Anomalous Hall magnetoresistance in a ferromagnet

The anomalous Hall effect, observed in conducting ferromagnets with broken time-reversal symmetry, offers the possibility to couple spin and orbital degrees of freedom of electrons in ferromagnets. In addition to charge, the anomalous Hall effect also leads to spin accumulation at the surfaces perpendicular to both the current and magnetization direction. Here, we experimentally demonstrate that the spin accumulation, subsequent spin backflow, and spin–charge conversion can give rise to a different type of spin current-related spin current related magnetoresistance, dubbed here as the anomalous Hall magnetoresistance, which has the same angular dependence as the recently discovered spin Hall magnetoresistance. The anomalous Hall magnetoresistance is observed in four types of samples: co-sputtered (Fe1−xMnx)0.6Pt0.4, Fe1−xMnx/Pt multilayer, Fe1−xMnx with x = 0.17–0.65 and Fe, and analyzed using the drift-diffusion model. Our results provide an alternative route to study charge–spin conversion in ferromagnets and to exploit it for potential spintronic applications.

2 Magnetoresistance (MR) in ferromagnetic (FM) materials and related heterostructures plays essential roles both in fundamental understanding of magnetism and electron transport in these structures and in various technological applications [1][2][3][4] . The most widely studied MR effects include anisotropic magnetoresistance (AMR), giant magnetoresistance (GMR) and tunnel magnetoresistance (TMR). These MR effects typically arise from spin-dependent transport of charge carriers either in the bulk or at the interfaces of these structures or the combination of both. Recently the discovery of several types of MR effects of different origins have triggered a renewed interest for spin-dependent MR; these include spin Hall magnetoresistance (SMR) in FM/heavy metal (HM) bilayers 5-10 , Rashba-Edelstein magnetoresistance (REMR) in Bi/Ag/CoFeB 11 , and Hanle magnetoresistance (HMR) in heavy metals 12 . One key aspect of these recently discovered MR effects is that they all originate from a two-step charge-spin conversion process, i.e., in the first step charge current is converted to spin current through either the spin Hall effect (SHE) 13,14 or the Rashba-Edelstein effect (REE) 15,16 , and in the second step part of the reflected spin current is converted back to charge current by the respective inverse effects. As inverse SHE (ISHE) always co-exists inside a material with SHE, the interplay of these two gives rise to an extra positive resistance contribution to a bulk conductor, which was first reported in the context of Hall effect in semiconductors 17 . In the proximity of surfaces/edges, part of the SHE generated spin current is cancelled out by the reflected spin current due to spin accumulation, resulting in a negative resistance contribution, as first derived by Dyakonov 18 . In the recently observed MR effects, the positive contribution does not play a role because it is insensitive to external field, while the negative contribution is modulated through controlling the amount of spin current reflection by either an adjacent magnetization (SMR and REMR) or an external magnetic field (HMR). Despite their small magnitude, these MRs are powerful tools to extract spin transport parameters, particularly spin-orbit torque (SOT) in FM/HM heterostructures 19-27 , which has important applications in three-terminal 20 , logic 28 , and sensing 29, 30 devices. Unlike conventional MR effect, in all these MR effects, the FM plays a relatively less important role as both 3 charge to spin and spin to charge conversions take place inside the HM layer. The FM only influences the conversion process indirectly through regulating the amount of spin current reflected back to FM/HM or FM/non-magnetic metal (NM) interfaces. From the application viewpoint, however, it will be of interest to investigate if a MR effect similar to SMR can be present in a FM alone, as this would allow additional flexibility in manipulating the charge-spin conversion process via controlling the magnetization of the FM directly.
Recently, anomalous Hall effect (AHE) in FM has attracted attention as an alternative mechanism for generating spin current or SOT in FM/NM multilayers. When a charge current flows in an FM in the longitudinal direction, spin-up and spin-down electrons are deflected to opposite transverse directions via extrinsic mechanisms like skew scattering and side-jump or intrinsic mechanism related to band structure of the material 31 . Due to the asymmetry in density of states at the Fermi level and charge transport in FM, both transverse charge and spin accumulations will occur at boundaries of the sample at steady state. The former acts on the entire sample, generating the AHE voltage; while the latter leads to a backflow of spin current that only affects the vicinity of the sample boundary. Taniguchi el al. 32, 33 have predicted theoretically the presence of AHE-related SOT in FM/NM/FM trilayers and magnetoresistance in FM/NM bilayers. Very recently, several experimental attempts have been made to detect the AHEinduced spin current through either spin injection experiment in Y3Fe5O12/Py heterostructure 34,35 or characterization of SOT in FM/NM/FM sandwich structures 36,37 . However, since all these experiments involve multiple layers, it is difficult to rule out completely contributions other than AHE to the predicted or observed MR or SOT. In this regard, here we report on a magnetoresistance induced by AHE and its inverse effect in a single FM layer, and refer it to as anomalous Hall magnetoresistance (AHMR). In order to observe the AHMR, one requires FM with a large AHE. Therefore, we focus on four types of FMs, i.e., co-sputtered (Fe1-xMnx)0.6Pt0.4, Fe1-xMnx/Pt multilayers 38,39 , Fe1-xMnx with x = 0.17 -0.65 and Fe. These materials are chosen because they allow to tune the saturation magnetization and thus the strength of AHE by simply adjusting the Mn composition (except for Fe). Fe1-xMnx itself can be tuned from ferromagnet to antiferromagnet by controlling the Mn composition. The inclusion of Pt further enhances the AHE in both the co-sputtered and multilayer samples. Magnetoresistance with SMR-like angular dependence is observed in all the four types of samples. We argue that the observed magnetoresistance is AHMR instead of SMR because all the samples behave as a single phase FM. Our argument is further substantiated by scaling analysis of the AHE and the relation between the MR and anomalous Hall angle. Based on the drift-diffusion formalism, we derive the analytical equations for MR in a single FM layer including AHE, and demonstrate that both the magnitude and thickness dependence of AHMR can be accounted for reasonably well using the analytical model.

Results
Angle dependent magnetoresistance. As depicted in Figs. 1a and 1b, the applied charge current (jc) in x-direction induces a transverse spin current ( s t ) via AHE with the flow direction given by × , where is the magnetization direction. The simultaneous action of inverse AHE will convert a portion of s t back to charge current ( c ′ ) that has a direction opposite to the original one, thereby increasing the overall resistance of FM (positive contribution). For the case wherein || , with the comparable scale of film thickness and spin diffusion length, the backflow of spin current largely cancels s t and reduces the extra resistance (negative contribution). Whereas, when || , with the large lateral size, such cancellation is confined in the proximity of the sample edges only, and s t inside the sample remains nearly constant.
To illustrate the difference in two cases, we illustrate in Figs. 1a and 1b the distribution of net s t in a colormap, the deeper the color the larger the net spin current. On the other hand, in the case of || (see Fig. 1c), there is no AHE. Therefore, when the magnetization rotates in the yz-plane, an angle dependent MR, i.e., AHMR, appears and its dependence is expected to be the same as that of SMR. However, it should be noted that in the case of AHMR, both positive and negative contributions come from a single 5 layer of FM material. The former is uniform throughout the sample, whereas the latter is dependent on the distribution of reflected spin current from the edges/surfaces, which is determined by the relative orientation of the magnetization with respect to the sample geometry and current direction.
To experimentally characterize the AHMR, we fabricated four types of samples (see Fig. 1d  to the lower end of spin Hall angles reported for Pt (which itself is scattered over a large range) 46 , but is two times as large as AH of NiFe/Pt bilayer 47 . In the case of NiFe/Pt, spin-orbit coupling at the interface has been cited as the cause for enhanced AH . In the present case, however, the enhancement of AH in (Fe1-xMnx)0.6Pt0.4 is presumably due to Pt atoms uniformly distributed in the alloy films. In addition to (Fe1-xMnx)0.6Pt0.4, we also show the results for Fe1-xMnx, Fe1-xPtx and Fe in Fig. 3e (represented by different symbols). We will discuss these results shortly after presenting the analytical model. Besides these samples, some other common ferromagnetic and antiferromagnetic materials including Co, NiFe, and Ir0.2Mn0.8 were also examined, but they all exhibit a AH at least one order of magnitude smaller, and therefore either very small or different MR(θzy) behavior was observed (see Supplementary Note 6). This is expected because the size of AH in transition metals typically follows the order: Fe >> Co > Ni [48][49][50] . As discussed in the Supplementary Note 7, field misalignment is not able to account for the magnitude of the measured MR(θzy) curves. Apart from the field misalignment, another possible source for the MR(θzy) observed is the geometric size effect (GSE) related AMR. However, if this is indeed the case, one would expect a same temperature dependence of MR(θzy) and MR(θzx). But, as shown in Supplementary Fig. 12, we observed a different temperature dependence for MR(θzy) and MR(θzx) in the FeMnPt and Fe samples, but same temperature dependence in the Py control sample which has a much smaller AHE. In view of these results, both field misalignment and GSE related AMR can be ruled out as the origin of the observed MR(θzy).

Derivation of AHMR.
In order to have a quantitative understanding of the results shown in Fig. 3e, we derive the analytical equation for MR in a single FM layer by including the AHE and its inverse effect (see Supplementary Note 8). As discussed, a transverse spin current s t is generated in the direction of × when the charge current flows in an FM, where is the magnetization direction. In the case of bulk, s t is uniformly distributed inside the sample, which gives an extra resistance due to the additional 9 opposite charge current induced by the inverse AHE. However, due to spin accumulation and backflow of spin current from the boundary, the situation changes when the sample has a finite dimension in the s t flowing direction. As derived in Supplementary Note 8, this will lead to a magnetoresistance that is given in the general form of where and s are the thickness and spin diffusion length of FM, respectively, is the polarization for longitudinal conductivity, AH is the anomalous Hall angle, and A is the AMR ratio. Apparently, Eq.
where c is the original applied charge current in x-direction, is the conductivity, e is the electron charge, and the rest of parameters are already defined as above. The second term in the brackets of Eqs.
(3) and (4) is resulted from the backflow of spin current induced by the spin accumulation described by Eq. (1). The same set of equations applies to the case when || except that d is replaced by the sample width (w), and the spatial distribution is along y-direction. In the calculations, we have used d = 10 nm, w = 100 μm, = 0.5, s = 3 nm, and AH = 0.03. The values used for polarization and spin diffusion length are within the range of those reported in FMs 51 . As shown in Fig. 4b for || , where the dashed lines are added as a reference to show the case when AHE is absent in the sample, due to the comparable scale of d and ls, the backflow spin current cancels s t largely throughout the sample (see Fig. 1a for illustration).
In contrast, in the case of || , the cancellation is mainly confined in the vicinity of the two side edges: s t in the remaining region remains almost intact because w >> ls (see Fig. 4e and Fig. 1b). It is this difference in the cancellation of s t that leads to the different degree of charge current correction, which consequently results in the different resistance for || and || : the origin of AHMR. On the other hand, the AHE is absent when || , and therefore no transverse spin current / spin accumulation nor redistribution of charge current occurs in this case. Although the AHE does not come into play when || , the conventional AMR still exits and gives rise to an increase in resistance, which is revealed by the 2 nd term in Eq. (1). Therefore, the AHMR, which is given by the 3 rd term of Eq. (1), has the same angular dependence as SMR, in qualitative agreement with the experimental data shown in Figs. 2b -2d and Fig. 3a. Notably, the size of AHMR ratio, given by ( AH ) 2 2 s tanh( 2 s ) , exhibits a quadratic relationship with AH , which is in good agreement with the ADMR data shown in Fig. 3e. These results affirm our argument that the ADMR in zy-plane is caused by the AHE and its inverse in the FM layer, i.e., the AHMR. It is apparent from Eq. (1) that, in addition to the experimentally determined anomalous Hall angle AH , the magnitude of AHMR is also directly dependent on and s , which are not available The range for and s are chosen to cover most of the ferromagnets. Therefore, besides (Fe1-xMnx)0.6Pt0.4, we also added the results for Fe1-xMnx, Fe1-xPtx, and Fe in the same figure (all have a thickness of 9 nm).
The anomalous Hall angle were varied by changing either the Mn or Pt composition (except for Fe).
Despite the variation in composition, the AHMR for all these Fe-based films indeed show a quadratic dependence on AH , as manifested in the dotted line, which is the fitting result for (Fe1-xMnx)0.6Pt0.4 obtained by using = 0.55 and s = 3.5 . As we will discuss shortly, similar range of values can also fit the thickness-dependence of AHMR as predicted by Eq. (1).
Thickness dependence of AHMR. Fig. 5a shows the experimentally observed thickness dependence of AHMR for (Fe0.71Mn0.29)0.6Pt0.4 with = 2 -15 nm. Instead of a monotonic decrease of AHMR with increasing d as predicted by the theoretical model, the experimental value increases sharply at small thickness, peaks at around d = 3 nm, and then decreases slowly as d increases. There are two possible reasons that cause the deviation from theoretical model at small thickness: one is the sharp increase of resistivity due to surface scattering and the other is the decrease of magnetization due to finite size effect. When the thickness of a thin film becomes smaller than or comparable to the electron mean free path, its resistivity scales with the thickness as ], here xx0 is the bulk resistivity, p is the specular reflectivity, 0 is the roughness, and lf is the electron mean free path 52 . For surface with finite roughness or small p, the resistivity will increase sharply when d < lf or 0 . Fig. 5b shows xx and Ms as a function of d. As expected, increases, whereas Ms decreases sharply at small thickness. It is interesting to note that Ms starts to decrease at a larger thickness than , understandably from the difference in length scale that governs the resistivity and magnetization of thin films. More discussion on the effect of thin film roughness can be found in Supplementary Note 9. In Fig. 3, we found that the relation AH ∝ s holds for most of the Mn composition range for FeMnPt, suggesting that the AHE is dominated by skew scattering 31 . For comparison, we plot, in Fig. 5c, AH / s as a function of for samples with different thicknesses. A nearly perfect linear relation is obtained when d > 3 nm. However, at d < 3 nm (see inset of Fig. 5c), a sublinear relation appears, suggesting gradual weakening of AHE in this region. This is understandable because, in this region, surface scattering dominates the electrical transport; but compared to bulk scattering, surface scattering may not be an efficient mechanism for AHE since it is mostly spin-independent. Surface effect was not taken into account when deriving Eq. (1); therefore, strictly speaking, it does not apply to the case when the film thickness becomes comparable to or smaller than the spin diffusion length which is usually larger than the mean free path. would also be of interest to probe the AHE-generated spin current directly using magneto-optical technique and correlate it with the MR data. We believe that the results described in this work demonstrate the importance of AHE as an alternate tool for studying spin-charge interconversion in magnetic materials and its potential in spintronic applications.

Methods
Sample preparation. All samples were deposited on SiO2(300 nm)/Si substrates using DC magnetron sputtering with a base and working pressure of 2×10 -8 Torr and 3×10 -3 Torr, respectively. Fe1-xMnx (or Fe1-xPtx) films were prepared by co-sputtering of Fe0.8Mn0.2 and Mn targets (or Fe and Pt targets). laser was employed to directly expose the substrates after coating the negative photoresist Microposit S1805. After exposure, the substrates were then soaked in developer MF319 to form the Hall bar pattern.
After deposition using sputtering, the photoresist was removed by the mixture of PG remover and acetone, and the metallic patterns are left on the substrates.
Characterization. Structural properties of the samples were characterized using a Rigaku X-ray diffraction (XRD) system with Cu Kα radiation. X-ray photoelectron spectroscopy (XPS) was performed on PHI Quantera II XPS Scanning Microprobe from Ulvac-PHI with a beam spot size of 50 μm. In addition, high resolution scanning transmission electron microscopy (STEM, a JEOL ARM200F) was employed to directly image the multilayer samples. Magnetic properties were characterized using a Quantum Design vibrating sample magnetometer (VSM) with the samples cut into a size of 4 mm × 3 mm. The resolution of the system is better than 6×10 -7 emu. The electrical measurements were also performed using the same Quantum Design system at a bias current of 100 μA.       in Supplementary Figures 2d -2f. According to this model, the temperature dependent magnetization of FM is given by where M(0) is the magnetization at T = 0 K, TC is the Curie temperature, s is the so-called shape parameter with a value in the range of 0 -2.5, and b is the critical exponent whose value is determined by the universality

Supplementary Note 2. Field dependent magnetoresistance (FDMR) measurement results
As a supplementary reference, FDMR results are shown in Supplementary Figures 3a -3l (9) with applied sweeping field in x-, y-and z-axis, respectively. The data are thus denoted as Hx, Hy, and Hz FDMR curves, respectively. As can be seen from Supplementary Figures 3a, 3d, 3g (Supplementary Figures 3b, 3e, 3h, 3k). On the other hand, when it is misaligned from z-axis towards y-axis, the magnetization is rotated in zy plane by the additional ycomponent, and a W shaped MR curve is observed (Supplementary Figures 3c, 3f, 3i, 3l). In view of these shapes and the magnetization positions, it can be inferred that x z y      with z  the resistivity when || m z. This relation is in agreement with the above Hx and Hy FDMR curves and ADMR results in Fig. 2 of the main text. To support the explanation, we performed macro-spin simulation for the Hz case following the approach described in our previous work 3 where θ and φ are the polar and azimuth angles of the magnetization, respectively; sin cos sin sin By using 3 2 10   hand, the fitting for Fe1-xMnx only serves as guide for eye due to the small number of data points. As mentioned above, the magnetic properties of Fe1-xMnx changes drastically when x approaches and exceeds 0.5. A more rigorous theoretical model is required to deal with AHE of such kind of materials with complex spin structures. Before ending this supplementary note, it is worth pointing out that, in the case of multilayers, MR(θzy) may also originate from SHE in the individual Pt layers 12 or interface scattering 13 , as we reported previously 3,4 . However, it is difficult to distinguish AHE and SHE contributions to MR(θzy) as both exhibit the same angular dependence.

Supplementary Note 5. Determination of anomalous Hall resistivity from Hall measurements
Hall resistivity in FM metals can be empirically written as 14 :  In addition to Fe based materials, we performed the same magnetoresistance measurements for control samples including Co(9), Py(9) and Ir0.2Mn0.8(9) thin films. Supplementary Figures 8a -8c  Previous reports suggested that such kind of behavior may come from the geometric size effect (GSE) [15][16][17] , which itself is still debatable as different mechanisms have been suggested such as electronic structure  Supplementary Figure 9. FDMR curves for control samples. a -c, Co (9); d -f, Py (9). The legends Hx, Hy and Hz denotes the FDMR curves obtained when the field is swept in x, y, and z-axis direction, respectively; and zx (or zy) in the parenthesis after Hz indicates the misalignment of Hz from z-axis towards x-axis (or y-axis).
The FDMR measurements were also performed for Co and Py to confirm the different relation of  The difference in MR prompted us to look into the AHE in these samples. In general, the Hall resistivity in these control is at least one order of magnitude smaller than that in Fe based systems with the same thickness. By using the method described in Supplementary Note 5, we separated the contribution of OHE   Figure 11. Simulated AMR contribution to MR(θzy) ratio due to field misalignment.
Before proceeding further, it is necessary to exclude other contributions besides AHE as the main cause for the MR(θzy). In this note, we first discuss the influence of field misalignment contribution.
During the zy plane ADMR measurement, the misalignment of either field or sample (which are relative to each other) can be represented by a small rotation around y-axis by and z-axis by . Assume that, at perfect alignment, the magnetization vector is given by: After the rotation around y-and z-axis, the magnetization vector is given by: 16 If the observed MR(θzy) is due to the misalignment of conventional AMR only, the angle dependent longitudinal resistivity should be given by: ∆ cos sin cos sin sin (7) where ∆ / is the AMR ratio, about 2 10 estimated from the ADMR results in Fig. 2  It should be noted that the misalignment angle above 5˚ is highly unlike in the present experimental setup. It is clearly that the size of the signal (about 10 -5 ) is always nearly two orders of the magnitude smaller than that observed in Fe based samples. There must be another mechanism that gives rise to the MR(θzy) signal, that is, the AHMR.
In addition, to exclude GSE related AMR as the origin of MR(θzy) in Fe based samples, temperature dependent ADMR measurements were performed on (Fe0.71Mn0.29)0.6Pt0.4(9), Fe(9) and Py (9)   reported and attributed to GSE in Co in the literature 17 . On the contrary, in Fe and Fe-based alloys, a totally opposite temperature dependence has been observed for MR(θzy) ratio, i.e., it decreases with the decrease of temperature. This suggest that MR(θzy) and MR(θzx) in these samples have a different origin.
As discussed in the main text, the AHMR ratio is given by . In both figures, is taken as 0.028, and the data in x-and y-axis are normalized to the minimum value in each axis. As can be seen, the increase of would lead to a decrease in AHMR ratio, whereas an opposite trend is obtained for . However, for a same increase by a factor of 4, the effect of on AHMR is about 10 times larger than that of . Therefore, the temperature dependence should be mainly determined by , which agrees with the general trend of experimental temperature-dependence of AHMR in the FeMnPt samples. In fact, has also been found to play an important role in determining the temperature dependence of SMR in W/CoFeB bilayers 23 . Although further systematic studies are required to quantitatively elucidate the temperature dependence of AHMR, which is out of the scope of this manuscript, from the aforementioned experimental results and analysis, one can rule out GSE related AMR as the origin of MR(θzy). Figure 13. Effect of , and on the temperature dependence of AHMR. a, Experimentally obtained temperature dependence of in (Fe0.71Mn0.29)0.6Pt0.4(9); b, Calculated dependence of AHMR ratio with = 0.2 -0.8, = 4.5 nm; c, Calculated dependence of AHMR ratio with = 0.55, = 2 -8 nm.  It is known that the percolated structure or significant surface roughness in very thin films can affect both the electrical and magnetic properties. In addition, any change in the surface condition after the sample was exposed to ambient may also affect its physical properties. As the films under investigation are polycrystalline in nature, it would be difficult to achieve layer-by-layer growth at atomic layer accuracy and therefore, the presence of a certain degree of roughness is unavoidable. We have previously investigated systematically the electrical properties of ultrathin metallic film 28 , including Al, Au, Cr, Cu, Ru, Ta, Co90Fe10, Ni81Fe19, and Ir20Mn80. Different materials indeed exhibit different level of roughness.

Supplementary Note 8. Derivation of anomalous Hall magnetoresistance (AHMR)
Except for Al, Au and Cu, the root-mean-square (RMS) roughness of remaining films at a thickness of 20 nm is generally below 0.2 nm. The resistivity of all these films show an upturn at small thickness, though the turning point is generally more than one order of magnitude larger than the roughness. This suggests that the thickness at which sharp upturn of resistivity appears is mainly determined by the electron mean free path, as we discussed in main text.
The averaged RMS roughness over 5 different areas with such a size is 0.26 nm and 0.33 nm for (Fe0.71Mn0.29)0.6Pt0.4 and Fe, respectively. As shown in Fig. 5b of the main text, the sharp upturn of resistivity in (Fe0.71Mn0.29)0.6Pt0.4 appears at about 3 nm, which is also about 10 times of the RMS roughness, in good agreement with previous studies. Therefore, the sharp upturn of resistivity kicks in when the thickness of the film becomes comparable to the electron mean free path rather than due to reaching the percolation threshold of forming discontinuous film. In fact, the resistivity values of (Fe0.71Mn0.29)0.6Pt0.4 and Fe with a thickness of 5 -20 nm are in the range of 3.9×10 3 -1.8×10 4 (Ω cm) -1 and 4.9×10 3 -2.3×10 4 (Ω cm) -1 , respectively. These values fall into the upper bound of bad metal and lower bound of good metal regime 14 . Therefore, we can say that both the (Fe0.71Mn0.29)0.6Pt0.4 and Fe films with a thickness of 5 -20 nm are continuous metallic films.
On the other hand, both surface and size-effect also affect magnetic properties of thin films, which typically would lead to decrease of saturation magnetization. There is no generic model to describe the thickness dependence of saturation magnetization in ultrathin films since both the surface and interface with substrate vary from sample to sample. As far as (Fe0.71Mn0.29)0.6Pt0.4 thin film is concerned, as shown in Fig. 5b of the main text, the magnetization began to decrease at a thickness of ~ 5 nm, which is larger than the thickness at which the resistivity shows a sharp upturn. This is understandable since they are governed by phenomena of different length scale. However, as explained in the main text, this does not affect the analysis and interpretation of the experimental data of films with d > 5 nm. For samples with d < 5 nm, the experimental data can be understood qualitatively if we take into account the thicknessdependent θAH obtained experimentally. However, we did not include the fitting results in Fig. 5a of the main text because thickness-dependence of is unknown both theoretically and experimentally. To substantiate the thickness dependence of AHMR, besides the results shown in Fig. 5 of the main text, we fabricated another batch of (Fe0.71Mn0.29)0.6Pt0.4 and extended the same thickness dependence study to Fe as well. Supplementary Figures 14a and 14b