Field-induced metal-to-insulator transition and colossal anisotropic magnetoresistance in a nearly Dirac material EuMnSb$_2$

How to realize applicably appreciated functionalities based on the coupling between charge and spin degrees of freedom is still a challenge in the field of spintronics. For example, anisotropic magnetoresistance (AMR) effect is utilized to read out the information stored by various magnetic structures, which usually originates from atomic spin-orbit coupling (SOC). However, the application of AMR in antiferromagnet-based spintronics is still hindered by rather small AMR value. Here, we discover a colossal AMR effect during the field-induced metal-to-insulator transition (MIT) in a nearly Dirac material EuMnSb$_2$ with an antiferromagnetic order of Eu$^{2+}$ moments. The colossal AMR reaches to an unprecedented value of 1.84$\times$10$^6$% at 2 K, which is four orders of magnitude larger than previously reported values in antiferromagnets. Based on density functional theory calculations, a Dirac-like band structure, which is strongly dependent on SOC, is confirmed around Y point and dominates the overall transport properties in the present sample with predominant electron-type carriers. Moreover, it is also revealed that the indirect band gap around Fermi level is dependent on the magnetic structure of Eu$^{2+}$ moments, which leads to the field-induced MIT and plays a key role on the colossal AMR effect. Finally, our present work suggests that the similar antiferromagnetic topological materials as EuMnSb$_2$, in which Dirac-like fermions is strongly modulated by SOC and antiferromagnetism, would be a fertile ground to explore applicably appreciated AMR effect.


transition (MIT) in a nearly Dirac material EuMnSbwith an antiferromagnetic order of Eu 2+
moments. The colossal AMR reaches to an unprecedented value of 1.8410 6 % at 2 K, which is four orders of magnitude larger than previously reported values in antiferromagnets. Based on density functional theory calculations, a Dirac-like band structure, which is strongly dependent on SOC, is confirmed around Y point and dominates the overall transport properties in the present sample with predominant electron-type carriers. Moreover, it is also revealed that the indirect band gap around Fermi level is dependent on the magnetic structure of Eu 2+ moments, which leads to the field-induced MIT and plays a key role on the colossal AMR effect. Finally, our present work suggests that the similar antiferromagnetic topological materials as EuMnSb2, in which Dirac-like fermions is strongly modulated by SOC and antiferromagnetism, would be a fertile ground to explore applicably appreciated AMR effect.

Ⅰ. INTRODUCTION
The manipulation of charge transport by spin degree of freedom in solid-state systems is at the core of spintronics. Recently, antiferromagnets have generated significant interest in the field of spintronics owing to their unique properties, such as zero stray magnetic field, ultrafast spin dynamics, and remarkable rigidity against external fields [1][2][3][4]. Anisotropic magnetoresistance (AMR), which is defined as the dependence of the resistivity on the direction of the magnetization with respect to current or crystal axes, could be utilized as electrical-readout to detect the switch of magnetization in antiferromagnetic (AFM)-based spintronic devices. AMR-based memory effects have been successfully demonstrated in several antiferromagnets, such as FeRh [5] and MnTe [6], opening perspective for both fundamental research and device technology of AFM spintronics [5][6][7][8][9][10][11]. In 3 principle, the conventional AMR effect, in which the electronic band structure is dependent on the spin orientation, is mainly associated with the magnetocrystalline anisotropy arising from the relativistic spin-orbit coupling (SOC). In this sense, the SOC-induced anisotropic response of band structure to external magnetic field and the detailed position of the Fermi level determine the AMR in practical antiferromagnets [12][13][14][15]. However, experimentally observed AMR is always limited to a few hundred percent, which is much smaller than the giant magnetoresistance (GMR) [16,17] or tunneling magnetoresistance (TMR), hindering its further practical application. Thus, the hunt for new materials with large AMR and new strategies toward enhancement of AMR are highly desired.
On the other hand, the rise of the magnetic topological materials provides a new fertile playground to explore the manipulation of quantum transport phenomena by spin degrees of freedom [2]. In recent years, the strong correlation between magnetism and topological band structure has been widely discussed in magnetic topological materials [18,19]. Especially, the topological band structure can be strongly dependent on the strength of SOC and the detailed magnetic structure in some magnetic topological materials [20][21][22], which leads to anisotropic response of band structure near the Fermi level to the change of magnetic structure. Therefore, the magnetic topological materials can potentially generate a large AMR, which has not yet been observed experimentally. Among various magnetic topological materials, the family of AMnPn2 (A = Ca, Sr, Ba, Eu or Yb, Pn = Sb or Bi) materials, in which the conducting two-dimensional (2D) Bi/Sb layers separated by the insulating Mn-Pn layers with magnetism can host Dirac-like fermions, exhibits great potential for strong correlation between the Dirac-like band structure and magnetism. In principle, the 2D Bi/Sb layers with square net structure can host 2D Dirac-like fermions [23,24], and quasi-2D massive Dirac-like fermions with SOC gap have been observed in AMnPn2 with Bi/Sb square net [23,[25][26][27][28][29][30][31], including SrMnBi2 [23,25] and 4 CaMnBi2 [26]. On the other hand, although theoretical study [32] has argued that the formation of zigzag chain structure in 2D Bi/Sb layers will destroy the linear band dispersion held by the square net structure, the existence of non-trivial Dirac fermions have also been verified in the AMnPn2 family materials with zigzag chain structure by quantum oscillation experiments, including SrMnSb2 [33], CaMnSb2 [34] and BaMnSb2 [35,36]. Angle resolved photoemission spectroscopy (ARPES) experiments have further confirmed a linearly Dirac-like dispersive band structure around Fermi level in EuMnSb2 and BaMnSb2 [35][36][37]. These results strongly indicate that holding a Dirac-like dispersion seems to be a generic nature of band structure in the AMnPn2 family materials. More interestingly, a considerable coupling between the magnetism and the Dirac-like fermions in the conducting Bi/Sb layer has been clearly observed by neutron scattering and Raman-scattering measurement [38][39][40][41] in various AMnPn2 family materials. Especially, when A represents magnetic ion Eu 2+ , the interaction between local moments and itinerant carrier could be significantly enhanced due to the highly tunable magnetic moments of Eu 2+ by external magnetic field. For instance, quantum Hall effect is observed in bulk antiferromagnet EuMnBi2, in which the field-controlled Eu 2+ magnetic order suppresses the interlayer coupling [28]. In contrast to the positive magnetoresistance (MR) observed in other members of this family, EuMnSb2 with zigzag chain structure in 2D Sb layers shows a large negative MR, especially for the temperature range below the AFM ordering temperature of Eu 2+ moments [37,42,43].
All these results strongly suggest a strongly modulated Dirac-like band structure by the magnetism of Eu 2+ moment in the EuMnPn2 materials.
Here, we discover a colossal AMR effect during the field-induced metal-to-insulator transition (MIT) in a nearly Dirac material EuMnSb2 with an antiferromagnetic order of Eu 2+ moment. The colossal AMR reaches to an unprecedented value of 1.8410 6 % at 2 K, which is four orders of 5 magnitude larger than previously reported values in antiferromagnets. Our density functional theory (DFT) calculations indicates that the field-induced MIT and colossal AMR effect are related to a Diraclike band structure around Y point, which is strongly modulated by SOC and antiferromagnetism.

A. Crystal growth and characterization
EuMnSb2 single crystals used in this study were grown by using Sn flux method. A mixture of Eu slugs, Mn pieces, Sb slugs and Sn shots in a molar ratio of 1:1:2:10 was loaded in an alumina crucible and sealed in an evacuated quartz tube under vacuum. The quartz tube was heated slowly to 1000℃ and held for 20h to homogenize the melt. Then it was cooled down to 600℃ at a rate of 2℃/h, where the flux was decanted using a centrifuge. The compositions of crystals were determined using an energy-dispersive X-ray spectrometer (EDS) mounted on the field emission scanning electronic microscope (FESEM), Sirion200. The single crystal X-ray diffraction (XRD) pattern was obtained by the X-ray diffractometer (SmartLab-9, Rigaku Corp.) with Cu Kα radiation and a fixed graphite monochromator.

B. Electrical transport and magnetic measurements
Electric transport measurements were carried out by using Quantum Design PPMS-14 equipped with a rotator module. Resistivity was measured using standard four-probe method and Hall resistivity was measured by standard Hall bar configuration. The rotation measurement was performed by using a commercial rotator in PPMS-14. Magnetization measurements were performed using Quantum Design MPMS-7.

C. Density functional calculations
Spin-polarized DFT+U calculations were performed within the Vienna ab initio Simulation Package (VASP) [44]. The generalized gradient approximation (GGA) [45] and the Perdew-Burke-Ernzerh (PBE) [45,46] of exchange-correlation functional were used. The electron-ion interaction was described with the projector augmented wave (PAW) method and Eu (4f, 5s, 5p, 6s), Mn (3d, 4s), Sb (5s, 5p) electrons were treated as valence states. An effective Hubbard U = 5 eV in the GGA+U scheme [47] was applied to the localized f (Eu) and d (Mn) electrons, and the spin-orbit coupling interactions were also taken into account in the DFT calculations. Ⅲ . RESULTS

A. Magnetic and transport properties
EuMnSb2 crystallizes in an orthorhombic structure with space group Pnma (No.62) as shown Fig.   1(d). The single crystal XRD pattern shows high quality of our sample with lattice parameter a = 22.56 Å (shown in Fig. S1 of the Supplementary Materials [48]), which is consistent with the previous report [37,42,43]. The Mn sublattice, sandwiched by two Sb atomic layers, exhibits C-type AFM order with magnetic moments aligned along the a axis below ~346 K [37,43]. So far, there is a discrepancy between the reported magnetic structures of Eu sublattice below 21 K. A-type AFM structure with magnetic moments along the c axis was suggested by powder neutron diffraction measurements [37], while neutron diffraction refinements performed on single crystals have found a canted A-type AFM order with Eu 2+ moments lying 41° away from the a axis in the ac plane [43]. To confirm the magnetic order of Eu sublattice in our EuMnSb2 single crystal, we have performed magnetization measurements. 7 The temperature-dependent magnetic susceptibilities χa and χbc are displayed in Fig. 1(a), with magnetic field (H) applied along the a axis and bc plane, respectively. χa shows a clear kink at 21 K, denoting the formation of AFM order of Eu sublattice. In Fig. 1(b), we display the isothermal magnetization of EuMnSb2 with magnetic field applied along the a axis and bc plane at 2 K, respectively. In both cases, the magnetization does not saturate up to 7 T. When H is applied along the a axis, a spin-flop transition is observed at H = 1.5 T (red curve), consistent with previous report [42].
The magnetization along the a axis is slightly smaller than that in the bc plane, indicating that the magnetic moments of Eu 2+ are more easily polarized in the bc plane. The small anisotropy of magnetization between the a axis and bc plane implies a preferred canted A-type AFM structure of Eu 2+ moments in EuMnSb2 as shown in Fig. 1(d). As we will discuss later, our DFT calculations also find that the total energy with canted A-type AFM structure of Eu 2+ moments is smaller than that with A-type AFM structure. Therefore, we consider the canted A-type AFM order as the magnetic ground state of Eu sublattice in the relevant discussion of the present manuscript. More magnetic susceptibility data can be found in Fig  which is larger than that in previous report [42]. The deviation of resistivity below 200 K from thermally activated resistivity behavior can be attributed to the short-range magnetic order of Eu 2+ 8 moments, which is also consistent with previous report [43]. We note that the resistivity of our EuMnSb2 sample at zero magnetic field shows insulating behavior below TN, which is quite different from that in previous report exhibiting a dip below TN [37,42,43]. In addition, the value of resistivity in our EuMnSb2 sample is also much larger than that in previously reported sample [37,42,43]. We ascribe the insulating behavior of resistivity to a lower carrier density in our EuMnSb2 sample, thus making the Fermi level locate inside the band gap. This will be discussed below in detail. When H is applied along the a axis, it induces a MIT at low temperature in our EuMnSb2 sample. With increasing magnetic field, the small kink in resistivity corresponding to TN moves to lower temperature and the upturn behavior below TN becomes weak with increasing magnetic field. This indicates that the band gap below TN decreases under magnetic field, which is consistent with the following band structure calculations. The resistivity ρxx with the magnetic field perpendicular and parallel to the current direction in the bc plane is shown in Fig. 1(e) and 1(f), respectively. ρxx with H//bc shows relatively weak negative MR behavior and remains insulating behavior at low temperature even at 14 T, which is quite different from the nearly metallic behavior with H//a.
The magnetic-field-dependent resistivity data shown in Fig. 1(c) reveals that EuMnSb2 exhibits a remarkable negative MR response at low temperatures. To proceed, we define MR as MR = [ ( ) − (0T)]/ (0T) and display the MR at different temperatures with magnetic field applied along the a axis in Fig. 1(g). The MR has been symmetrized with respect to positive and negative magnetic fields to remove the Hall contribution. EuMnSb2 shows a large negative MR below TN and above about 5 T. The negative MR behavior weakens with temperature increasing above TN. A large negative MR with maximum value of -99.9987% at 2 K and under H= 14 T is observed, which is larger than the value in previously reported EuMnSb2 by far [37,42]. To gain more sight into the 9 emergence of the negative MR, we have plotted the temperature-dependent MR at various magnetic fields applied along the a axis in Fig. 1(h). The large negative MR response sets in when the magnetic moments of Eu 2+ form AFM order, and disappears quickly above TN, indicating a clear correspondence between the negative MR and magnetic order of Eu sublattice. Previously, negative MR has been observed in some low-carrier density magnetic Eu-based materials [49][50][51][52]. The negative MR was ascribed to the formation of magnetic polaron, for example, in Eu5In2Sb6 [51] and EuB6 [52].
Nevertheless, ac magnetic susceptibility measurement in EuMnSb2 [42] does not support the existence of magnetic polaron, which excludes the magnetic polaron as possible origin of the negative MR.
Instead, the germane correlation between the charge-transport and magnetization implies that the large negative MR observed in EuMnSb2 is related to the magnetic order of Eu 2+ moments, which can be further demonstrated by the following band structure calculations.

B. Anisotropic magnetoresistance
Conspicuously, the MR also shows extremely anisotropic when the magnetic field rotates from the a axis, as shown in Fig. 2(a). The set-up geometry of our measurements is illustrated in the inset of Fig. 2(a). Here the electric current I is applied in the bc plane while magnetic field H is rotated from the a axis by angle θ with keeping perpendicular to I. All curves with varied angles show a negative MR, and the resistivity at high field increases drastically with θ increasing. Fig. 2(b) displays the AMR = [ρxx(θ)-ρxx(0°)]/ρxx(0°) [53,54] as a function of θ at 2 K under several representative magnetic fields.
It clearly shows a remarkably large AMR with maximum value 1.8410 6 % at 12 T, which exceeds previously reported AMR values in AFM materials by far. A complete AMR contour map with magnetic field larger than 5 T is presented in Fig. 2(c). The AMR shows two-fold symmetry and the 10 center located at H= 12 T and θ= 90°. Fig. 2(d) displays the magnetic-field-dependent AMR at 90°, and one can see that the AMR shows a relatively small value below 5 T, above which it increases rapidly and shows the maximum at 12 T. The AMR data at low magnetic fields are shown in Fig. S4 of the Supplementary Materials [48]. In fact, the relatively smaller AMR value at 5 T already reaches to ~800%.
To reveal the temperature dependence of the colossal AMR, we have measured the AMR at various temperatures under H= 12 T, as depicted in Fig. 2(e). The two-fold AMR decreases monotonically with increasing temperature. Fig. 2

C. Electronic structure calculations
To investigate the origin of the colossal AMR and the topological nature of our EuMnSb2 sample, we have calculated the electronic band structure with different Eu 2+ magnetic orders. First, we perform the band structure calculation of EuMnSb2 in the AFM state of Eu sublattice without SOC [ Fig. 3(a)].
The band structure at Y point shows a linear band dispersion with a tiny gap, indicating EuMnSb2 is a massive Dirac semimetal without considering SOC, just similar to the other members in the AMnPn2 family. Next, we performed band structure calculations considering canted A-type AFM and c-axisoriented A-type AFM orders of Eu sublattice with SOC, as shown in Fig. 3(b) and 3(c), respectively.
The resulted total energy in the former case is about 10 meV less than that in the latter, indicating that 11 the Eu sublattice in EuMnSb2 favors the canted A-type AFM order at zero magnetic field. As mentioned earlier, this finding is consistent with our magnetic susceptibility measurements.  Fig. 3(e) and 3(f), respectively. Remarkably, a strong modulation of the band structure with the reorientation of Eu 2+ moments is found: When the Eu 2+ moments are aligned along the a axis, the indirect band gap reduces to 23.3 meV; when the Eu 2+ moments are oriented in the c axis, the indirect band gap is found to be 62.0 meV, which is a bit larger than the value with Eu moments along the a axis. The change of band structure by the spin orientation of Eu 2+ moments in EuMnSb2 leads to significant magneto-transport response. At zero field, EuMnSb2 shows a relatively large band gap ~ 148.9 meV, resulting in semiconducting behavior of the resistivity below TN. When the magnetic field is applied along the a axis or c axis, the Eu 2+ moments orient toward the field direction progressively, giving rise to a large reduction of the band gap. This leads to the negative MR observed below TN. 13 Owing to the exchange interaction between Eu 2+ and Mn 2+ moments, a large magnetic field is needed to fully polarize the Eu 2+ moments (e.g., about 22 T in EuMnBi2 [28]). Therefore, the MR does not saturate up to 14 T as shown in Fig. 1(g) and 2(a), and should further increase until the Eu 2+ moments are fully polarized. To further establish the relationship between the band gap and magnetic structure, we have calculated the band gap values based on various magnetic orders of Eu sublattice with magnetic moments oriented θ away from the c axis in the ac plane, as shown in Fig. S7  rotating from the c axis to a axis. When the magnetic field applied along the a axis increases, the Eu 2+ moments trend to gradually polarize along the a axis (increasing θ), giving rise to a reduction of the band gap. As all the Eu 2+ moments are oriented along the a axis ferromagnetically (θ = 90°) , the smallest band gap with a value ~23.3 meV is achieved. We note that while the heuristic calculation considers A-type AFM order with Eu 2+ moments aligned along the c axis as the starting point, which differs from the canted A-type AFM order in real case, it serves to motivate the general evolution of band gap against external magnetic field in EuMnSb2. In addition, the fitted excitation gap with EA = 75.9 meV above 200 K is smaller than the calculated band gap of 148.9 meV in canted AFM state but larger than that in field-induced AF state. This indicates that the short-range magnetic correlation in high-temperature paramagnetic state above TN can also give rise to a reduction in the band gap, which results in the transport behavior above TN.

Ⅳ. DISCUSSION
We  Fig. 2(a)), which leads to a colossal AMR effect.
When the applied external field further increases above 12 T, the Fermi level is also across the conduction band around Y point with H//c. Then, a significant MR effect also appears with H//c, which leads to the decreasing of AMR effect above 12 T [ Fig. 2(d)]. Therefore, we think that there are two key factors to account for above colossal AMR effect. One is the magnetic-structure-induced anisotropic reduction of band gap. The other is that the position of Fermi level should be across the conduction band during the anisotropic reduction of band gap by applying magnetic field, which actually leads to a characteristic behavior for MIT in resistivity.
The second factor mentioned above also offers a natural explanation on the different behavior between our sample and the samples used in previous report. The zero-field resistivity of our EuMnSb2 sample shows a different low-temperature behavior from that in previous reports, as shown in Fig. S8 of the Supplementary Materials [48]. In previous reported EuMnSb2 grown by flux or floating-zone method [37,42,43], the low-temperature resistivity shows a metallic or weak insulating behavior.
However, in our EuMnSb2 single crystal, there is a strong insulating behavior at low temperature, especially below TN. Then, the value of resistivity at 2K in our samples becomes almost three order of magnitude larger than that in literatures. Our EuMnSb2 sample has much lower carrier density 9.6710 17 cm -3 (as shown in Fig. S9