GaAs(1 1 0) films

Journal Pre-proofs Strong current-direction dependence of anisotropic magnetoresistance in single crystalline Fe/GaAs(110) films F.L. Zeng, C. Zhou, M.W. Jia, D. Shi, Y. Huo, W. Zhang, Y.Z. Wu PII: DOI: Reference:

S0304-8853(19)33005-7 https://doi.org/10.1016/j.jmmm.2019.166204 MAGMA 166204

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Journal of Magnetism and Magnetic Materials

Received Date: Revised Date: Accepted Date:

28 August 2019 24 November 2019 25 November 2019

Please cite this article as: F.L. Zeng, C. Zhou, M.W. Jia, D. Shi, Y. Huo, W. Zhang, Y.Z. Wu, Strong currentdirection dependence of anisotropic magnetoresistance in single crystalline Fe/GaAs(110) films, Journal of Magnetism and Magnetic Materials (2019), doi: https://doi.org/10.1016/j.jmmm.2019.166204

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© 2019 Published by Elsevier B.V.

Strong current-direction dependence of anisotropic magnetoresistance in single crystalline Fe/GaAs(110) films F. L. Zeng1, C. Zhou1, M. W. Jia1, D. Shi1, Y. Huo1, W. Zhang2 and Y. Z. Wu1,3* 1 Department of Physics, State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China 2 Department of Physics, Oakland University, Rochester MI 48309, USA 3 Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China

Abstract The longitudinal and transverse resistivities of single crystalline Fe(110) film are both experimentally studied as functions of the magnetization orientation and the current orientation with respect to the crystalline axes. Unusual dependences and symmetries are revealed but cannot be described well by established conventional models. Furthermore, the anisotropic magnetoresistance ratios differ by more than one order of magnitude for currents along different crystalline directions. Analytical expressions for the resistivities are derived by using a phenomenological model based on a series expansion of the resistivity tensor with respect to the direction cosines of the magnetization up to the fourth order. The experimental data can be fitted well by these expressions, and the resistivity coefficients obtained from the fitting are consistent with the symmetries of the current-direction dependence measurements. The resistivity coefficients increase with temperature due to electron-phonon scattering. Our studies suggest that the spin-dependent transport properties could exhibit strong magnetocrystalline effect, which paves a new way for designing future spintronics devices by taking advantage of the material crystallinity.

Keywords: Anisotropic magnetoresistance, Planar Hall effect, Fe

1

1. INTRODUCTION Magnetoresistance (MR), the effect of magnetism on charge transport, brings about many novel phenomena and fascinating applications in spintronics [1]. The giant magnetoresistance (GMR) [2,3] and the tunneling magnetoresistance (TMR) [4-7], which exhibit large magnitudes of MR, have been extensively explored because of intriguing physics as well as their current and future applications in industries. The conventional anisotropic magnetoresistance (AMR) [8-13] along with many newly discovered related MR effects such as spin-Hall MR [14,15], Rashba MR [16], spinorbit MR [17,18], Hanle MR [19], and unidirectional MR [20,21] are also essential to the spintronics technology. The interplay between these effects in single or multilayered structures offers new physics and potential applications in spin-dependent transport [22,23]. At the same time, new challenges arise in the accurate understanding of these complex MR effects. Many MR effects are related to spin-orbit coupling (SOC), and therefore they strongly depend on the electronic structure of the materials. Thus, it is crucial to explore the relationship between MR effects and the electronic structure in the materials. In this regard, crystalline systems offer an ideal platform for elucidating the physical origins of various MR effects, as well as providing tailored MR properties for the relevant applications. Out of the many MR effects, AMR is one of the most fundamental magnetotransport properties in ferromagnetic (FM) materials, which describes the dependence of the longitudinal resistivity 𝜌𝑥𝑥 on the magnetization orientation (𝐌) relative to the current direction (𝐉) [9]. The mechanism of AMR is usually attributed to the s-d scattering influenced by the SOC [9-13]. Phenomenologically, the AMR in polycrystalline FM films can be expressed as 𝜌𝑥𝑥 = 𝜌⊥ + (𝜌∥ − 𝜌⊥ )cos2 𝜑𝑀 , where 𝜑M is the angle between 𝐌 and 𝐉 , and 𝜌∥ and 𝜌⊥ are the resistivity for 𝐌||𝐉 (𝜑𝑀 = 0°)

and

𝐌 ⊥ 𝐉 (𝜑𝑀 = 90°) , respectively. Therefore, AMR in

polycrystalline materials has a clear two-fold symmetry with respect to the magnetization orientation. Moreover, the transverse resistivity 𝜌𝑥𝑦 also depends on the magnetization orientation, which is usually called the planar Hall effect (PHE). The 2

PHE in polycrystalline FM film can be expressed as 𝜌𝑥𝑦 = (𝜌∥ − 𝜌⊥ )sin𝜑𝑀 cos𝜑𝑀 , so the PHE also has a two-fold symmetry with the same 𝜑𝑀 -dependent amplitude as its AMR counterpart. Unlike polycrystalline FM materials, single-crystalline FM materials have much more complex transport properties due to the crystallographic orientations, and therefore are better systems for exploring the intrinsic mechanism of AMR. In singlecrystalline FM systems, the AMR is related not only to the direction of the magnetization, but also to the orientation of the current with respect to the crystal axes [8,9]. As a result, the AMR in single crystalline films deviates from the regular cos2 𝜑𝑀 dependence, particularly an additional four-fold symmetry could be observed in some FM systems [8,24-29]. The additional terms emerge as a result of the SOC, which reflects the effect of the crystalline axes. The AMR effect in single-crystalline FM systems has been theoretically proposed [8,9] and experimentally studied in manganite [28], diluted magnetic semiconductor [30,31], magnetite Fe3O4 [24-27,32], and transition metals such as Co [33] or Ni [29] films. In the past, most studies on the current-direction dependent AMR effect were performed on the (001) film of a cubic bulk FM material, thus the AMR effect as a function of magnetization direction angle shows a clear in-plane four-fold symmetry due to the lattice structure [8,24-29]. In a recent work [34], the current-direction dependent AMR effect in a Fe/GaAs(001) system displayed a symmetry transition from two-fold to four-fold upon increasing Fe thickness, which was attributed to the emerging interfacial spin-orbit field with the 𝐶2𝑣 symmetry at the Fe/GaAs(001) interface. However, it should be noted that the Fe(110) system also has a 𝐶2𝑣 crystalline symmetry, but arising from the bulk. It is still not clear how such bulk 𝐶2𝑣 symmetry influences the AMR properties, despite a few earlier studies showing the deviation of AMR properties in Fe(110) films away from the conventional cos2 𝜑𝑀 relation [35,36]. In addition, the PHE is expected to strongly depend on the current orientation as well, but has not been experimentally explored. Here, we systematically investigated the current-direction dependence of both AMR and PHE in single crystalline Fe films grown on GaAs(110) substrates. Both 3

longitudinal and transverse AMR curves show clear deviations from the conventional cos2 𝜑𝑀 and

sin𝜑𝑀 cos𝜑𝑀

dependence,

respectively.

The

current-direction

dependence contains both two-fold and four-fold symmetries for AMR, however, only four-fold symmetry can be observed for PHE. We further show that the currentdirection dependence of both AMR and PHE can be well explained by the phenomenological model with a series expansion of the resistivity tensor with respect to the direction cosines of the magnetization. The resistivity coefficients are found to increase with the temperature, which can be attributed to the increase of electronphonon scattering. Moreover, the measured AMR ratios could show a difference more than one order of magnitude for different current directions. Such a strongly currentdirection dependent AMR effect provides a plausible way to tailor on-demand MR properties in single crystalline FM materials.

2. EXPERIMENT Fe/GaAs(110) films are prepared by molecular beam epitaxy in an ultrahigh vacuum (UHV) chamber with a base pressure of 2 × 10−10 Torr [37-39]. To perform transport measurement, un-doped GaAs(110) substrates with high resistivity 3 × 1014 μΩ ∙ cm are used. In the UHV chamber, the GaAs(110) substrates are cleaned by bombardment with 1 keV Ar+ ions for 1 hour, and the sample is rotated during the Ar+ bombardment. Then the GaAs(110) substrates are annealed at 600 °C for 45 minutes before thin film deposition. The smooth single crystalline GaAs(110) surface could be proven by the in-situ reflective high energy electron diffraction (RHEED) pattern, as shown in Fig. 1(a). Then, a 10-nm-thick Fe layer is epitaxially deposited at room temperature with a typical growth rate of 0.3 nm/min. The RHEED patterns in Fig. 1(a) also show the high quality epitaxial growth of Fe film with the epitaxial relationship Fe(110)[001]||GaAs(110)[001] [40,41]. Before being taken out from the UHV chamber, the sample is covered with a 6 nm MgO capping layer to prevent oxidization. The Fe (110) films are patterned into standard Hall bars by photolithography and 4

lift-off techniques for electrical transport measurements. Both the Ar+ ion-milling and contact deposition are performed in sputtering system. In order to study the currentdirection dependence of AMR properties, a current-direction dependent Hall bar structure is designed and patterned as shown in Fig. 1(d). All the Hall bars have the identical width of 100 μm and length of 300 μm, but orient along the different crystalline directions. The contacts were made of a 10-nm-thick Cr layer covered by a 30-nm-thick Au film. Thus, the current-direction dependent transport measurement could be performed on these samples under the same preparation condition, with 𝛼𝐽 denoting the angle between the current and the Fe[1̅10] crystalline axis. The AMR measurements are conducted in a Dewar system cooled by liquid nitrogen, and the magnetic field could be applied along an arbitrary in-plane direction by a vector magnet. The longitudinal and transverse resistances are measured by the standard lock-in technique. A Keithley 6221 current source provides an alternating current (AC) and SR830 lock-in amplifiers are used to detect the longitudinal and transverse voltages simultaneously.

3. RESULTS AND DISCUSSION The magnetic properties of the Fe film are firstly characterized by the longitudinal magneto-optic Kerr effect (MOKE) at room temperature. Fig. 1(b) displays the typical magnetic hysteresis loops of the Fe(110) film with the field along Fe[1̅10], [001] and [1̅11] directions. The easy axis of the system is clearly identified along the Fe[001] direction by the square-shape loop. It should be noted that the saturation Kerr signals are different in the hysteresis loops because of the crystalline effect in the system with the 𝐶2𝑣 symmetry [39]. Moreover, the magnetic anisotropy in the Fe film could be quantified by MOKE with a rotating field (ROT-MOKE) [37,42]. Fig. 1(c) shows the measured magnetic torque 𝑙(𝜙𝑀 ) as a function of magnetization angle 𝜙𝑀 , which is defined as the angle between the magnetization and Fe[1̅10] direction. The 𝑙(𝜙𝑀 ) curve indicates that the Fe(110) film contains a dominating uniaxial magnetic anisotropy 𝐾𝑈 and a weak four-fold anisotropy 𝐾4 . Thus, the anisotropy fields 𝐻𝑈 = 5

2𝐾𝑈 /𝑀𝐹𝑒 and 𝐻4 = 2𝐾4 /𝑀𝐹𝑒 can be obtained by fitting the equation 𝑙(𝜙𝑀 ) = 𝐻𝑈 sin2𝜙𝑀 /2 − 𝐻4 sin4𝜙𝑀 /4 [42]. The fitted 𝐻𝑈 is −601 ± 3 Oe with easy axis along the ⟨001⟩ direction, and the fitted 𝐻4 is −267 ± 7 Oe with easy axis along the ⟨1̅10⟩ or ⟨001⟩ direction. After patterning the Fe(110) film into the Hall bar structures shown in Fig. 1(d), we perform systematic AMR measurements with different current directions. Figs. 2(a) and 2(b) show the longitudinal and transverse resistivities respectively for current along Fe[1̅10] direction by sweeping the in-plane magnetic field with different 𝜑𝐻 . The longitudinal resistivity 𝜌𝑥𝑥 shows the largest difference between 0° and 90° , and the transverse resistivity 𝜌𝑥𝑦 shows the largest difference between 45° and 135°. The magnetoresistance measurement in Fig. 2 indicates that the Fe(110) sample can be saturated at 2 kOe even when the field is applied along the hard axis. Above the saturation field, the resistivity decreases slightly with the field, due to the suppression of the electron-magnon scattering by the external magnetic field [43]. To characterize the current-direction AMR, we systematically measure longitudinal resistivity 𝜌𝑥𝑥 and transverse resistivity 𝜌𝑥𝑦 as a function of magneticfield angle 𝜑𝐻 for all the Hall bars, with different 𝛼𝐽 . The strength of the applied rotating field is 2.5 kOe, which ensures the single-domain in the sample while rotating the magnetization in the film plane. The magnetization angle 𝜑𝑀 can be calculated from the given 𝜑𝐻 based on the anisotropic field 𝐻𝑈 and 𝐻4 , under the given 𝐻 and 𝜑𝐻 [37,42]. This transformation between 𝜑𝐻 and 𝜑𝑀 is crucial to quantitatively determine the current-direction AMR effect. Figs. 2(c) and 2(d) show the typical 𝜌𝑥𝑥 and 𝜌𝑥𝑦 curves as a function of 𝜑𝑀 measured at 300 K. It is clear that both 𝜌𝑥𝑥 (𝜑𝑀 ) and 𝜌𝑥𝑦 (𝜑𝑀 ) show a strong current-direction dependence. The 𝜌𝑥𝑥 curves show the symmetrical behavior only for 𝛼𝐽 = 0° , 90° and 180° , but contain clear asymmetrical 𝜑𝑀 -dependence for other 𝛼𝐽 values with the current direction away from the crystal axis. As indicated by the short vertical dashed lines, the maximums of 𝜌𝑥𝑥 (𝜑𝑀 ) curves clearly deviate from 180 angle. Therefore, those curves cannot be described by the conventional AMR relation of cos 2 𝜑𝑀 . Moreover, the 𝜌𝑥𝑦 (𝜑𝑀 ) 6

curves also strongly deviate from the PHE relation of sin𝜑𝑀 cos𝜑𝑀 , because the maximums of 𝜌𝑥𝑦 curves, indicated by the short vertical dashed lines in Fig. 2(d), deviate from 45°. Figs. 3(a) and 3(b) display the typical 𝛼𝐽 -dependence of 𝛥𝜌𝑥𝑥 and 𝛥𝜌𝑥𝑦 𝑚𝑎𝑥 respectively measured at different temperatures with 𝛥𝜌𝑥𝑥(𝑥𝑦) defined as 𝜌𝑥𝑥(𝑥𝑦) − 𝑚𝑖𝑛 𝜌𝑥𝑥(𝑥𝑦) .

𝛥𝜌𝑥𝑥 shows a mixture of two- and four-fold symmetry with the minimum

value at 𝛼𝐽 = 90°, and the maximum value at 𝛼𝐽 ≈ 35° and 145°, which is close to the ⟨111⟩ directions. In contrast, 𝛥𝜌𝑥𝑦 shows a clear four-fold symmetry with respect to 𝛼𝐽 . Although the 𝛼𝐽 -dependent 𝛥𝜌𝑥𝑥 and 𝛥𝜌𝑥𝑦 curves are independent of temperature, the amplitudes of both 𝛥𝜌𝑥𝑥 and 𝛥𝜌𝑥𝑦 curves decrease monotonously with decreasing temperature. It should be noted that 𝛥𝜌𝑥𝑥 is not equal to 𝛥𝜌𝑥𝑦 for all the single crystalline devices, which is significantly different with the conventional AMR behavior in the polycrystalline systems. Besides the current-direction effect of the AMR ratio, the average resistivity 𝜌𝑥𝑥 also strongly depends on the current direction, as shown in Fig. 3(c). The 𝜌𝑥𝑥 (𝛼𝐽 ) curve exhibits a clear two-fold symmetry with the maximum for 𝐉||⟨1̅10⟩ and the minimum for 𝐉||⟨001⟩. The 𝜌𝑥𝑥 (𝛼𝐽 ) curves can be fitted by a cos2𝛼𝐽 function, and the difference of the resistivities for 𝛼𝐽 = 0° and 𝛼𝐽 = 90° can be 20% at 300 K. This current-direction dependent resistivity was also observed in ultra-thin Fe/GaAs(001) films [34], which was attributed to the interface 𝐶2𝑣 orbit symmetry at the Fe/GaAs(001) interface. On the other hand, the Fe/GaAs(110) system contains the bulk 𝐶2𝑣 orbit symmetry, which can give rise to the observed two-fold symmetric resistivity. Fig. 3(d) displays the calculated AMR ratio 𝛥𝜌𝑥𝑥 /𝜌𝑥𝑥 , which shows a similar 𝛼𝐽 dependence as 𝛥𝜌𝑥𝑥 (𝛼𝐽 ) . Since both 𝛥𝜌𝑥𝑥 and 𝜌𝑥𝑥 have similar temperature dependences, the AMR ratio shows little temperature-dependence. It should be noted that the AMR ratio measured at 90 K is ~0.39% for 𝛼𝐽 ≈ 45°, which is about 13 times larger than that for 𝛼𝐽 = 90° (𝐉||[001]) of 0.03%. Such a huge current-direction 7

dependence of the AMR ratio could pave the way towards tailoring the AMR effect through the crystalline orientation in spintronics applications. To better understand the magnetoresistance behavior in FM materials, a phenomenological model has been proposed based on the symmetry analysis [9,44]. The component 𝜌𝑖𝑗 of the resistivity tensor is a function of the direction cosines (𝑚𝑖 ) of the magnetization, and can be expanded in ascending power [9,44]: 𝜌𝑖𝑗 = 𝑎𝑖𝑗 + 𝑎𝑘𝑖𝑗 𝑚𝑘 + 𝑎𝑘𝑙𝑖𝑗 𝑚𝑘 𝑚𝑙 + 𝑎𝑘𝑙𝑚𝑖𝑗 𝑚𝑘 𝑚𝑙 𝑚𝑚 + 𝑎𝑘𝑙𝑚𝑛𝑖𝑗 𝑚𝑘 𝑚𝑙 𝑚𝑚 𝑚𝑛 + ⋯,(1) where 𝑎𝑖𝑗 , 𝑎𝑘𝑖𝑗 , 𝑎𝑘𝑙𝑖𝑗 , 𝑎𝑘𝑙𝑚𝑖𝑗 , 𝑎𝑘𝑙𝑚𝑛𝑖𝑗 , … are the components of the galvanomagnetic tensor, and 𝑖 and 𝑗 can be in any of the three orthogonal ⟨100⟩ directions [9,44]. In this study, we only considered the expansion parameters up to the fourth order. The expansion parameters should be restricted by the 𝑂ℎ symmetry of bulk system with cubic structure. After a series of symmetry operations, the matrix tensor elements are significantly simplified. Moreover, the tensor elements are restricted by the Onsager relation 𝜌𝑖𝑗 (𝒎) = 𝜌𝑗𝑖 (−𝒎). Finally, a general from of the resistivity tensor is obtained [9,44]. Based on the Ohm’s law, the longitudinal and transverse resistivity can be written as 𝜌𝑥𝑥 = 𝒋 ⋅ 𝝆 ⋅ 𝒋 and 𝜌𝑥𝑦 = 𝒕 ⋅ 𝝆 ⋅ 𝒋 . Here, unit vector 𝒋 indicates the current direction with the components 𝑗1 , 𝑗2 and 𝑗3 projected to the three basic vectors of [100], [010] and [001], and 𝒕 indicates the in-plane transverse direction perpendicular to 𝒋. In this study, the magnetization and current directions are limited in the (110) plane. Then, the projected longitudinal resistivity 𝜌𝑥𝑥 (𝜑𝑀 , 𝛼𝐽 ) and the transverse resistivity 𝜌𝑥𝑦 (𝜑𝑀 , 𝛼𝐽 ) in the (110) plane can be expressed as (see supplemental materials): 𝜌𝑥𝑥 (𝜑𝑀 , 𝛼𝐽 ) = 𝐴0 + 𝐴1 cos2𝛼𝐽 + (𝐵0 + 𝐵1 cos2𝛼𝐽 )cos(2𝜑𝑀 + 2𝛼𝐽 ) +𝐵2 sin2𝛼𝐽 sin(2𝜑𝑀 + 2𝛼𝐽 ) +(𝐷0 + 𝐷1 cos2𝛼𝐽 )cos(4𝜑𝑀 + 4𝛼𝐽 ) +𝐷2 sin2𝛼𝐽 sin(4𝜑𝑀 + 4𝛼𝐽 ),

(2)

𝜌𝑥𝑦 (𝜑𝑀 , 𝛼𝐽 ) = −𝐴1 sin2𝛼𝐽 − 𝐵1 sin2𝛼𝐽 cos(2𝜑𝑀 + 2𝛼𝐽 ) +𝐵2 cos2𝛼𝐽 sin(2𝜑𝑀 + 2𝛼𝐽 ) 8

−𝐷1 sin2𝛼𝐽 cos(4𝜑𝑀 + 4𝛼𝐽 ) +𝐷2 cos2𝛼𝐽 sin(4𝜑𝑀 + 4𝛼𝐽 ).

(3)

Phenomenologically, the parameters 𝐴0 , 𝐴1 , 𝐵0 , 𝐵1 , 𝐵2 , 𝐷0 , 𝐷1 and 𝐷2 are determined by the components of the galvanomagnetic tensor 𝑎𝑖𝑗 , 𝑎𝑘𝑖𝑗 , 𝑎𝑘𝑙𝑖𝑗 , 𝑎𝑘𝑙𝑚𝑖𝑗 , 𝑎𝑘𝑙𝑚𝑛𝑖𝑗 (see supplemental materials), which should be temperature-dependent. Eq. (2) indicates that the two-fold term in the 𝜌𝑥𝑥 (𝜑𝑀 ) curves can be attributed to the parameters 𝐵0, 𝐵1 and 𝐵2, and the four-fold term is related to the parameters 𝐷0 , 𝐷1 and 𝐷2 . Those coefficients in Eqs. (2) and (3) can be quantified by fitting the experimental curves. Next, we will show that the measured 𝜌𝑥𝑥 (𝜑𝑀 , 𝛼𝐽 ) and 𝜌𝑥𝑦 (𝜑𝑀 , 𝛼𝐽 ) curves can be well fitted by the phenomenological model. First, we simplified Eqs. (2) and (3) into the following equations: 𝜌𝑥𝑥 (𝜑𝑀 , 𝛼𝐽 ) = 𝑘0𝑥𝑥 + 𝑘1𝑥𝑥 cos(2𝜑𝑀 + 2𝛼𝐽 ) + 𝑘2𝑥𝑥 sin(2𝜑𝑀 + 2𝛼𝐽 ) +𝑘3𝑥𝑥 cos(4𝜑𝑀 + 4𝛼𝐽 ) + 𝑘4𝑥𝑥 sin(4𝜑𝑀 + 4𝛼𝐽 ), 𝑥𝑦

𝑥𝑦

(4)

𝑥𝑦

𝜌𝑥𝑦 (𝜑𝑀 , 𝛼𝐽 ) = 𝑘0 + 𝑘1 cos(2𝜑𝑀 + 2𝛼𝐽 ) + 𝑘2 sin(2𝜑𝑀 + 2𝛼𝐽 ) 𝑥𝑦

𝑥𝑦

+𝑘3 cos(4𝜑𝑀 + 4𝛼𝐽 ) + 𝑘4 sin(4𝜑𝑀 + 4𝛼𝐽 ). 𝑥𝑦

Here, the coefficients 𝑘𝑖𝑥𝑥 and 𝑘𝑖

(5)

(𝑖 = 0, 1, 2, 3 and 4) should have the sin2𝛼𝐽 or

cos2𝛼𝐽 dependence based on Eqs. (2) and (3), but are treated as fitting constants firstly. 𝑥𝑦

The detailed expressions of 𝑘𝑖𝑥𝑥 and 𝑘𝑖 are 𝑘0𝑥𝑥 = 𝐴0 + 𝐴1 cos2𝛼𝐽 , 𝑘1𝑥𝑥 = 𝐵0 + 𝑥𝑦

𝐵1 cos2𝛼𝐽 , 𝑘2𝑥𝑥 = 𝐵2 sin2𝛼𝐽 , 𝑘3𝑥𝑥 = 𝐷0 + 𝐷1 cos2𝛼𝐽 , 𝑘4𝑥𝑥 = 𝐷2 sin2𝛼𝐽 , 𝑘0 = 𝑥𝑦

𝑥𝑦

𝑥𝑦

−𝐴1 sin2𝛼𝐽 , 𝑘1 = −𝐵1 sin2𝛼𝐽 , 𝑘2 = 𝐵2 cos2𝛼𝐽 , 𝑘3𝑥𝑥 = −𝐷1 sin2𝛼𝐽 , 𝑘4 = 𝐷2 cos2𝛼𝐽 . We can fit all the measured 𝜌𝑥𝑥 (𝜑𝑀 ) and 𝜌𝑥𝑦 (𝜑𝑀 ) curves with different 𝛼𝐽 using Eqs. (4) and (5), as demonstrated by the solid lines in Figs. 2(c) and 2(d). 𝑥𝑦

Subsequently, the 𝛼𝐽 -dependent 𝑘𝑖𝑥𝑥 and 𝑘𝑖

can be obtained and well fitted by the

sin2𝛼𝐽 or cos2𝛼𝐽 dependence, as shown in Figs. 4(a) and 4(b). By fitting the 𝑥𝑦

𝑘𝑖𝑥𝑥 (𝛼𝐽 ) and 𝑘𝑖 (𝛼𝐽 ) curves, we can extract the coefficients 𝐵0 , 𝐵1 , 𝐵2 , 𝐷0 , 𝐷1 9

and 𝐷2 . The coefficients 𝐴0 and 𝐴1 can also be extracted from the 𝜌𝑥𝑥 (𝛼𝐽 ) curves shown in Fig. 3(c). Table 1 lists all the extracted coefficients, and the superscripts 𝑥𝑥 and 𝑥𝑦 indicate the coefficients extracted from the 𝜌𝑥𝑥 and 𝜌𝑥𝑦 measurements, respectively. It is clear that the coefficients 𝐵0, 𝐵1 and 𝐵2, related to the two-fold AMR term, are one order of magnitude larger than the coefficients 𝐷0 , 𝐷1 and 𝐷2 , related to the fourfold AMR term. According to the phenomenological model in Eqs. (2) and (3), the coefficients 𝐵𝑖𝑥𝑥 (𝐷𝑖𝑥𝑥 ) (𝑖 = 1 and 2) determined from the AMR measurements 𝑥𝑦

𝑥𝑦

should be exactly the same as 𝐵𝑖 (𝐷𝑖 ) determined from the PHE measurements. 𝑥𝑦

𝑥𝑦

However, in Table 1, the magnitude of 𝐵𝑖 (𝐷𝑖 ) is always larger than that of 𝐵𝑖𝑥𝑥 (𝐷𝑖𝑥𝑥 ) with a ratio of ~1.14. This difference can be attributed to the imperfect current flow in our devices. In the phenomenological model, the derivation of Eqs. (2) and (3) is based on the assumption that the current flows uniformly along the Hall bar for 𝜌𝑥𝑥 and 𝜌𝑥𝑦 measurements. However, in reality, the current flow may penetrate into the voltage electrodes, as shown by the inset in Fig. 4(b). Thus the measured transverse voltage 𝑉𝑥𝑦 is larger than the voltage drop between the edges of the current bar, indicated by 𝑉12 in the inset of Fig. 4(b), which can explain the difference between 𝑥𝑦

𝑥𝑦

𝐵𝑖 (𝐷𝑖 ) and 𝐵𝑖𝑥𝑥 (𝐷𝑖𝑥𝑥 ). In order to determine 𝜌𝑥𝑦 more accurately, the current leaking into the electrode should be avoided, thus the electrode width should be much smaller than the width of the Hall bar. Nevertheless, our studies show that the phenomenological model can describe not only the current-direction dependent AMR effect, but also the current-direction dependent PHE effect, which had been given much less attention in past studies on current-direction dependent MR properties. We performed the temperature-dependent measurement of 𝜌𝑥𝑥 (𝜑𝑀 , 𝛼𝐽 ) and 𝜌𝑥𝑦 (𝜑𝑀 , 𝛼𝐽 ), and extracted all the coefficients as a function of temperature. Figs. 4(c) 𝑥𝑦

𝑥𝑦

and 4(d) show the extracted 𝐵𝑖𝑥𝑥 (𝐷𝑖𝑥𝑥 ) and 𝐵𝑖 (𝐷𝑖 ) as a function of temperature. The magnitudes of most extracted coefficients decrease with decreasing temperature 10

due to electron-phonon scattering, as 𝐷𝑖 coefficients increase with decreasing temperature. For 𝐵0𝑥𝑥 , the magnitude even increases a little with a decreasing 𝑥𝑦

𝑥𝑦

temperature. Moreover, the measured coefficients 𝐵𝑖 (𝐷𝑖 ) are always larger than 𝐵𝑖𝑥𝑥 (𝐷𝑖𝑥𝑥 ) due to the current leaking effect. As shown in Figs. 4(c) and (d), the four-fold terms 𝐷0 , 𝐷1 and 𝐷2 are one order smaller than the two-fold terms 𝐵0, 𝐵1 and 𝐵2. Thus, if neglecting the four-fold terms, the 𝛼𝐽 -dependent magnetoresistivity 𝛥𝜌𝑥𝑥 and 𝛥𝜌𝑥𝑦 can be simplified as: 𝛥𝜌𝑥𝑥 (𝛼𝐽 ) = √𝐵02 + (𝐵12 + 𝐵22 )/2 + (𝐵12 − 𝐵22 )cos4𝛼𝐽 /2 + 2𝐵0 𝐵1 𝑐𝑜𝑠 2 𝛼𝐽 ,

(6)

𝛥𝜌𝑥𝑦 (𝛼𝐽 ) = √(𝐵12 + 𝐵22 )/2 + (𝐵12 − 𝐵22 )cos4(𝛼𝐽 + 𝜋/4)/2.

(7)

So, the 𝛥𝜌𝑥𝑥 (𝛼𝐽 ) curve shows the combination of four-fold symmetry and two-fold symmetry, and the 𝛥𝜌𝑥𝑦 (𝛼𝐽 ) curve only possesses a pure four-fold symmetry, in good agreement with the experimental results. Eq. (6) clearly shows that the four-fold contribution in the 𝛥𝜌𝑥𝑥 (𝛼𝐽 ) curve is attributed to 𝐵1 and 𝐵2 , but the two-fold contribution is determined by 𝐵1 and 𝐵0 . Eq. (7) also shows that the four-fold symmetric axes in the 𝛥𝜌𝑥𝑥 (𝛼𝐽 ) and 𝛥𝜌𝑥𝑦 (𝛼𝐽 ) curves have a 45° offset, consistent with the experimental data in Fig. 3. By calculating the 𝛥𝜌𝑥𝑥 (𝛼𝐽 ) and 𝛥𝜌𝑥𝑦 (𝛼𝐽 ) curves with the fitted parameters 𝐵0, 𝐵1 and 𝐵2 shown in Figs. 4(c) and 4(d), we can nicely reproduce the experimental curves at all temperatures, as presented by the solid lines in Figs. 3(a) and 3(b). Moreover, the calculated AMR ratio by 𝛥𝜌𝑥𝑥 (𝛼𝐽 )/(𝐴0 + 𝐴1 cos2𝛼𝐽 ) also agrees very well with the experimental data in Fig. 3(d). Therefore, our results demonstrate that the phenomenological model can describe well both current-direction dependent AMR and PHE in single-crystalline thin films. On the other hand, the phenomenological model can only describe the current-direction dependence and magnetization-orientation dependence of magnetoresistance, but it cannot directly provide the values of 𝐴𝑖 , 𝐵𝑖 and 𝐷𝑖 in Eqs. (2) and (3). Thus, further efforts are necessary to theoretically understand the origin of these AMR coefficients based on the electronic structures. The current-direction dependent AMR effect in single crystalline Fe(110) film was 11

also studied by R. P. van Gorkom et. al [36] for the temperature between 4.2 K and 230 K, but the measurements were only performed for the current along two directions with 𝛼𝐽 = 70° and 𝛼𝐽 = 90°. They found that the AMR coefficients obtained from the Döring equations [8] had a strong reduction when the temperature increases from 4.2 K to 100 K. This was attributed to the transition from defect-dominated scattering to phonon-dominated scattering, each having its own anisotropic magnetoresistance contribution. In our studies, we systematically varied the current orientation angle in different devices, and developed the analytical expressions for the resistivity based on the phenomenological model. The current-direction dependent AMR effect at room temperature from a 20 nm Fe film is similar with that from the 10 nm Fe film (see supplementary material), thus the two-fold AMR effect observed here in Fe(110) film is a bulk effect, which is different from the interfacial origin of the two-fold symmetric current-direction dependent AMR effect in the Fe/GaAs(001) system [34]. Since the phenomenological model based on the bulk cubic symmetry can well fit with the experimental data in Figs. 3 and 4, the observed two-fold and four-fold angular dependent AMR should originate from the cubic lattice of bcc Fe. The anisotropic current-orientation effect of AMR in single crystalline system has been experimentally explored in several systems [8,24-33], and all can be well fitted by the phenomenological model. However, there were very few theoretic studies to understand the microscopic origin of those coefficients in the phenomenological model. In conventional AMR theory, AMR stems from the anisotropic spin mixing due to the spin-orbit coupling [8-13], which should strongly associate with the band structure in the single crystalline system, thus the observed current-direction dependence of AMR is closely linked to the crystal symmetry. Our experimental results call for the further theoretical studies on the relation between the current direction and the crystalline axis based on the electron band structures. Moreover, the huge difference in AMR ratios with the current along different directions in Fig. 3(d) could be used to control the spindependent transport properties by varying the current flow orientation. For example, the spin rectification effect is expected to be proportional to the AMR ratio [45,46], so 12

it is expected that the spin rectification effect in a spin dynamic measurement should depend strongly on the direction of the microwave current.

4. CONCLUSION In summary, we systematically studied the current-direction dependence of anisotropic magnetoresistance and planar Hall effect in single crystalline Fe(110) films. The longitudinal and transverse resistivities with the current away from the crystal axis clearly deviate from conventional cos2𝜑𝑀 and sin𝜑𝑀 cos𝜑𝑀 dependences. The AMR amplitude 𝛥𝜌𝑥𝑥 shows both two-fold and four-fold symmetries with respect to the current direction, but 𝛥𝜌𝑥𝑦 only exhibits a four-fold symmetry. Our studies show that both longitudinal and transverse current-direction dependent AMRs can be well described by a phenomenological model. The fitted resistivity coefficients are found to increase with temperature because of electron-phonon scattering. The measured AMR ratios for the currents along different crystalline directions exhibit a difference that exceeds one order of magnitude. Such strongly current-direction dependent AMR indicates a way to design future spintronics devices by adjusting the current orientation with respect to the crystalline axis. Our studies suggest that other spin-dependent transport properties in single-crystalline films, such as the spin Hall effect and the spin rectification effect, may also depend on the current direction.

ACKNOWLEGMENTS The authors thank C. Won and Y. Ji for valuable discussions. This project was supported by the National Key Basic Research Program of China (Grant No. 2015CB921401), National Key Research and Development Program of China (Grant No. 2016YFA0300703), National Natural Science Foundation of China (Grants No. 11474066, No. 11734006 and No. 11434003), and the Program of Shanghai Academic Research Leader (No. 17XD1400400).

13

Fig. 1. (a) RHEED patterns of GaAs(110) substrate and a 10 nm Fe(110) film with the incident electron along GaAs[1̅10] direction. (b) The hysteresis loops measured by longitudinal MOKE with the field along Fe[1̅10], [001] and [1̅11] directions. (c) Torque curve 𝑙(𝜙𝑀 ) obtained with the Rot-MOKE method. The red line is the fitting curve. (d) Schematic of device for measuring the current-direction dependent AMR. The representative Hall bar structure indicates the current angle 𝛼𝐽 with respect to Fe[1̅10] direction as well as the field angle 𝜑𝐻 and the magnetization angle 𝜑𝑀 with respect to current direction.

14

Fig. 2. The magnetic-field dependence of (a) longitudinal resistivity 𝜌𝑥𝑥 − 𝜌𝑥𝑥 (𝐻 = 0) and (b) transverse resistivity 𝜌𝑥𝑦 with different 𝜑𝐻 for the current along Fe[1̅10] direction (𝛼𝐽 = 0°). (c) 𝜌𝑥𝑥 − 𝜌𝑥𝑥 (𝜑𝑀 = 0°) and (d) 𝜌𝑥𝑦 curves (offset for clarity) as a function of magnetization angle 𝜑𝑀 with different current direction angle 𝛼𝐽 . The in-plane rotating filed is 2.5 kOe. The solid lines are the theoretical fits based on the phenomenological model.

15

Fig. 3. The current-direction 𝛼𝐽 dependence of (a) 𝛥𝜌𝑥𝑥 , (b) 𝛥𝜌𝑥𝑦 , (c) 𝜌𝑥𝑥 and (d) AMR ratio 𝛥𝜌𝑥𝑥 /𝜌𝑥𝑥 at different temperatures. The symbols represent the experimental data. The solid lines in (a) and (b) correspond to the calculated lines based on Eqs. (6) and (7). The solid lines in (c) are the fitting results with the cos2𝛼𝐽 function.

16

𝑥𝑦

Fig. 4. (a-b) The fitted parameter 𝑘𝑖𝑥𝑥 and 𝑘𝑖

(𝑖 = 1, 2, 3 and 4) are plotted as a

function of 𝛼𝐽 . The symbols are fitted parameters and the solid lines are fitting curves by the theoretical model. The inset of (b) indicates the current-leaking effect for the PHE measurement. The temperature dependence of the extracted parameters (c) 𝐵𝑖𝑥𝑥 , 𝑥𝑦

𝑥𝑦

𝐷𝑖𝑥𝑥 (𝑖 = 0, 1 and 2) and (d) 𝐵𝑖 , 𝐷𝑖 .

17

𝑥𝑦

Table 1. Summary of the extracted coefficients by fitting the 𝑘𝑖𝑥𝑥 (𝛼𝐽 ) and 𝑘𝑖 (𝛼𝐽 ) curves in Figs. 4(a) and 4(b). The superscripts 𝑥𝑥 and 𝑥𝑦 denote the parameters from the 𝜌𝑥𝑥 and 𝜌𝑥𝑦 measurements respectively. Coefficient

Value

Coefficient

Value

𝐴xx 0

25.3 ± 0.2 μΩ cm

𝐵0xx

8.8 ± 0.3 nΩ cm

𝐴1xx

3.1 ± 0.2 μΩ cm

𝐷0xx

−1.4 ± 0.1 nΩ cm

𝐵1xx

20.2 ± 0.3 nΩ cm

𝐵1

xy

23.0 ± 0.4 nΩ cm

𝐵2xx

46.8 ± 0.6 nΩ cm

𝐵2

xy

53.2 ± 0.6 nΩ cm

𝐷1xx

−0.5 ± 0.1 nΩ cm

𝐷1

xy

−0.7 ± 0.1 nΩ cm

𝐷2xx

−4.4 ± 0.1 nΩ cm

𝐷2

xy

−5.0 ± 0.1 nΩ cm

18

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20

The anisotropic magnetoresistance (AMR) and planar Hall effect (PHE) are systematically investigated in single crystalline Fe(110) films Both AMR and PHE strongly depend on the applied current directions with respect to the crystalline axes The AMR ratio could have 13 times difference by varying the current directions The current-orientation dependence of AMR shows a mixture of two- and four-fold symmetry but PHE only shows the four-fold symmetry The observed strong current-direction effect on AMR and PHE can be well explained by the phenomenological model

21