Two-dimensional (2D) layered materials with magnetic behavior (both ferromagnetic and antiferromagnetic) have received considerable attention in recent years because of their unique properties and potential application in spintronics and information storage technology [1–4]. The magnetic properties of intrinsic magnetic 2D layer materials can be tuned via van der Waals engineering, electrostatic gating, and strain engineering [5–9]. The magnetic properties of intrinsic nonmagnetic 2D can be introduced through the formation of vacancies or defects, adsorption of atoms, and doping transition metal atoms or nonmetal atoms [10–13].

Anisotropic magnetoresistance (AMR) effect, which is defined as magnetoresistance changes with an angle between magnetization and current [14], first reported by William Thomson [15], has been used in many fields like magnetic sensors and AMR-magnetic random-access memory (MRAM) devices [16]. The physical mechanism is very complicated, as stated in Refs. [17, 18]. Traditionally, AMR can be explained by spin-orbit coupling. If the direction of the magnetic field is perpendicular to the current, then the electronic orbits are parallel with the current and lead to a small cross-section for scattering, which results in a low resistance state. On the contrary, while the direction of the magnetic field is parallel to the current, the electronic orbits are perpendicular to the current, and the cross-section for scattering is larger, resulting in a high resistance state [19]. In recent studies of many new materials, such as Nb₃SiTe₆ [20], Dirac semimetal Cd₃As₂ [21], topological type-II Weyl semimetal like WTe₂ [22], and A-type antiferromagnetic NaCrTe₂ [23], this rule has been broken. The low resistance state exists when the applied fields are parallel to the current.

FeNbTe₂ is a 2D layered material with intralayer ferromagnetism and interlayer antiferromagnetism [24]; Jing Li et al. synthesized FeNbTe₂ single crystals and studied the transport properties. The research shows that FeNbTe₂ has a space group of Pmna (No. 53, Figure 1A), and its resistance increases with decreasing temperature, showing semiconductor characteristics. A detailed study on the magnetic properties of polycrystalline FeNbTe₂ was conducted by Jian H. Zhang et al. in 1997 [25]. The measurement χ a c ( = d M / d H ) confirms the existence of a spin glass state in the polycrystalline FeNbTe₂ sample with a transition at T _f = 44 K. The most recent study was reported in 2019 [26]. Chong Xiao et al. made FeNbTe₂ flake micro-nano devices and performed magnetoelectric transport measurements. They demonstrated an intrinsic unsaturated negative magnetoresistance (nMR) of FeNbTe₂. Angle-resolved photoemission spectroscopy (ARPES) measurement, the electronic transport measurement, and DC/AC magnetic susceptibility prove that the existence of intrinsic unsaturated nMR is due to the comprehensive effect of Anderson localization and a spin glass state. In this article, we systematically studied the anisotropic magnetoresistance of synthetic FeNbTe₂ single crystals.

FeNbTe₂ single crystals were synthesized by a chemical vapor transport (CVT) process [24, 26]. All reagents, iron (99.99%), niobium (99.99%), and tellurium (99.99%) were purchased from Aladdin (Shanghai, China). The materials were mixed in a molar ratio of 1:1:2 and sealed in a 16-mm Δ 150-mm vacuum quartz tube, and 10 mg of iodine was used as a transport agent. In a two-temperature zone tube furnace, the sealed vacuum quartz tube was heated to 850°C (high temperature zone) and 750°C (low temperature zone) for 12 h, kept for 10 days, and then cooled to room temperature in 12 h. The structure of FeNbTe₂ single crystals was characterized by an X-ray diffractometer (XRD, Advance D8). To investigate the elemental composition and crystal structure, transmission electron microscopy (TEM) measurements were performed, in which the elemental composition was given by energy-dispersive spectrum (EDS) mapping, and the crystal structure was detected by selected-area electron diffraction (SAED) and high-resolution TEM (HRTEM).

Magnetic and transport measurements were performed in a Quantum Design Physical Property Measurement System with a 9 T magnet (PPMS-9T). The magnetic properties of the bulk sample were measured by using a vibrating sample magnetometer (VSM) integrated into the PPMS-9T with an accuracy of 10^–7 emu. The resistivity of the thin flake on its (x, y) plane was measured by the standard four-terminal method, with a rotator sample holder for controlling the relative orientation between the magnetic field and the crystal. The FeNbTe₂ material was prepared into micro-nano devices for transport measurement. The device was prepared by transferring FeNbTe₂ flake onto the Au (50 nm)/Cr (5 nm) electrodes as shown in Figure 1B, in which the flake was mechanically exfoliated from the single-crystalline bulk sample. Au (50 nm)/Cr (5 nm) electrodes were fabricated onto 285 nm SiO₂/Si substrates by using a standard ultraviolet radiation photolithographic procedure. As shown in Figure 1C, a standard rectangular coordinate system is established on the flake, with the flake plane as the xy plane. The y-axis is parallel to the current direction, the z-axis points to the normal direction of the sheet surface, and the y-axis is perpendicular to the xz plane. The angular dependence of MR was measured by rotating the sample in the magnetic field; θ and ψ are defined as the angle between the magnetic field and the z axis on the yz plane, and the magnetic field and the x axis on the xy plane, respectively, as shown in Figure 1C.

The quality of the FeNbTe₂ single crystal was characterized by XRD, which is shown in Figure 1D. The sharp diffraction peaks and the (0l0) peaks show the interlaminar orientation of the single crystal along the b axis. The optics image of a FeNbTe₂ crystal is shown in the inset, the size is about 3 mm × 1.5 mm. The TEM characterization results are as follows: energy-dispersive spectra (EDS) mapping of elementals Fe, Nb, and Te is shown in Figure 1E. The distribution of the three elements is uniform, and the stoichiometric ratio of Fe, Nb, and Te is close to 1: 1: 2, consistent with the composition of the FeNbTe₂ crystal. The single-crystalline structure is further confirmed by selected-area electron diffraction (SAED) and high-resolution TEM (HRTEM), as shown in Figure 1F and Figure 1G. SAED shows clear spots, which demonstrates the single crystal nature of the FeNbTe₂ nanoplate. The HRTEM in Figure 1G image displays distinct lattice fringes with interplanar spacings of 0.394 and 0.625 nm, which are assigned to the orthorhombic FeNbTe₂ with lattice parameters of a = 0.792 and c = 0.624 nm [24].

Figure 2A shows the measured zero field cooled (ZFC) and field cooled (FC) temperature dependence of magnetization in an applied field of 0.1 to 1 T. With a field of 1 T, the ZFC–FC curves overlap with each other, which is quite different from the date, measured under 0.1 T. At an applied field of 0.1 T, when the temperature is higher than 22 K, the ZFC and FC magnetization decreases monotonically with the increase in temperature. When it is below 22 K, a distinct λ -shaped bifurcation appears between the two curves of ZFC and FC, which is generally considered one of the characteristics of spin glass [11]. Figure 2B displays the magnetization of FeNbTe₂ versus magnetic field H up to 1 T, measured at different temperatures. The curve measured at 10 K is similar to the curve at 20 K, which corresponds to the 22 K bifurcation point mentioned earlier. The coercivity H_C is ≈ 100 Oe at 10 K and decreases to ≈60 Oe at 20 K. At a higher temperature of about 40 K, the M–H curve behaves in a more paramagnetic linear correlation [27]. The magnetic moment saturation cannot be observed in a magnetic field up to 7 T at 10 K, as shown in the inset of Figure 2B.

To investigate the magnetic transport properties, the MR-H measurements were performed at 10 K with the magnetic field parallel to the x-axis and z-axis, respectively. As shown in Figure 2C, MR is normalized to the zero-field value R 0 by M R ( % ) = ( R ( H ) − R 0 ) / R 0 × 100 % ; the magnetoresistance decreases with increasing field, which is typically nMR. The value of nMR at 10 K under 7 T is ≈ −0.32%, consistent with previous reports [26]. It can also be seen that, compared with the rate of change with the magnetic field parallel to the x-axis, the MR measured with the magnetic field parallel to the z-axis is smaller.

When the magnetic field is applied along the z-axis and x-axis, the value of nMR differs by 30%; this shows the existence of anisotropy in this sample. The magnetic anisotropy of the FeNbTe₂ was investigated by angle-dependent magnetic resistance measurements; the magnetic field rotates along the y-axis and z-axis.

We first consider the AMR measurement with the magnetic field rotating within the xz plane. Here, two kinds of measurements were performed, one is the measurement of the angle-dependent magnetoresistances with a fixed magnetic field (9 T) at different temperatures, and the other is the measurement with different magnetic fields at isothermal conditions of 10 K. The results are normalized by M R ( % ) = ( R − R ( θ = 0 ) ) / R ( θ = 0 ) × 100 % , as shown in Figure 3A and Figure 3B, respectively.

In Figures 3A, B, the MR curves take on a similar shape to a cosine square function of θ. The curve can be fitted with the formula M R = − R m c o s 2 ( θ + π / 2 ) . It can be seen from the figures that, under different conditions, the resistance when θ = 0 is higher than the resistance when   θ = π / 2 . This is contrary to the principle of the traditional AMR [14, 15], while is similar to that of some new materials, Nb₃SiTe₆, Dirac semimetal Cd₃As₂, topological type-II Weyl semimetal WTe₂ and A-type antiferromagnet NaCrTe₂ [20–23]. As can be seen in Figure 3A, FeNbTe₂ flake shows obvious out-of-plane anisotropy up to 50 K, the anisotropic MR is about 0.14% at 10 K, and gradually decreases with increasing temperature. At a temperature of 10 K, the AMR is obvious under different magnetic fields. The AMR increases nearly linearly with the magnetic field increase, which is shown in Figure 3B.

Next, we consider the AMR measurement at different temperatures with the fixed magnetic field of 9 T and rotating within the xy plane. The normalized results are displayed in Figure 4A and Figure 4B. As shown in Figure 4A, the measurement was performed at temperatures ranging from 10 to 50 K. Different from the law of out-of-plane anisotropy; the variation amplitude of in-plane anisotropy is not strictly temperature-dependent. The temperature where the largest variation appears is about 40 K, with an amplitude of about 0.06%, and the curve flipped between 20 K and 40 K. At temperatures of 10 and 20 K, when the magnetic field is applied parallel to the x-axis, ψ = 0 rad, the resistance is minimum, and when ψ = π/2, the resistance is maximum, which is consistent with the spin-orbit coupling induced density of states and spin-related anisotropic scattering characteristics. When the magnetic moment is perpendicular to the direction of the current, the orbit of electronics is parallel to the direction of the current and is in a low resistance state, and when the magnetic moment is parallel to the direction of the current, it is in a high resistance state [28]. When the temperature rises to 40 K, the measurement curve is flipped along the y-axis, and the resistance is maximum when the magnetic field direction is perpendicular to the current. The curves can be fitted by the formula M R = ∓ R m c o s 2 ( ψ + π / 2 )   .

In view of the inversion of the curve within the temperature range of 20–27 K, a detailed measurement is carried out. As shown in Figure 4B, the curve gradually reverses as the temperature increases, and the reversal phenomenon is more obvious at the temperature of 24 K, which is close to the measured critical temperature of 22 K of the spin glass state of the sample. The reverse phenomenon may be attributed to the fierce competition between the ferromagnetic and antiferromagnetic states near the critical temperature of the spin glass state.

In short, we have explored the AMR properties of the FeNbTe₂ single crystal in this article. The measured results show that the out-of-plane AMR is contrary to the principle of the traditional AMR and is related to the intralayer ferromagnetic and interlayer antiferromagnetic configuration (A-type antiferromagnet). The in-plane AMR is consistent with the traditional AMR theory when the temperature is lower than 25 K, but it is contrary to the traditional AMR principle at higher temperatures.