 0.3 Mn 0.7 Al. Figure 7(a,b) present the angular dependent FMR measurements. They can be fitted very well with equation (2) and the fitted twofold and fourfold anisotropy fields are listed in Table 2. For Co 2 Fe 0.7 Mn 0.3 Al, the fitted results agree very well with VSM measurements using the analysis mentioned above. The good agreement once again proves the validity of the model used in the split loop analysis. From the angular dependent FMR measurements in Fig. 7 7 Al did not show any split hysteresis loop. Therefore, the quantitative comparison between FMR and VSM measurements is not applicable. Besides, we found that the effective magnetization decreases from 15300 Oe to 14273 Oe and twofold anisotropy field decreases from 145 Oe to 12 Oe with the Mn composition increasing from 0 to 0.7. Recent X-ray magnetic circular dichroism measurements show that Co, Fe, Mn all exhibit net ferromagnetic states and contribute ferromagnetism to the films 37 . Since the magnetic moment of Mn atom is generally larger than that of Fe atom, it would be expected that the magnetization of the system would increase with increasing Mn concentration if Mn, Fe and Co atoms are completely ferromagnetic coupled. Our FMR measurements, however, show the opposite behavior. This strongly suggests that there must be antiferromagnetic interaction among the system. The evolution of H 2 and H 4 with the changing Mn concentration could be associated with the competition between ferromagnetic Ruderman-Kittel-Kasuya-Yoshida exchange and antiferromagnetic superexchange, as reported by Şaşıoğlu et al. 44,45 , Through the FMR linewidth measurements, we also obtain the damping factor of samples with

Summary
Combining VSM and FMR measurements on the full-Heusler alloy Co 2 Fe 1-x Mn x Al epitaxially grown on GaAs(001), three different models for the interpretation of split hysteresis loop are checked. The most suitable model is identified as the one that assumes the switching fields centered at the point where the energy landscape shows equal minima near the easy axis and the field supported hard axis. Our studies reveal that H 2 decreases rapidly with increasing Mn concentration and almost vanishes at x = 0.7, while H 4 shows much less concentration dependence. The decreasing effective magnetization with adding Mn component strongly suggests the existence of antiferromagnetic coupling among the system.

Methods
The Co 2 Fe 1-x Mn x Al samples were prepared by molecular-beam epitaxy on GaAs (001) at 553 K. All the films have the same thickness of 45 nm and the Mn composition x varies from 0 to 0.7. Before being taken out of the ultrahigh vacuum chamber, the films were protected by 2 nm of aluminum capping layer. The crystal structure and the quality of order were analyzed by double-crystal X-ray diffraction as described previously 27 . The hysteresis loops along different directions were obtained via VSM. The FMR measurements were performed with a Vector Network Analyzer, which generates microwave with tunable frequencies (20 MHz to 20 GHz). The VNA also detects the rf signal via the change of the forward transmission coefficient in scattering parameters, S 21 . The external magnetic field H was applied within the film plane. The orientation of magnetic field was controlled by a servo motor with high accuracy of positioning (error margin < 0.15°).