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Absence of topological Hall effect in Fex{}_{x}Rh100−x{}_{100-x} epitaxial films:
revisiting their phase diagram

Xiaoyan Zhu Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    Hui Li Cavendish Laboratory, University of Cambridge, J J Thomson Avenue, Cambridge, CB3 0HE, UK    Jing Meng Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    Xinwei Feng Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    Zhixuan Zhen Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    Haoyu Lin Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    Bocheng Yu Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    Wenjuan Cheng Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    Dongmei Jiang Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    Yang Xu Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    Tian Shang Corresponding author: tshang@phy.ecnu.edu.cn tshang@phy.ecnu.edu.cn Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China Chongqing Key Laboratory of Precision Optics, Chongqing Institute of East China Normal University, Chongqing 401120, China    Qingfeng Zhan Corresponding author: qfzhan@phy.ecnu.edu.cn qfzhan@phy.ecnu.edu.cn Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China
Abstract

A series of Fex{}_{x}Rh100−x{}_{100-x} (30≤x≤5730\leq x\leq 57) films were epitaxially grown using magnetron sputtering, and were systematically studied by magnetization-, electrical resistivity-, and Hall resistivity measurements. After optimizing the growth conditions, phase-pure Fex{}_{x}Rh100−x{}_{100-x} films were obtained, and their magnetic phase diagram was revisited. The ferromagnetic (FM) to antiferromagnetic (AFM) transition is limited at narrow Fe-contents with 48≤x≤5448\leq x\leq 54 in the bulk Fex{}_{x}Rh100−x{}_{100-x} alloys. By contrast, the FM-AFM transition in the Fex{}_{x}Rh100−x{}_{100-x} films is extended to cover a much wider xx range between 33% and 53%, whose critical temperature slightly decreases as increasing the Fe-content. The resistivity jump and magnetization drop at the FM-AFM transition are much more significant in the Fex{}_{x}Rh100−x{}_{100-x} films with ∼\sim50% Fe-content than in the Fe-deficient films, the latter have a large amount of paramagnetic phase. The magnetoresistivity (MR) is rather weak and positive in the AFM state, while it becomes negative when the FM phase shows up, and a giant MR appears in the mixed FM- and AFM states. The Hall resistivity is dominated by the ordinary Hall effect in the AFM state, while in the mixed state or high-temperature FM state, the anomalous Hall effect takes over. The absence of topological Hall resistivity in Fex{}_{x}Rh100−x{}_{100-x} films with various Fe-contents implies that the previously observed topological Hall effect is most likely extrinsic. We propose that the anomalous Hall effect caused by the FM iron moments at the interfaces nicely explains the hump-like anomaly in the Hall resistivity. Our systematic investigations may offer valuable insights into the spintronics based on iron-rhodium alloys.

††preprint: Preprint: , 23:39

I INTRODUCTION

The CsCl-ordered equiatomic iron-rhodium (Fe-Rh) alloy undergoes a first-order magnetic phase transition from the high-temperature ferromagnetic (FM) state to the low-temperature antiferromagentic (AFM) state near room temperature [1, 2]. Such a transition leads to a significant drop in the magnetization and a jump in the electrical resistivity, which can be applied to the spintronic devices. The FM-AFM transition in Fe-Rh alloys can be easily tuned by external control parameters, such as chemical substitution [3], epitaxial strain [4, 5, 6], and magnetic or electric fields [7, 8, 9, 10]. Many exotic properties have been found in Fe-Rh alloys, which are closely related to their FM-AFM transition. The spin–orbit torque efficiency can be significantly tuned by varying the temperature across the FM-AFM transition in Fe-Rh-based heterostructures [11]. The large magnetocaloric effect can be controlled by ferroelectric domains in Fe-Rh film near the FM-AFM transition [12]. Since the FM-AFM transition presents near the room temperature, therefore, Fe-Rh alloys represent one of the ideal candidate materials for spintronic applications, such as memory resistor [13], heat-assisted magnetic recording [14], and magnetic refrigeration [12].

The bulk Fe-Rh alloys exhibit a rich phase diagram when varying the Fe- or Rh concentrations. We summarize the phase diagram of bulk Fe-Rh alloys in Fig. I(a). On the Rh-rich side, the γ\gamma-PM indicates the paramagnetic (PM) phase with a face-centered cubic (FCC) crystal structure, where both Rh- and Fe atoms occupy the same sites [see Fig. I(b)]. For the intermediate Fe concentration (<< 48%), the Fe-Rh alloys adopt the mixed α\alpha- and γ\gamma-phases. While the γ\gamma-phase remains PM, the α\alpha-phase becomes FM below certain temperatures [denoted as (α\alpha+γ\gamma)-(FM+PM) in Fig. I(a)]. When increasing the Fe-content above 48%, the Fe-Rh alloys show a pure α\alpha-phase with a body-centered cubic (BCC) crystal structure. In particular, the Fe-Rh alloys with 48-54% Fe-content undergo multiple magnetic transitions, from high-temperature PM state (marked as α\alpha-PM) to the FM state (α\alpha-FM), and then finally to the low-temperature AFM state (α\alpha-AFM). The FM-AFM transition temperature decreases as increasing the Fe-content. For these alloys, the Fe and Rh atoms occupy the corner- and center sites, respectively [see Fig. I(c)]. It is noted that the α\alpha-FM and α\alpha-AFM are also known as ordered α\alpha’-phase and α\alpha”-phase. On the Fe-rich side, both Rh- and Fe atoms occupy the same sites [see Fig. I(d)], and the Fe-Rh alloys behave similarly to pure Fe metal, exhibiting a FM ground state below the Curie temperature (∼\sim1000 K). In the α\alpha-AFM phase, the Fe-moments are aligned with a collinear G-type magnetic structure, exhibiting a typical magnetization value of ∼\sim3.1 μB\mu_{\mathrm{B}} [15]. However, there is no net moment on the Rh site. While in the α\alpha-FM phase, both Fe-moments (∼\sim3.2 μB\mu_{\mathrm{B}}) and Rh-moments (∼\sim0.9 μB\mu_{\mathrm{B}}) are aligned ferromagnetically along the <001>-direction [16].

In addition to the exotic properties related to the FM-AFM transition in the Fe-Rh alloys [17, 18, 19], the topological Hall effect (THE) has been observed recently in equiatomic Fe-Rh thin films [20]. In general, the THE is often considered as the hallmark of spin textures with a finite scalar spin chirality, e.g., magnetic skyrmions [21, 22]. The THE in Fe-Rh film is proposed to be attributed to the emergence of noncollinear spin texture arising from the competition among different exchange interactions in its AFM state. The interfacial inhomogeneity in the magnetic thin film could lead to an inhomogeneous anomalous Hall effect (AHE), whose signal resembles the THE [23, 24, 25, 26]. Considering that a large amount of the remaining FM phase persists in the AFM state of Fe-Rh film in the previous work [20], the origin of THE requires further investigation. In addition, while most of the thin-film studies focus on equiatomic Fe-Rh films (i.e., ∼\sim50% Fe-content), less is known for Fe-Rh films with different Fe-contents.

Here, we revisit the phase diagram of Fe-Rh epitaxial thin films by varying the Fe- or Rh-contents, and report a comprehensive study of their magnetic and transport properties by means of magnetization-, electrical resistivity-, and Hall resistivity measurements. Different from the bulk alloys, the Fe-Rh films show a pure α\alpha-phase in a wide Fe-concentration range (i.e., 33 to 53%). The absence of topological Hall resistivity in our high-quality epitaxial Fe-Rh films with different Fe-contents excludes its possible nontrivial origin. We propose that the anomalous Hall resistivity caused by the remaining FM moments could give rise to a THE-like signal in the Hall resistivity.

Refer to caption
Figure 1. : (a) Phase diagram of bulk Fe-Rh alloys and epitaxial Fe-Rh films. Data of bulk alloys were taken from Refs. 16, 27. The star symbols represent the FM-AFM magnetic transition temperatures (left axis) for Fe-Rh films, while the sphere symbols show the resistivity jump |\lvertΔ\Deltaρ\rho|\rvert (right axis) for these films against the Fe-content. Crystal structures for (b) γ\gamma-phase (FCC, FmFm-3mm, No. 225) and (c) α\alpha-phase (BCC, PmPm-3mm, No. 221) Fe-Rh. The crystal structure of disordered α\alpha-phase is shown in panel (d). Different from the ordered α\alpha-phase in (c), the Fe or Rh atoms occupy the same site in the disordered α\alpha-phase and γ\gamma-phase.

II EXPERIMENTAL DETAILS

A series of Fex{}_{x}Rh100−x{}_{100-x} (30≤x≤5730\leq x\leq 57) films with a thickness of ∼\sim50 nm were epitaxially grown on (001)-oriented MgO substrates by magnetron co-sputtering Fe and Rh targets in an ultrahigh vacuum chamber with a base pressure lower than 1×\times10−8{}^{-8} Torr. To remove surface contamination, MgO substrates were pre-annealed at 600 ∘{}^{\circ}C for 1 h in the vacuum. Afterwards, the substrates were heated up to 700 ∘{}^{\circ}C, where both Fe and Rh atoms were deposited under a 3 mTorr-Ar pressure. During the deposition, MgO substrates were continuously rotated to improve homogeneity. After the deposition, Fex{}_{x}Rh100−x{}_{100-x} films were annealed in situ at 750 ∘{}^{\circ}C for an extra hour to improve their crystallinity. Finally, a 3-nm-thick Ta cap layer was deposited at room temperature to avoid oxidation.

The crystal structure and the epitaxial nature of Fex{}_{x}Rh100−x{}_{100-x} films were characterized by Bruker D8 Discover high-resolution x-ray diffractometer (HRXRD). The thickness of films were determined by x-ray reflectivity (XRR). The measurements of electrical resistivity (ρ\rho), Hall resistivity (ρxy\rho_{\mathrm{xy}}), and magnetization were performed on a Quantum Design physical property measurement system (PPMS) and a magnetic property measurement system (MPMS), respectively. For the transport measurements, the Fex{}_{x}Rh100−x{}_{100-x} films were patterned into a Hall-bar geometry (central area: 0.2 mm ×\times 4 mm; electrodes: 0.4 mm ×\times 0.65 mm) by using a shadow mask during the growth. To avoid spurious resistivity contributions due to misaligned Hall probes, the longitudinal contribution to the Hall resistivity ρxy\rho_{\mathrm{xy}}, was removed by an anti-symmetrization procedure, i.e., ρxy(H)\rho_{\mathrm{xy}}(H) = [ρxy(H)\rho_{\mathrm{xy}}(H) – ρxy\rho_{\mathrm{xy}}(–HH)]/2. Similarly, in the case of longitudinal electrical resistivity ρ\rho measurements, the spurious transverse contribution was removed by a symmetrization procedure, i.e., ρ(H)\rho(H) = [ρ(H)\rho(H) + ρ\rho(–HH)]/2.

III RESULTS AND DISCUSSIONS

III.1 x-ray diffraction and lattice parameters

We estimated the composition of Fex{}_{x}Rh100−x{}_{100-x} films using the following model:

x=n⋅NA=(m/MA)⋅NA=[(β⋅v⋅t⋅S)/MA]⋅NA.x=n\cdot N_{A}=(m/M_{\mathrm{A}})\cdot N_{\mathrm{A}}=[(\beta\cdot v\cdot t\cdot S)/M_{\mathrm{A}}]\cdot N_{\mathrm{A}}. (1)

Here, nn, mm, NAN_{A}, and MAM_{A} are molar number, mass, Avogadro constant, and molar mass; vv and tt are deposition rate and time; β\beta and SS represent the density and surface area of the film, respectively. The deposition rate vv was controlled by adjusting the DC sputtering power of Fe and Rh targets, which was further calibrated by XRR measurements. Table III.1 lists the sputtering power of Fe- and Rh targets for different Fex{}_{x}Rh100−x{}_{100-x} films. For instance, to produce the Fe30{}_{30}Rh70{}_{70} film, the PFeP_{\mathrm{Fe}} and PRhP_{\mathrm{Rh}} were set to 20 and 15 W, respectively. We compared the magnetization- and electrical-resistivity results of the Fe49{}_{49}Rh51{}_{51} film prepared by co-sputtering method with the one grown from Fe50{}_{50}Rh50{}_{50} alloy target, both films show almost identical behaviors, suggesting that the above model gives correct Fe and Rh contents. Seven Fex{}_{x}Rh100−x{}_{100-x} films with xx ranging from 30 to 57 were deposited.

Table 1.:  Summary of the sputtering power of Fe (PFeP_{\mathrm{Fe}}) and Rh (PRhP_{\mathrm{Rh}}) targets for Fex{}_{x}Rh100−x{}_{100-x} thin-film growth and the estimated Fe-content for the obtained films. The deviation of Fe-content is about 2%. Except Fe30{}_{30}Rh70{}_{70} film, all other Fe-Rh films adopt a pure α\alpha-phase.
PFeP_{\mathrm{Fe}} (W) 20 25 30 35 40 45 35
PRhP_{\mathrm{Rh}} (W) 15 15 15 15 15 15 10
xx (Fe-content) 30 33 36 44 49 53 57
Refer to caption
Figure 2. : (a) Representative XRD patterns of Fex{}_{x}Rh100−x{}_{100-x} films for xx = 30, 33, 49, and 57. The insets show enlarged plots of (001)- and (002)-reflections of Fe33{}_{\mathrm{33}}Rh67{}_{\mathrm{67}} film. (b) φ\varphi-scan measurements for some selected Fex{}_{x}Rh100−x{}_{100-x} films. The intensity is plot on the logarithmic scale.

The HRXRD measurements were performed to check the crystal structure and the epitaxial nature of the deposited Fex{}_{x}Rh100−x{}_{100-x} films. Figure III.1(a) shows representative XRD patterns for Fex{}_{x}Rh100−x{}_{100-x} films with xx = 30, 33, 49, and 57. For xx = 30, the (002)-reflection of Fe3{}_{3}Rh7{}_{7}-phase is clear, which adopts a γ\gamma-phase [28] and is consistent with the bulk phase diagram in Fig. I. Since the γ\gamma-phase Fe30{}_{30}Rh70{}_{70} is paramagnetic, its magnetic- and electrical transport properties will not be discussed here. No sign of the α\alpha-phase can be identified in this film. When increasing the Fe-content up to 33%, the γ\gamma-phase disappears, in the meanwhile, α\alpha-phase starts to show up [see insets in Fig. III.1(a)]. For 33≤x≤5733\leq x\leq 57, all Fex{}_{x}Rh100−x{}_{100-x} films show a pure α\alpha-phase, exhibiting distinct (001)- and (002) reflections. This is obviously different from the bulk materials. In the bulk form, Fe-Rh alloys (with x << 48) show mixed γ\gamma- and α\alpha-phases. The absence of foreign phases or misorientation suggests the good quality of our deposited Fex{}_{x}Rh100−x{}_{100-x} films. It is noted that the α\alpha-phase Fex{}_{x}Rh100−x{}_{100-x} films were epitaxially grown on the MgO substrates with an in-plane 45∘{}^{\circ} rotation, i.e., FeRh[110](001)——MgO[100](001) [5], which was further checked by φ\varphi-scan measurements [see Fig. III.1(b)]. It is noted that, for xx = 33, the intensities of the XRD reflections are rather low due to the increased mismatch between the film and the substrate, its epitaxial nature cannot be verified by the φ\varphi-scan measurements. For xx = 33, the epitaxy is less good than the rest of the films, and it might be polycrystalline in nature but with preferred (00ll) orientation.

Refer to caption
Figure 3. : Out-of-plane lattice parameters for the Fex{}_{x}Rh100−x{}_{100-x} films and bulk counterparts as a function of the Fe-content. The star symbols represent the current work, while the other symbols stand for the previous studies, which were taken from Refs. 28, 29, 30. The dashed line is a guide to the eyes.

We estimated the out-of-plane lattice constant (i.e., cc-axis) for the α\alpha-phase Fex{}_{x}Rh100−x{}_{100-x} (33≤x≤5733\leq x\leq 57) films according to the XRD patterns. As shown in Fig. III.1, for xx = 49, the lattice parameter (2.989 Å) is almost identical to the value of Fe50{}_{50}Rh50{}_{50} film (2.988 Å) grown by using a Fe50{}_{50}Rh50{}_{50} alloy target [31], which further proves that the above model [see Eq. (1)] estimates the proper Fe- or Rh concentration. The obtained lattice parameter linearly decreases as increases the Fe-content xx (see star symbols). While in the previous studies, the lattice parameters are clearly more scattered (see square and triangle symbols). Such linear xx-dependent lattice parameters again confirm that our Fex{}_{x}Rh100−x{}_{100-x} films are very homogeneous and have a better quality.

III.2 Magnetic properties of Fex{}_{x}Rh100−x{}_{100-x} films

Refer to caption
Figure 4. : Temperature-dependent magnetization collected in a field of μ0H\mu_{0}H = 0.1 T (a) and zero-field electrical resistivity (b) for Fex{}_{x}Rh100−x{}_{100-x} (33≤x≤5733\leq x\leq 57) films. The inset shows the magnetization data for xx = 57 film.
Refer to caption
Figure 5. : Temperature dependence of the magnetization (up-panels) and the electrical resistivity (bottom-panels) for Fex{}_{x}Rh100−x{}_{100-x} (33≤x≤5733\leq x\leq 57) films. All the data were collected in a magnetic field of μ0H\mu_{0}H = 5 T during the cooling (solid lines) and heating processes (dashed lines). To better compare the results of different films, both the magnetization and electrical resistivity data are normalized to the values between 0 and 1.

The Fex{}_{x}Rh100−x{}_{100-x} (33≤x≤5733\leq x\leq 57) films were first characterized by temperature-dependent magnetization M(T)M(T) and electrical resistivity ρ(T)\rho(T). For the α\alpha-phase Fex{}_{x}Rh100−x{}_{100-x} (33≤x≤5333\leq x\leq 53) films, there is a clear anomaly around 380 K, which is attributed to the FM-AFM transition. Since the onset of this transition is above 400 K in zero-field condition in Fex{}_{x}Rh100−x{}_{100-x} films, the entire transition can not be detected up to 400 K. However, the magnetic field can efficiently tune such a FM-AFM transition, and thus, the full transition can be clearly seen in a field of 5 T. For xx = 57, the bulk sample undergoes a FM transition at very high temperature (∼\sim1000 K) (see Fig. I). As shown in the inset of Fig. III.2(a), the magnetization of Fe57{}_{57}Rh43{}_{43} film resembles the pure Fe film [32], and there is no magnetic transition below 400 K. The magnetization of Fe57{}_{57}Rh43{}_{43} film (∼\sim1200 emu/cc) is almost 10 times larger than the Fex{}_{x}Rh100−x{}_{100-x} (33≤x≤5333\leq x\leq 53) films in their AFM state (∼\sim100 emu/cc). It is noted that in all the Fex{}_{x}Rh100−x{}_{100-x} films, the upturn feature below 10 K is most likely attributed to the PM contribution of MgO substrate [33].

Figure III.2(b) presents the zero-field temperature-dependent electrical resistivity ρ(T)\rho(T) for Fex{}_{x}Rh100−x{}_{100-x} (33≤x≤5733\leq x\leq 57) films. All the films show a typical metallic behavior below 350 K, the electrical resistivity decreases as lowering the temperature. Similar to the magnetization results, the resistivity jump at FM-AFM transition is not completed up to 400 K for xx = 44, 49, 53. For xx = 33, though the resistivity anomaly is very weak, it is still can be observed (see Fig.5). While for xx = 57, there is no clear anomaly in the studied temperature range, consisting with its magnetization data [see inset in Fig. III.2(a)].

To better track the FM-AFM transition of Fex{}_{x}Rh100−x{}_{100-x} films, the M(T)M(T) and ρ(T)\rho(T) were also collected upon heating and cooling the temperature in a field of μ0\mu_{0}HH = 5 T. For 33≤x≤5333\leq x\leq 53, the M(T)M(T) exhibits a significant drop below 400 K upon cooling (see solid lines in the up panels in Fig. III.2), indicating that these Fex{}_{x}Rh100−x{}_{100-x} films undergo a magnetic phase transition from high-TT FM state to low-TT AFM state. Such a FM-AFM transition is clearly reflected also in ρ(T)\rho(T) data. As shown by solid lines in the bottom panels of Fig. III.2, in contrast to the M(T)M(T) data, the ρ(T)\rho(T) undergoes a significant jump near the FM-AFM transition. In the FM state, both the Fe- and Rh moments are aligned along the cc-axis [16], resulting in a low-resistivity state. While in the AFM state, the enhanced magnetic scattering leads to a high-resistivity state. When increasing the magnetic field, the Fe moments are forced to ferromagnetically align again, accompanied by a resistivity drop at the metamagnetic transition [34]. Upon heating, all the Fex{}_{x}Rh100−x{}_{100-x} films undergo an AFM-FM transition, reflected by a jump in the magnetization or a drop in the electrical resistivity (see dashed lines in Fig. III.2). While for xx = 57 [see Fig. III.2(f)], similar to the results in Fig. III.2, no trace of magnetic transition can be identified in a field of 5 T, consistent with its FM nature in the studied temperature range.

Refer to caption
Figure 6. : Temperature-dependent magnetization (a) and electrical resistivity (b) collected under various magnetic fields up to 5 T for Fe49{}_{49}Rh51{}_{51} film. Both the magnetization and electrical resistivity data are normalized to the values between 0 and 1. (c) The magnetic transition temperatures TtT_{\mathrm{t}} (i.e., TtmidT_{\mathrm{t}}^{\mathrm{mid}}) determined from magnetization- and electrical-resistivity measurements vs. Fe-content. Open- and solid symbols represent the TtT_{\mathrm{t}} determined from the measurements upon cooling- and heating processes, respectively. (d) The transition width Δ\DeltaTtT_{\mathrm{t}} (= TtonsetT_{\mathrm{t}}^{\mathrm{onset}} - TtoffsetT_{\mathrm{t}}^{\mathrm{offset}}) of the FM-AFM (or AFM-FM) transition vs. the Fe-content.
Refer to caption
Figure 7. : The magnetization at TtonsetT_{\mathrm{t}}^{\mathrm{onset}} and TtoffsetT_{\mathrm{t}}^{\mathrm{offset}}, and their difference Δ\DeltaMM [= M(Ttonset)M(T_{\mathrm{t}}^{\mathrm{onset}}) - M(Ttoffset)M(T_{\mathrm{t}}^{\mathrm{offset}})] for various Fex{}_{x}Rh100−x{}_{100-x} films (33≤x≤5333\leq x\leq 53). The data were obtained from the magnetization collected during the cooling process (see details in Fig. III.2). For the magnetization at TtonsetT_{\mathrm{t}}^{\mathrm{onset}} and TtoffsetT_{\mathrm{t}}^{\mathrm{offset}}, the background signal from the MgO substrates was subtracted according to MFeRh(Ttonset)M_{\mathrm{FeRh}}(T_{\mathrm{t}}^{\mathrm{onset}}) = Mraw(Ttonset)M_{\mathrm{raw}}(T_{\mathrm{t}}^{\mathrm{onset}}) - MMgO(Ttonset)M_{\mathrm{MgO}}(T_{\mathrm{t}}^{\mathrm{onset}}). The data extracted from Ref. 20 were also presented.

Figures III.2(a) and III.2(b) plot the normalized temperature-dependent magnetization and electrical resistivity collected under various magnetic fields up to 5 T between 250 and 400 K for Fe49{}_{49}Rh51{}_{51} film. When increasing the magnetic field, the FM-AFM (or AFM-FM) transition is suppressed to lower temperatures. As indicated by the arrows, the TtonsetT_{\mathrm{t}}^{\mathrm{onset}}, TtmidT_{\mathrm{t}}^{\mathrm{mid}}, and TtoffsetT_{\mathrm{t}}^{\mathrm{offset}} are defined as the onset, middle, and offset of the magnetic transition temperatures, respectively. We found that TtT_{\mathrm{t}} is suppressed at a rate of -7 K/T by the external magnetic field for Fe49{}_{49}Rh51{}_{51} film. The Tt(H)T_{\mathrm{t}}(H) exhibits a linear field dependence up to 10 T [34]. To obtain zero-field magnetic transition temperatures, TtT_{\mathrm{t}} values determined from 5 T-data (see Fig. III.2) were extrapolated to zero field using the above rate. The estimated zero-field TtT_{\mathrm{t}} values (here we choose TtmidT_{\mathrm{t}}^{\mathrm{mid}}) are summarized in Fig. III.2(c) as a function of Fe-content. The TtT_{\mathrm{t}} determined from magnetization- and electrical-resistivity measurements are highly consistent. When increasing the Fe-content, TtT_{\mathrm{t}} determined during the heating process slightly decreases from 400 K for xx = 33 to 385 K for xx = 53. The TtT_{\mathrm{t}} determined during the cooling process exhibits an almost identical trend, yielding a xx-independent transition width Δ\DeltaTtT_{t} [see Fig. III.2(d)]. Such Δ\DeltaTt(x)T_{t}(x) indicates that the first-order FM-AFM transition exists in the Fex{}_{x}Rh100−x{}_{100-x} (33≤x≤5333\leq x\leq 53) films with a much wider Fe-content than the bulk alloys. For the latter case, it is limited only at 48≤x≤5448\leq x\leq 54 (see details in Fig. I).

To quantitatively describe the FM-AFM transition in Fex{}_{x}Rh100−x{}_{100-x} (33≤x≤5333\leq x\leq 53) films, their magnetization at different magnetic states are summarized in Fig. III.2. The M(Ttonset)M(T_{\mathrm{t}}^{\mathrm{onset}}) represents the magnetization at TtonsetT_{\mathrm{t}}^{\mathrm{onset}} (i.e., FM state), while M(Ttoffset)M(T_{\mathrm{t}}^{\mathrm{offset}}) is the magnetization at TtoffsetT_{\mathrm{t}}^{\mathrm{offset}} (i.e., AFM state). Both M(Ttonset)M(T_{\mathrm{t}}^{\mathrm{onset}}) and M(Ttoffset)M(T_{\mathrm{t}}^{\mathrm{offset}}) reach a maximum value as the Fe-content increases up to 49%. According to the XRD results [see Fig. III.1(a)], all the Fex{}_{x}Rh100−x{}_{100-x} (33≤x≤5333\leq x\leq 53) films show a pure α\alpha-phase at room temperature, which is completely different from the bulk alloys (see Fig. I). Therefore, in the Fex{}_{x}Rh100−x{}_{100-x} films, the larger magnetization value indicates a larger FM phase concentration at T≥TtonsetT\geq T_{\mathrm{t}}^{\mathrm{onset}}, and vice versa. As can be clearly seen in Fig. III.2, the magnetization of xx = 49 and 53 films is significantly larger than that of x<49x<49, implying that most of the Fe moments stay PM in the latter cases. In the AFM state (i.e., T≤TtoffsetT\leq T_{\mathrm{t}}^{\mathrm{offset}}), the magnetization is mainly attributed to the pinned Fe moments at the Ta/Fex{}_{x}Rh100−x{}_{100-x} or Fex{}_{x}Rh100−x{}_{100-x}/MgO interfaces [35, 36]. As a consequence, the smaller magnetization value indicates a larger AFM phase concentration at T≤TtoffsetT\leq T_{\mathrm{t}}^{\mathrm{offset}}, and vice versa. We also summarized the magnetization drop Δ\DeltaMM, a measure of the FM-AFM transition, versus the Fe-content in Fig. III.2. Similar to the M(Ttonset)M(T_{\mathrm{t}}^{\mathrm{onset}}), the Δ\DeltaMM also reaches a maximum value at xx = 49, which is significantly larger than the rest of Fex{}_{x}Rh100−x{}_{100-x} films.

For the Fe-Rh-based spintronic applications, the Fex{}_{x}Rh100−x{}_{100-x} films with a large Δ\DeltaMM value are preferred [13, 14, 12], since it could also give rise to a pronounced jump in the electrical resistivity. The estimated resistivity jumps Δ\Deltaρ\rho [= ρ(Ttoffset)\rho(T_{\mathrm{t}}^{\mathrm{offset}}) - ρ(Ttonset)\rho(T_{\mathrm{t}}^{\mathrm{onset}})] [see details in Fig. III.2(b)] of Fex{}_{x}Rh100−x{}_{100-x} (33≤x≤5333\leq x\leq 53) films are summarized in the phase diagram (see Fig. I). Indeed, for xx = 49 and 53, the Δ\Deltaρ\rho values are significantly larger than that of x<49x<49. For instance, the Δ\Deltaρ\rho = 55 µΩ\mathrm{\Omega}cm for xx = 49, while it is less than 10 µΩ\mathrm{\Omega}cm for xx = 33. Our results demonstrate that the Fex{}_{x}Rh100−x{}_{100-x} films with Fe-content up to 53% exhibit magnetic and transport properties that are comparable to the ideal 49%-case. While for x≤44x\leq 44, the FM-AFM transition is less pronounced, leading to small Δ\DeltaMM and Δ\Deltaρ\rho values.

Refer to caption
Figure 8. : Magnetoresistivity up to 9 T for (a) x=44x=44, (b) x=49x=49, and (c) x=53x=53 collected at various temperatures covering both the FM- and AFM states. (d) Temperature-dependent 9-T MR values for the above three films. The magnetic field was applied along the out-of-plane direction. The MR was calculated following MR = [ρ(H)\rho(H) - ρ(0)\rho(0)]/ρ(0)\rho(0), where ρ(0)\rho(0) is the zero-field electrical resistivity. The shaded region highlights the coexistence of AFM- and FM phases, where a giant MR appears.

III.3 Magnetoresistivity and Hall resistivity

Refer to caption
Figure 9. : Field-dependent Hall resistivity ρxy(H)\rho_{\mathrm{xy}}(H) collected at various temperature below 400 K up to 9 T for Fex{}_{x}Rh100−x{}_{100-x} films with (a) x=44x=44, (b) xx = 49, and (c) xx = 53.

The field-dependent longitudinal- and transverse resistivity were measured in a wide temperature range for Fex{}_{x}Rh100−x{}_{100-x} films. Since the films with xx = 44, 49, and 53 exhibit a more pronounced magnetic phase transition, here, the field-dependent measurements were focused on these films. Figure III.2(a)-(c) plot the magnetoresistivity (MR) collected at various temperatures with the magnetic field up to 9 T. The MR values of Fex{}_{x}Rh100−x{}_{100-x} films at μ0H\mu_{0}H = 9 T are summarized in Fig. III.2(d). All three films exhibit similar temperature-dependent MR in the studied temperature range. In the AFM state (T≤250T\leq 250 K), the MR is positive, which is mainly attributed to the enhanced magnetic scattering by applying external magnetic field. Once the magnetic field destroys the AFM phase and fully polarizes the Fe moments, the MR exhibits a significant drop near the metamagnetic transition [34, 37]. As the temperature increases close to room temperature, where the FM phase shows up, the MR becomes negative, a typical feature for the ferromagnets. While in the mixed AFM- and FM states, a giant MR was observed, whose value reaching almost 50% at TT = 350 K for xx = 53. Such a giant MR is related to the field-induced metamagnetic transition in Fex{}_{x}Rh100−x{}_{100-x} films, as observed in their AFM state [37].

The Fe50{}_{50}Rh50{}_{50} film has been found to exhibit a topological Hall effect in a wide temperature range [20], which is often attributed to the topological spin textures in magnetic materials [38, 39]. Since the Fe-Rh alloys have a simple G-type AFM structure, the appearance of THE is rather puzzling. To further investigate the possible THE in Fex{}_{x}Rh100−x{}_{100-x} films, we also performed systematic Hall-resistivity measurements. As shown in Fig. III.2, the ρxy(H)\rho_{\mathrm{xy}}(H) were collected at various temperatures covering both the AFM- and FM states for Fex{}_{x}Rh100−x{}_{100-x} (xx = 44, 49, and 53) films. In the FM state, the ρxy(H)\rho_{\mathrm{xy}}(H) is dominated by the anomalous Hall effect (see 380- K curves in Fig. III.3). While in the AFM state (i.e., T<T< 300 K), in contrast to the previous work [20], all the ρxy(H)\rho_{\mathrm{xy}}(H) curves exhibit almost a linear field dependence at μ0H≤\mu_{0}H\leq 6 T, definitely excluding the possible THE in our Fex{}_{x}Rh100−x{}_{100-x} films. While for μ0H>\mu_{0}H> 6 T, the ρxy\rho_{\mathrm{xy}} becomes nonlinear, which is clearly reflected by the 300- K curves in Fig. III.3. Such nonlinear ρxy(H)\rho_{\mathrm{xy}}(H) is attributed to the field-induced metamagnetic transition in Fex{}_{x}Rh100−x{}_{100-x} films. The metamagnetic transition field is about 8.3 T at 300 K, which increases when decreasing temperature, reaching 9.8 T at TT = 291 K [34]. Therefore, the ρxy\rho_{\mathrm{xy}} is always dominated by the ordinary Hall effect (OHE) at μ0H≤\mu_{0}H\leq 9 T for T<T< 250 K. However, once the magnetic field is larger than the metamagnetic transition field, the ρxy(H)\rho_{\mathrm{xy}}(H) resembles the typical features of AHE in ferromagnets. Interestingly, in the mixed AFM- and FM states, our Fex{}_{x}Rh100−x{}_{100-x} films exhibit a clear hump-like anomaly in the ρxy(H)\rho_{\mathrm{xy}}(H). As shown in Fig. III.3, a clear hump can be observed at μ0H∼\mu_{0}H\sim 3 T at 350 K. Such a hump-like anomaly resembles the topological Hall resistivity reported in the previous work [20]. However, such an anomaly is absent in the AFM state, implying that its origin is very unlikely the noncollinear spin textures. On the contrary, this anomaly can be reproduced, assuming anomalous Hall resistivity with different origins existing in the Fex{}_{x}Rh100−x{}_{100-x} films (see details in the discussion section).

III.4 Discussion

First, we discuss the magnetic phase diagram of Fex{}_{x}Rh100−x{}_{100-x} films. After optimizing the growth conditions, we could produce phase-pure Fex{}_{x}Rh100−x{}_{100-x} films with a wide xx range (i.e., Fe-content). For the bulk case, the Fex{}_{x}Rh100−x{}_{100-x} alloys adopt the mixed α\alpha- and γ\gamma-phases for 33≤x≤4833\leq x\leq 48 [see details in Fig. I(a)]. While the γ\gamma-phase remains PM, the α\alpha-phase becomes FM below certain temperatures. For x>x> 48, the Fex{}_{x}Rh100−x{}_{100-x} alloys show a pure α\alpha-phase with the Curie temperatures between 600 and 1000 K. For some particular Fe concentrations, i.e., 48≤x≤5448\leq x\leq 54, the Fex{}_{x}Rh100−x{}_{100-x} alloys undergo multiple magnetic transitions, from PM state to the FM state, and then finally to the AFM state. Different from the bulk alloys, Fex{}_{x}Rh100−x{}_{100-x} films show significantly different structural and magnetic properties. For the Fe-deficient case, Fe30{}_{30}Rh70{}_{70} film adopts a γ\gamma-phase, and there is no magnetic transition below 400 K, implying its PM nature. For 33≤x≤5733\leq x\leq 57, all Fex{}_{x}Rh100−x{}_{100-x} films show a pure α\alpha-phase. For the bulk alloys, no additional magnetic transition has been found below the Curie temperature for 33≤x≤4833\leq x\leq 48. While in the case of films, there is a distinct FM-AFM transition for 33≤x≤5333\leq x\leq 53, whose critical temperature TtT_{\mathrm{t}} determined during the heating process slightly decreases from 400 K for xx = 33 to 385 K for xx = 53 [marked as α\alpha-(AFM+PM) in Fig. I(a)]. In the case of Fe-rich films (i.e., xx ≥\geq 57 ), though they show pure α\alpha-phase, the FM-AFM transition is absent, and all the films host a FM ground state. It is noted that the FM-AFM transition exists in the Fex{}_{x}Rh100−x{}_{100-x} films with a wide Fe-content, however, there is a large portion of remaining PM phase for x≤44x\leq 44, which is reflected by a reduced resistivity jump Δρ\Delta\rho and a magnetization drop Δ\DeltaMM (see details in Fig. I and Fig. III.2). In addition, in all the Fex{}_{x}Rh100−x{}_{100-x} films, the Fe moments pinned at the interfaces also contribute to the magnetization in the AFM state and could lead to a hump-like anomaly in the Hall resistivity. The absence of AHE in the AFM state proves that our Fex{}_{x}Rh100−x{}_{100-x} films have negligible remaining FM contribution (see Fig. III.3).

Refer to caption
Figure 10. : Schematic plots of different contributions to the Hall resistivity for the Fex{}_{x}Rh100−x{}_{100-x} films in the mixed AFM- and FM states (a) or in the AFM state (b). In the mixed FM- and AFM states, ρxySUM\rho_{\mathrm{xy}}^{\mathrm{SUM}} = ρxyO\rho_{\mathrm{xy}}^{\mathrm{O}} + ρxypin\rho_{\mathrm{xy}}^{\mathrm{pin}} + ρxyFM\rho_{\mathrm{xy}}^{\mathrm{FM}} +ρxyAFM\rho_{\mathrm{xy}}^{\mathrm{AFM}}, while in the AFM state, ρxySUM\rho_{\mathrm{xy}}^{\mathrm{SUM}} = ρxyO\rho_{\mathrm{xy}}^{\mathrm{O}} + ρxypin\rho_{\mathrm{xy}}^{\mathrm{pin}} + ρxyAFM\rho_{\mathrm{xy}}^{\mathrm{AFM}}. Here, ρxypin\rho_{\mathrm{xy}}^{\mathrm{pin}}, ρxyAFM\rho_{\mathrm{xy}}^{\mathrm{AFM}}, and ρxyFM\rho_{\mathrm{xy}}^{\mathrm{FM}} all denote the anomalous-Hall resistivity.

Now we discuss the possible THE in Fex{}_{x}Rh100−x{}_{100-x} films. Based on the experimental observations in Fig. III.3, we show the schematic plots in Fig. III.4 to discuss the Hall resistivity in Fex{}_{x}Rh100−x{}_{100-x} films. The linear ρxy(H)\rho_{\mathrm{xy}}(H) is caused by the OHE (marked as ρxyO\rho_{\mathrm{xy}}^{\mathrm{O}}), whose negative slope suggests that the electron carriers are dominant in the Fex{}_{x}Rh100−x{}_{100-x} films. The ρxypin\rho_{\mathrm{xy}}^{\mathrm{pin}}, ρxyAFM\rho_{\mathrm{xy}}^{\mathrm{AFM}}, and ρxyFM\rho_{\mathrm{xy}}^{\mathrm{FM}} all denote the anomalous Hall resistivity, which are attributed to the ferromagnetically pinned Fe moments at the interfaces, AFM magnetization, and FM magnetization, respectively. In general, both ρxypin\rho_{\mathrm{xy}}^{\mathrm{pin}} and ρxyFM\rho_{\mathrm{xy}}^{\mathrm{FM}} are proportional to the magnetization (i.e., ρxypin\rho_{\mathrm{xy}}^{\mathrm{pin}} ∝\propto MpinM^{\mathrm{pin}}, ρxyFM\rho_{\mathrm{xy}}^{\mathrm{FM}} ∝\propto MFMM^{\mathrm{FM}}), typical for ferromagnets [40]. Here, MpinM^{\mathrm{pin}} and MFMM^{\mathrm{FM}} are the magnetization attributed to the pinned Fe moments at the interfaces and the FM regions of the film, both of which saturate when increasing the magnetic field up to 1 T. While the ρxyAFM\rho_{\mathrm{xy}}^{\mathrm{AFM}} is proportional to ρ2\rho^{2}MM, ρ\rhoMM, or their combinations, depending on the intrinsic or extrinsic mechanism [41]. Here we use ρ\rhoMM to produce ρxyAFM\rho_{\mathrm{xy}}^{\mathrm{AFM}}, while the ρ2\rho^{2}MM leads to similar behaviors. We assume that the AFM magnetization MAFMM^{\mathrm{AFM}} is linear in the low-field region but undergoes a metamagnetic transition at higher magnetic field, whose critical field increases as lowering the temperature. For example, the metamagnetic transition field is close to 5 T near room temperature but increases to 7.5 T at 290 K [34]. As shown in Fig. III.4(a), in the mixed AFM- and FM states, the AHE due to FM and AFM magnetization (i.e., ρxyFM\rho_{\mathrm{xy}}^{\mathrm{FM}} and ρxyAFM\rho_{\mathrm{xy}}^{\mathrm{AFM}}) is dominant, and as a consequence, the total Hall resistivity ρxySUM\rho_{\mathrm{xy}}^{\mathrm{SUM}} shows a step-like feature, typical for the magnets that undergo a metamagentic transition. The ρxySUM\rho_{\mathrm{xy}}^{\mathrm{SUM}} qualitatively agrees very well with ρxy(H)\rho_{\mathrm{xy}}(H) collected at 350 K. In the AFM state, as shown in Fig. III.4(b), since the OHE is dominant, the observed ρxy(H)\rho_{\mathrm{xy}}(H) is almost linear in field. While the ρxypin\rho_{\mathrm{xy}}^{\mathrm{pin}} could cause a hump-like anomaly in the ρxySUM\rho_{\mathrm{xy}}^{\mathrm{SUM}}, resembling the observed topological Hall resistivity in Ref. [20]. However, such an anomaly is clearly absent in our Fex{}_{x}Rh100−x{}_{100-x} films with different Fe-contents (see Fig. III.3). Since the ρxypin\rho_{\mathrm{xy}}^{\mathrm{pin}} is attributed to the FM iron moments pinned at the interfaces, such a hump-like anomaly in the ρxy(H)\rho_{\mathrm{xy}}(H) should strongly depend on the thin-film quality. We summarized the magnetization values from Ref. [20] in Fig. III.2 to compare with our films. Though the remaining magnetization M(Ttoffset)M(T_{\mathrm{t}}^{\mathrm{offset}}) is comparable to our Fe49{}_{49}Rh51{}_{51} film, the FM-state magnetization M(Ttonset)M(T_{\mathrm{t}}^{\mathrm{onset}}) is two times smaller than our film. As a consequence, the ρxy(H)\rho_{\mathrm{xy}}(H) is significantly affected by the ρxypin\rho_{\mathrm{xy}}^{\mathrm{pin}} in the previous work. In addition, since the ρxypin\rho_{\mathrm{xy}}^{\mathrm{pin}} has opposite sign against the ρxyO\rho_{\mathrm{xy}}^{\mathrm{O}}, the large contribution of ρxypin\rho_{\mathrm{xy}}^{\mathrm{pin}} might cause the sign change in the ρxy(H)\rho_{\mathrm{xy}}(H) when cooling the film down to lower temperatures. Indeed, such a sign change was observed in the previous work, the slope of ρxy(H)\rho_{\mathrm{xy}}(H) becomes positive below 80 K [20]. While in our Fex{}_{x}Rh100−x{}_{100-x} (xx = 44, 49, and 53) films, the ρxy(H)\rho_{\mathrm{xy}}(H) is always negative in the AFM state, which again proves that the remaining FM magnetization at the interfaces has little effect in our Fex{}_{x}Rh100−x{}_{100-x} films. To conclude, the observed THE in Fex{}_{x}Rh100−x{}_{100-x} films is most likely an extrinsic effect. The other techniques, such as resonant x-ray scattering or Lorentz transmission electron microscopy, are highly desirable to search for possible topological magnetic phases in Fex{}_{x}Rh100−x{}_{100-x} family.

IV CONCLUSION

To summarize, we grew a series of epitaxial Fex{}_{x}Rh100−x{}_{100-x} (30≤x≤5730\leq x\leq 57) films on MgO substrates. By systematic x-ray diffraction-, magnetization-, and electrical resistivity measurements, we established the structural and magnetic phase diagram of Fex{}_{x}Rh100−x{}_{100-x} films. For xx ≤\leq 30, Fex{}_{x}Rh100−x{}_{100-x} films are PM and adopt a γ\gamma-phase. For x≥33x\geq 33, all films show a pure α\alpha-phase. In the bulk Fex{}_{x}Rh100−x{}_{100-x} alloys, the FM-AFM transition is limited only at 48≤x≤5448\leq x\leq 54. While the FM-AFM transition persists in the Fex{}_{x}Rh100−x{}_{100-x} films with 33≤x≤5333\leq x\leq 53, and the transition temperature slightly decreases from 400 K for xx = 33 to 385 K for xx = 53. As further increases the Fe-content (i.e., x>53x>53), the FM-AFM transition no longer exists, and Fex{}_{x}Rh100−x{}_{100-x} films are FM in the studied temperature range. The resistivity jump and magnetization drop at the FM-AFM transition are much more pronounced in the Fex{}_{x}Rh100−x{}_{100-x} films with ∼\sim50% Fe-content than in the Fe-deficient films, the latter have a large amount of PM phase. The magnetoresistivity is positive and weak in the AFM state, while it becomes negative when the FM phase shows up, and a giant MR appears in the mixed AFM- and FM states. The Hall resistivity measurements reveal trivial behaviors in the Fex{}_{x}Rh100−x{}_{100-x} films, which is dominated by the OHE in the AFM state and by the AHE in the mixed- or FM state, respectively. Our results demonstrate that the previously observed topological Hall resistivity is absent in our Fex{}_{x}Rh100−x{}_{100-x} (xx = 44, 49, and 53) films. We proposed that the AHE caused by the FM iron moments at the interfaces could explain the hump-like anomaly in the Hall resistivity. To conclude, the observed THE in Fex{}_{x}Rh100−x{}_{100-x} films can be explained by extrinsic mechanisms rather than the presence of noncollinear spin textures.

Acknowledgements.
The authors thank G. T. Lin and J. Ma for their assistance during the transport measurements. This work was supported by the Natural Science Foundation of Shanghai (Grants No. 21ZR1420500 and 21JC1402300), Natural Science Foundation of Chongqing (Grant No. 2022NSCQ-MSX1468), the National Natural Science Foundation of China (Grants No. 12174103 and 12374105). Y.X. acknowledges support from the Shanghai Pujiang Program (Grant No. 21PJ1403100).

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