We report a dual-component anomalous Hall effect (AHE) in polycrystalline Fe_{3}Ga_{4} thin films grown on STO (001) and Al_{2}O_{3} substrates. Systematic magnetic and magnetotransport measurements reveal an AHE consisting of positive and negative contributions that coexist across a wide range of temperatures and magnetic phases. We find that both magnitudes are nearly equal in the low-temperature ferromagnetic (FM) phase, but that their relative ratio is reduced upon heating through the antiferromagnetic helical spin-spiral state where they compete with metamagnetism and topological Hall effects, maintaining finite values at least up to the high-temperature FM phase.

The anomalous Hall effect (AHE) is a rich phenomenon offering a method for ascertaining the presence and the type of ferromagnetic (FM) ordering in an electrical conductor. It appears as voltage orthogonal to both the magnetization and applied current with mechanisms strongly tied to spin–orbit interactions. With the recent realization that transverse crystal-momentum can originate from the accumulation of a Berry phase in reciprocal space, the use of the AHE has gained renewed interest as a probe in systems with non-trivial topology and non-collinear spin ordering.^{1–10} A particularly interesting use-case is in the determination of the spin-structure of metallic, metamagnetic (MM) Fe_{3}Ga_{4}. The magnetic phase diagram of Fe_{3}Ga_{4} exhibits five distinct temperature regions: a ferromagnetic (FM) state below a first-order transition temperature at $ T 1$, an antiferromagnetic (AFM) state between $ T 1$ and $ T 2$, reemergence to ferromagnetic state again between $ T 2$ and $ T 3$, and a magnetic state thought to be FM-like between $ T 3$ and $ T 4$, above which the material becomes paramagnetic (see Fig. 1).^{11}

Over the years, several theories have been proposed to account for the competition between these states, the observation of field-induced MM transitions in the AFM state, and the interplay between localized and itinerant moments. The earliest explanation attributed the behavior to the coexistence of FM and AFM according to the theory of itinerant-electron models of Moriya–Usami^{12} and Isoda.^{13} Much later, work on high-quality single-crystalline Fe_{3}Ga_{4} revealed a deviation from the characteristic AHE behavior during the field-induced MM transition that was attributed to the emergence of a topological Hall effect (THE).^{11} This, together with unpolarized neutron diffraction and electronic structure simulations, led to the conclusion that the AFM state was a FM incommensurate spin-density wave (ISDW) having a modulation wavevector propagating along the $c$-axis with spins polarized along the $a$-axis.^{14} However, a close parallel was recently drawn between the THE in Fe_{3}Ga_{4} and the Kagome-net magnet YMn_{6}Sn_{6} that hosts a THE facilitated by fluctuation-driven chirality.^{15,16} It was then shown that this model of an AFM helical spin-structure (HSS) state propagating along the $c$-axis with moments rotating in the $ab$-plane, which converts to conical ordering during the field-induced MM transition and ultimately leads to a field-polarized FM state at high-field, reproduces most aspects of the magnetic behavior in the intermediate state between $ T 1$ and $ T 2$.^{17,18}

Nevertheless, there are still outstanding questions regarding the nature of the low- and high-temperature FM phases and their transitions, such as the change in sign of the AHE observed when crossing $ T 1$ but not $ T 2$, and whether Fe_{3}Ga_{4} can be prepared in a technologically useful form.^{19,20} In order to address these issues, we have performed systematic magnetotransport and magnetic measurements on polycrystalline single-phase Fe_{3}Ga_{4} thin films. This geometry allows Hall measurements with current and magnetic field orientations that were previously unfeasible on polycrystalline ingots and single-crystalline boules.

The Fe_{3}Ga_{4} films were prepared by DC magnetron sputtering from a single composition 2-inch diameter Fe_{3}Ga_{4} sintered target using 75 W of power on to single-crystal (001) strontium titanate (STO) and Al_{2}O_{3} substrates held at 600 °C in a 4 mTorr pure Ar atmosphere. Before film deposition, the chamber was cleared of flakes and contaminants by depositing Ti as a passivation layer, giving a base pressure of 1.0 $\xd7$ 10^{−8 }Torr. The films were grown to a thickness averaging 156 nm into the bulk-like polycrystalline strain regime (i.e., not epitaxial). Hall bars were fabricated by high energy Ar ion milling using a Kapton shadow mask and were $\u223c$6 mm long by $\u223c$1 mm wide with contacts made to side arms using Au wires and Ag paint. Resistivity measurements were performed using a standard four-wire technique in a Quantum Design (QD) Dynacool PPMS with typical AC excitation currents of 1 mA at a frequency of 18.3 Hz. Magnetization measurements were performed with a QD MPMS3 SQUID magnetometer. The results discussed later refer to films grown on STO substrates unless indicated otherwise. Details of the structural and chemical characterization of the films (TEM, SAED, XRD, and WXRF) can be found in the supplementary material.

The Hall resistivity as a function of perpendicular magnetic field, $ \rho x y(H)$, for several temperatures are shown in Fig. 2. The data have been symmetrized to remove any longitudinal magnetoresistance component by using a careful procedure (see the supplementary material) that considers sample magnetic hysteresis as well as trapped remnant flux in the superconducting solenoid supplying the applied field. For consistency, the Hall effect and magnetization experiments were performed on the same sample, and under identical temperatures and field orientations. At high fields, $ \rho x y(H)$ exhibits linear behavior with a positive slope due to the ordinary Hall (OH) effect ( $ \rho x y 0$), indicating hole-like conduction. At low fields, we observe prominent hysteresis that closely mimics the field dependence of the out-of-plane magnetization, a well-known feature of the anomalous Hall (AH) effect ( $ \rho x y A$). Interestingly, however, at $T<$ 100 K, the direction of the $ \rho x y(H)$ *hysteresis loops briefly invert* relative to $M H,$ eventually *reversing* with increasing field to match the behavior of the magnetization. This indicates that the films possess a multi-component AH effect with both positive and negative contributions. Notably, at $T=$ 100 K, where the film is transitioning from the low-T FM to the intermediate-T AFM phase, this results in multiple crossing points and high-field hysteresis loops not found in the corresponding $M H$ loop. These multiple contributions to Hall resistivity are indicative of a rich spin landscape around this metamagnetic transition and separating them can lend some answers as to the nature of this unique effect.

$ R S(T)$ is shown in Fig. 2 lower-right inset, using values fit from the high-field region (small $ \mu 0M/B$). Notice that $ R S$ approaches zero at $T\u2264$ 75 K, coincidently where it is reported to change sign in single-crystal Fe_{3}Ga_{4} samples. We do not believe that the sign change seen in single crystals or the dual-component AHE observed in our films stem from morphological factors, such as the AH sign-changes induced by competition between bulk and interface/surface scattering contributions, as these occur in small-grain (<50 nm) clustered and thin-film multilayer systems.^{21,22} Moreover, these systems do not show a coexistence of positive and negative contributions, but rather a conversion from one to the other with changing temperature. It is thus apparent that evaluation of the AH coefficient alone is insufficient to describe the multicomponent character of the AH effect in Fe_{3}Ga_{4}. Indeed, we must also consider that $ R S$ is, itself, a function of the temperature- and field-dependent longitudinal resistivity, as $ R S\u2009\u223c\u2009\rho + \rho 2$. Here, the linear-term corresponds to asymmetric spin–orbit scattering (skew-scattering) and the quadratic-term corresponds to side-jump processes and intrinsic band structure (Berry's curvature) effects. We can neglect skew-scattering in our case, however, since it is only expected to become significant at resistivities below $\u223c$1 $\mu \Omega \u2009cm$( $\u223c$100× smaller than the value of our samples). This is further justified by our quadratic fit of $ R S$ vs $\rho $, which yielded an insignificant linear term.

*intrinsic*AH coefficient, a parameter with a closer connection to the electronic structure than $ R S$.

^{23}Now, by using the previously determined $ R 0$ values along with the magnetoresistance and magnetization data, we can plot the AH conductivity $ \sigma x y A\u2261( \rho x y\u2212 \rho x y 0)/ \rho 2$ vs $M$ at each temperature and perform linear fits to obtain $ S H(T)$ from the slopes. Such plots are shown in Fig. 3 for three temperatures typical of the Hall effect observed in each magnetic phase. Focusing on the $T=$ 5 K curve, we find that the AH effect is indeed linearly dependent on $M$, with a hysteretic intrinsic AH coefficient obtained from the slope of the curve that sharply changes sign from –8.1 $\xd7$ 10

^{−5}to 1.5 $\xd7$ 10

^{−4}V

^{−1}at $M=$ 0.16 T when increasing field to positive magnetic saturation, and at $M=$ 0.27 T when returning from positive magnetic saturation.

Overall, the hysteretic features of the three curves in Fig. 3 are disparate, yet acquire positive linearity with $M$ near saturation. The value of $ S H$ in this high-field range as a function of temperature is displayed in the Fig. 3 inset. The positive, linear-dependence on $M$ consistent at all temperatures enables further parsing of the Hall resistivity, subtraction of the positive AH contribution, and isolation of the negative AH and non-collinear (topological) components for closer evaluation. In Fig. 4, we use the obtained $ R 0$ and high-field $ S H$ value to decompose the total Hall resistivity at 5 K and reveal the field-dependence of the residual Hall resistivity $ \rho x y T = \rho x y\u2212 \rho x y 0\u2212 \rho x y A$, where $ \rho x y A = S H \rho 2 \mu 0M$. Since no metamagnetic transitions are observed at this low T, $ \rho x y T$ solely represents the negative AH contribution.

A comparison of the positive ( $ \rho x y A$) and residual Hall resistivity $( \rho x y T$) magnitudes as a function of temperature at $H=$ 2 T is shown in Fig. 5. Though both components generally increase with temperature, their growth rates are different. This is quantified in the Fig. 5 inset where we show their absolute relative ratio $ \rho x y T / \rho x y A$ at 2 T, exposing several enlightening features. We see that the ratio is at or near unity below 50 K, reflecting the visibility of both components in the low- $T$ FM phase. The ratio also steadily drops as the temperature increases from $\u223c$50 to $\u223c$150 K, near the middle of the AFM phase. Finally, at $T\u2265$ 150 K, it has settled to a value of $\u223c$1/3. According to reports on Fe_{3}Ga_{4} single crystals, a negative AH effect is expected at $T\u2265$ 100 K when $I\u2225\u2009c$-axis and $H\u2225\u2009ab$-plane.^{11} On average, 1/3 of the grains of our polycrystalline films will have this configuration, suggesting that $ \rho x y T$ originates from these crystallites. Nevertheless, the hysteresis loop of $ \rho x y T$ is quite square compared to $ \rho x y A$ in our films, which seems at odds with the fact that the $c$-axis of Fe_{3}Ga_{4} is the magnetic easy axis.^{11,17} However, if we attribute the positive contribution only to those crystallites oriented with $I\u2225\u2009ab$-plane, where $H$ can lie along the $a$-, $b$-, or $c$-axes, the greater field-response variety could combine to result in a stretched hysteresis loop.

Regarding the approach of $| \rho x y T / \rho x y A|$ to unity at $T\u2264$ 100 K, we note that without complete multi-axis Hall measurements of single-crystal Fe_{3}Ga_{4}, it is difficult to fully ascribe the positive and negative contributions in our films to a particular intrinsic anisotropy. While the single-crystal Hall measurements had well-defined geometry for the direction of $I\u2225\u2009c$-axis, the monoclinic structure and shape of the crystals limited the known orientation of $H$ to somewhere in the $ab$-plane. The need to resolve the AH behavior for $H$ within the $ab$-plane is emphasized by recent anisotropic magnetic measurements on single crystals that revealed a unique two-step metamagnetic transition from AFM to the field polarized-FM state for $H\u2225\u2009b$-axis. Thus, it is possible that even with $I\u2225\u2009c$-axis, there is a difference in the sign of the AH effect for $H\u2225\u2009a$ or $b$. Differing $T$-dependence of the components' magnitudes, together with a loss of the THE in the low- $T$ FM state could then change the dominant contributor to the total AH resistivity, leading to the change in sign of $ R S$ observed in single-crystal Fe_{3}Ga_{4} when cooling below $\u223c$100 K, or to a balance of the components in our films. Interestingly, aspects of a sign-tunable AHE demonstrated in heterostructures based on SrRuO_{3} ferromagnetic films, whose origin has been ascribed to intrinsic Berry curvature connected to the multiorbital character of the electronic structure,^{24–26} were recently explained by Kimbell *et al.* in terms similar to ours, that of a multi-channel intrinsic AHE.^{27} Nevertheless, future studies, to obtain a comprehensive understanding of these multi-component AHE characteristics, should include Berry curvature calculations carried out across the several phase transitions of Fe_{3}Ga_{4} while assuming a substrate material as well as a predicted spin-space group for all three phases, which to this point is yet to be determined for the Fe_{3}Ga_{4} phase.

## SUPPLEMENTARY MATERIAL

See the supplementary material for TEM and SEM characterization, Hall data symmetrization procedure details, longitudinal magnetotransport characterization, magnetization characterization, and additional Hall effect measurements.

This work has been funded by the Office of Naval Research, under the NRL 6.1 Base Program and by ONR Grant No. N0001423WX02132.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Joseph C. Prestigiacomo:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). **Michelle E. Jamer:** Conceptualization (equal); Investigation (supporting); Validation (equal); Writing – review & editing (equal). **Patrick G. Callahan:** Data curation (equal); Formal analysis (equal); Methodology (equal); Resources (equal); Validation (equal); Writing – review & editing (equal). **Steven P. Bennett:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Validation (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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