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Gokul Acharya, Bimal Neupane, Chia‐Hsiu Hsu, Xian P. Yang, David Graf, Eun Sang Choi, Krishna Pandey, Md Rafique Un Nabi, Santosh Karki Chhetri, Rabindra Basnet, Sumaya Rahman, Jian Wang, Zhengxin Hu, [Bo Da](https://orcid.org/0000-0002-0785-8662), Hugh O. H Churchill, Guoqing Chang, M. Zahid Hasan, Yuanxi Wang, Jin Hu

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[Insulator‐to‐Metal Transition and Isotropic Gigantic Magnetoresistance in Layered Magnetic Semiconductors](https://mdr.nims.go.jp/datasets/1a2830e9-fec0-4495-a018-d7c07ce7395c)

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1  Insulator-to-Metal Transition and Isotropic Gigantic Magnetoresistance in Layered GdPS  Gokul Acharya1, Bimal Neupane2, Chia-Hsiu Hsu3,4 Xian P. Yang5, David Graf6, Eun Sang Choi6, Krishna Pandey7, Md Rafique Un Nabi1,8, Santosh Karki Chhetri1, Rabindra Basnet1,9, Sumaya Rahman1, Jian Wang10, Zhengxin Hu11,12, Bo Da11, Hugh Churchill1,7,8, Guoqing Chang3, M. Zahid Hasan5,13,14, Yuanxi Wang2, Jin Hu1,7,8  1Department of Physics, University of Arkansas, Fayetteville, AR, USA 2Department of Physics, University of North Texas, Denton, TX, USA 3Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore 4Quantum Materials Science Unit, Okinawa Institute of Science and Technology (OIST) Okinawa 904-0495, Japan  5Laboratory for Topological Quantum Matter and Advanced Spectroscopy (B7), Department of Physics, Princeton University, Princeton, NJ, USA. 6National High Magnetic Field Lab, Tallahassee, FL, USA 7Materials Science and Engineering Program, Institute for Nanoscience and Engineering, University of Arkansas, Fayetteville, AR, USA 8MonArk NSF Quantum Foundry, University of Arkansas, Fayetteville, Arkansas, USA   2  9Department of Chemistry & Physics, University of Arkansas at Pine Bluff, Pine Bluff, Arkansas 71603, USA 10Department of Chemistry and Biochemistry, Wichita State University, Wichita, KS, USA 11Center for Basic Research on Materials, National Institute for Materials Science (NIMS), Tsukuba-city, Ibaraki, Japan 12Department of Chemistry, University of Cambridge, Cambridge, United Kingdom 13Princeton Institute for Science and Technology of Materials, Princeton University, Princeton, NJ, USA. 14Lawrence Berkeley National Laboratory, Berkeley, CA, USA.   Magnetotransport, the response of electrical conduction to external magnetic field, acts as an important tool to reveal fundamental concepts behind exotic phenomena and plays a key role in enabling spintronic applications. Magnetotransport are generally sensitive to magnetic field orientations1–8. In contrast, efficient and isotropic modulation of electronic transport, which is useful in technology applications such as omnidirectional sensing, is rarely seen, especially for pristine crystals. Here we propose a strategy to realize extremely strong modulation of electron conduction by magnetic field which is independent of field direction. GdPS, a layered antiferromagnetic semiconductor with resistivity anisotropies, supports an field-driven insulator-to-metal transition with a paradoxically isotropic gigantic negative magnetoresistance insensitive to magnetic field orientations. This isotropic magnetoresistance originates from the combined effects of a near-zero spin-orbit coupling of   3  Gd3+-based half-filling f-electron system and the strong on-site f-d exchange coupling in Gd atoms. Our results not only provide a novel material system with extraordinary magnetotransport that offers a missing block for antiferromagnet-based ultrafast and efficient spintronic devices9,10 but also demonstrate the key ingredients for designing magnetic materials with desired transport properties for advanced functionalities.  Electronic transport is one of the most widely adopted techniques to uncover physical properties of materials. Owing to its sensitivity to density of states and carrier scatterings, applying a magnetic field modulates electronic transport via various mechanisms that are directly associated with the fundamental properties of materials. The competition of diverse contributing mechanisms leads the electrical resistivity to change with the direction and strength of the applied magnetic field. A variety of magnetotransport phenomena have been well-explored, such as weak (anti)localization2 quantum oscillations3 giant magnetoresistance (GMR)4, and colossal magnetoresistance (CMR)5. This venerable subject remains rather vibrant in contemporary condensed matter and materials physics research, not only because it provides an effective approach to understand the electronic properties of materials, but also owing to emergence of novel phenomena, such as chiral magnetic effect6–8 and extremely large positive magnetoresistance (MR) 7,8,11,12 in topological quantum materials.   Efficient modulation of electron conduction by magnetic field, which is manifested into resistance increase or decrease and characterized by MR, is scientifically interesting and technologically useful. Generally, MR is sensitive to the direction of the magnetic field with respect to the applied current. Examples include ordinary orbital MR due to Lorentz force that is sensitive to the angle   4  between field and current1, GMR caused by spin dependent scattering in magnetic multilayers and hence naturally anisotropic13, and chiral magnetic effects induced by the magnetic field component parallel to the current orientation6–8. In contrast, strong MR independent of magnetic field direction is much less common. Ferromagnetic metals and alloys show small MR anisotropy of a few percent, but the amplitude of MR is too small to be technological useful14. The traditional CMR effect in correlated materials is considered to be more isotropic than GMR effects, but it generally varies by at least 30% percent depending on magnetic field direction15,16. Recently, a few non-oxide materials with strong negative MR have been discovered, such as EuMnSb217, EuTe18, Mn3Si2Te619,20, and EuCd2P221. These materials all display substantial field-direction dependent MR.  Magnetotransport phenomena typically arise from modulations in carrier density and/or scattering by the magnetic field. Therefore, the dominance of anisotropic magnetotransport is not surprising, given that both the magnetic field and electron transport are directional. Anisotropic and strong MR has already found applications in electronic compasses, position and motion sensing, and hard disk drives. On the other hand, isotropic MR, if realized, can provide omnidirectional sensing capabilities, serving as a valuable complement to anisotropic MR-based devices. This synergy allows for the acquisition of comprehensive information for magnetic field sensing and magnetic imaging. Here we propose a strategy to create isotropic colossal MR independent of magnetic field direction in compounds with vanishing magnetic anisotropy, through an insulator-to-metal transition driven by the same-atom d-f exchange splitting. We present a magnetic semiconductor GdPS as such a case, in which the zero orbital angular momentum for f-electrons yields a near-zero magnetic anisotropy. The substantial magnetization from half-filled f-orbitals strongly splits   5  the Gd d-bands through an on-site exchange interaction, and results in a metallic state with well-defined Fermi Surface with unprecedented negligible anisotropy for MR.  Strategy toward colossal isotropic MR Isotropic magnetotransport is challenging from the perspective of electron dynamics as mentioned above. A more feasible approach is to couple magnetism with carrier scattering and/or scattering density in an isotropic manner. This approach requires isotropic magnetization which is usually challenging in crystalline materials because magnetic orders are generally characterized by magnetic easy and hard axes. Fortunately, this task is addressable since the mechanism of magnetic anisotropy has been well-explored. Spin-orbital coupling (SOC) and crystal field interactions play key roles in determining magnetic anisotropy of bulk crystalline materials22. Electron orbitals are oriented by the lattice due to electrostatic interaction, so that the coupling between electron orbits and spins gives rise to easy-axis or easy-plane anisotropy. Strong SOC and field angle- dependent magnetization can be utilized to create anisotropic MR 20. Here, minimizing SOC and the crystal field effect may lead to isotropic magnetism. Since this work aims to develop a universal approach to create isotropic magnetotransport, we look into the commonly seen magnetism arising from d- or f-electrons. d-electrons are generally more sensitive to crystal fields. Even though orbital angular momentum for d orbitals is usually quenched or partially quenched, SOC still creates residual orbital moment and strong magnetic anisotropy22,23. For f electrons that are well-screened from the crystal field, however, they are characterized by even stronger SOC and hence significant magnetic anisotropy is expected, which has been known to be a key enabling factor of rare earth-based permanent magnet22. An exception applies to the vanishing orbital angular momentum L for half-filled f-orbitals, where SOC is effectively zero and the magnetic anisotropy is mainly   6  attributed to shape anisotropy22. Therefore, starting from the common 3+ valence for rare earth ions, Gd3+ with a half-filled 4f 7 configuration offers the opportunity to realize isotropic magnetism.   Indeed, small magnetic anisotropic energy (MAE) has been found in some Gd compounds such as ferromagnetic GdN (0.45 µeV per Gd) and GeFe2 (9 µeV per Gd)24, but strong isotropic MR have yet been discovered. The next challenge is to establish an approach to couple transport with magnetism to realize isotropic and strong MR. In magnetic materials, in addition to CMR and GMR in a few materials that do not demonstrate isotropic MR as mentioned above, weak MR due to the field-modulation of spin scattering for charge carriers is more generally seen25,26. To realize strong and isotropic MR, field-induced metal-to-insulator (or vice versa) transition originating from electronic band reconstruction provides an effective approach. This can be realized via exchange splitting of electronic bands that results in gap closing or opening. Exchange interaction is sensitive to the net moment but not the direction of the moments. Therefore, as illustrated in Fig. 1, starting from an antiferromagnetic (AFM) ground state with zero net moment, finite magnetization induced by magnetic field turns on exchange splitting and modulates electronic bands. For isotropic magnetization, this band structure modulation would be isotropic with respect to the direction of the applied magnetic field. Because f-electrons are usually localized, here we pursue exchange interaction between f and orbitals near the Fermi energy (EF) from the same Gd3+ ions to maximize exchange splitting27. Since the exchange interaction splits spin up and down bands, it is easier to start from a gapped system and close the band gap by splitting bands near the EF, as opposed to opening a gap for a metallic system which may involve band structure complications. With this approach, field driven insulator-to-metal transition is expected.     7  Isotropic magnetism in GdPS To realize the above scenario, the band gap of the candidate material should be within the band shifts achievable by reasonable exchange splitting, while sufficient large to minimize the effect of inevitable defects, self-doping, and thermally excited carriers. Therefore, the candidate material should be an AFM semiconductor possessing a moderate band gap, f-electron magnetism and d-bands near EF. GdPS is a system meeting all the above needs with Gd3+ to ensure isotropic magnetism. GdPS was discovered decades ago28 but only the electronic structure has been explored theoretically29. This material adopts a layered crystal structure consisting of flat phosphorus planes sandwiched by Gd-S layers (Fig. 2a). It is structurally similar to the tetragonal ZrSiS29 with Si square-net planes30,31, with the difference that GdPS undergoes an orthorhombic structure distortion where the P plane fragments into an arm-chair chain-like structure, as shown in Fig. 2b. Earlier theoretical studies29 has shown that such a structural distortion plays a vital role in modulating the electronic phase. As shown in Fig. 2c, in tetragonal ZrSiS, the C2v symmetry leads to a Dirac nodal-line state that is characterized by linearly dispersed bands crossing near the EF, which is slightly gaped by weak SOC (~10 meV) from Si30,31. In GdPS (Fig. 2d), however, the arm-chair chain configuration of P atoms loses one mirror plane as compared to the square lattice, which strongly gaps the Dirac crossing29 (see Extended Data Fig. 1 for more details) and eventually leads GdPS to be a semiconductor.   DFT calculations (Supplementary Note 2) reveal that GdPS adopts an AFM ground state with a checkerboard (Néel)-type magnetic structure. Though the predicted magnetic structure is difficult to verify by neutron scattering due to the high neutron absorption of Gd, the AFM ground state is consistent with our magnetization measurements on GdPS single crystals. The AFM transition   8  temperature TN displays a slight sample variation between 7 and 7.5 K, which should be ascribed to the light P vacancies (see Methods). At the AFM ground state, DFT calculations estimate a moderate bandgap of ~0.5 eV (Fig. 2e), which is promising for realizing the desired insulator-to-metal transition as will be shown below.   The vanishing SOC of Gd3+, as well as the very weak ligand SOC from the light elements P and S, leads to the needed isotropic magnetism. As shown in Fig. 3a, despite the layered crystal structure, magnetizations measured under in-plane (H//ab) and out-of-plane (H//c) magnetic fields are almost identical for the temperature below (1.5 K) and above (10 K) TN. The isotropic magnetization persists to higher magnetic fields and saturates above ~15 T for both field orientations, implying a transition from an AFM to a polarized ferromagnetic (FM) state. The saturation moment of ~7.2 µB/Gd is consistent with the 8S7/2 multiplet for Gd3+. The moment polarization persists even above TN, with a nearly unchanged saturation moment at 10 K, as shown in the lower panel of Fig. 3a. Because sample rotation is difficult during high field magnetization measurements, tunnel diode oscillator (TDO) was adopted to examine the moment polarization with field orientations (Fig. 3b). As shown in Fig. 3c, Hp at 0.5 K displays very weak angular dependence. The observed isotropic magnetism in GdPS is consistent with the calculated small MAE of 30 µeV per Gd for the AFM ground state (see Methods). The MAE can be further reduced to only 2 µeV per Gd in the polarized FM state. The combined experimental and theoretical results indicate that the Gd moments may rotate freely with the external magnetic field, which is important for the isotropic MR as shown below.   Isotropic gigantic MR and Insulator-to-metal transition in GdPS   9  Given the large Gd3+ moment, the field-driven spin-polarization leads to strong exchange splitting that modifies the electronic structure. As shown in Fig. 2e, in the AFM ground state, the conduction band edge of the semiconducting GdPS consist of Gd d-bands around Γ and P p-bands around Y and T. In the polarized FM state, the strong net moment from Gd sublattice leads to exchange interactions that splits both bands, where the Gd d-band splitting at the conduction band edge is as large as 0.5 eV (Figs. 2f and 2g) owing to the same-atom d-f exchange, as compared to a much smaller band splitting of 0.1 eV for the P p-bands. The strong Gd d-band splitting at Γ lowers the energy of the spin-polarized band, replacing the P p-band edge at Y as the new conduction band minimum that approaches the EF. In real materials, when P vacancies increases EF, a Fermi surface from Gd d-band centered at Γ can emerge, leading to an insulator-to-metal transition which has indeed been observed in our magnetotransport experiments described below. Such a process is only determined by the strength of the exchange splitting, thus it is isotropic with respect to magnetic field due to isotropic magnetization of GdPS.  As shown in Fig. 4a, the in-plane resistivity (ρxx) measured with current following parallel to the phosphorus plane of GdPS (I // ab) increases with cooling, a typical non-metallic behavior for the gapped electronic structure in the AFM ground state. Applying a perpendicular magnetic field (H⊥ab) changes resistivity drastically, and eventually leads to a metallic-like temperature dependence with decreased resistivity upon cooling. The metal state is further verified by quantum oscillations under high magnetic fields as will be discussed later. Such an insulator-to-metal transition is accompanied by gigantic negative MR. At T = 2 K, resistivity drops drastically up on increasing field as shown in Fig. 4b. The corresponding MR, quantified by the field-induced resistivity change normalized to its zero-field value, reaches 97.3%. When normalized to the 9T   10  resistivity value following previous reports17,32,33, the MR is as high as 3,659%. The gigantic negative MR is reproducible in all samples we measured, reaching up to 99% (Extended Data Fig. 2 and Table 1). The strong negative MR occurs in a wide temperature and field region. As shown in Fig. 4b, at 2K, 50% reduction of resistivity is easily achievable under a magnetic field of 3 T, which is much lower than the spin polarization field of 15 T. Also, heating up to 100 K which is well above the magnetic ordering temperature, MR remains as high as nearly 50% at 9 T. This further confirms the exchange splitting scenario that is driven by net magnetization without a need for magnetic ordering or full moment polarization, making GdPS useful for practical applications where operation at high temperature and low field are preferable. As expected, MR in GdPS is isotropic. Figure 4c depicts the angular dependent MR at T = 5 K measured by rotating magnetic field both perpendicular to (-dependence) and within (φ-dependence) the phosphorus plane, which reveals that MR is essentially independent of the field orientation. This isotropic MR is reproducible and persists at high temperatures beyond TN, as shown in the Extended Data Fig. 3. To better quantify the isotropy of MR, we define the isotropic index by the ratio of MR maximum and minimum in the angular dependent measurements. Such definition takes the strength of MR into account so that it characterizes both isotropy and amplitude of MR. The calculated isotropic index is as small as 0.8% and 0.2% at 9T for out-of-plane (-) and in-plane (φ-) rotation, respectively.   Similar isotropic and gigantic MR is also present in out-of-plane electronic transport, where the electrical current following perpendicularly to the phosphorus plane, as shown in Figs. 4d-e. Compared to in-plane transport, the zero-field out-of-plane resistivity ρzz at low temperatures is much larger than ρxx by a factor of 35 (Fig. 4d), consistent with the layered crystals structure of   11  GdPS. However, ρzz is much more sensitive to magnetic field and drops strongly under a small magnetic field. As shown in Fig. 4e, at 2 K, negative MR up to 90% for ρzz can be induced by a small field of 1 T. Further increasing field to 9 T boosts MR to 99.99%, which converts to 740,167% when normalizing to the 9 T resistivity value. Similar to ρxx, strong negative MR in ρzz persists to high temperatures with a much greater value of 93.8% at 100 K and 9 T (Fig. 4e), and displays isotropic field orientation dependence with isotropic index < 0.01% for both out-of-plane and in-plane rotations (Fig. 4f). The isotropic MR in ρzz also persists at high temperatures, as shown in the Extended Data Fig. 4.  The isotropic and gigantic MR in both in-plane and out-of-plane magnetotransport drastically differs from other strong but anisotropic MR compounds such as EuMnSb217, EuTe18, Mn3Si2Te619,20, and EuCd2P221. The only known example of large, isotropic MR material is a disordered topological insulator with delicate composition tuning to allow a formation of a percolation of the current paths under field34. Therefore, GdPS represents the first example of isotropic gigantic MR in pristine crystals that originates from a well-defined insulator-to-metal transition. Quantitatively comparing in-plane and out-of-plane transport at low temperatures, one can find that ρzz is much larger at 0 T but becomes comparable with ρxx at 9 T. This suggests that electronic transport becomes more isotropic under a magnetic field, which can be understood by the evolution of electronic structure during the insulator-to-metal transition. Electronic structure at zero magnetic field can be probed by angle-resolved photoemission spectroscopy (ARPES). As shown in Figs. 5a-c, the ARPES observations are qualitatively consistent with the calculated band dispersions in Fig. 2e. At the Y point, a small electron pocket is seen, which can be attributed to P vacancies that electron-does GdPS. Despite its presence, this electron pocket is not large enough   12  to support full metallicity, as the non-metallic transport up to room temperature at zero field (Fig. 4a) implies our GdPS is a lightly doped semiconductor.   The electronic structure under magnetic fields, though unattainable by ARPES, can be resolved by quantum oscillations. Both Shubnikov–de Haas (SdH) oscillations in magnetotransport (Fig. 5e) and de Haas-van Alphen (dHvA) oscillations in TDO measurements (Fig. 3b and Extended Data Fig. 7) have been observed in our high field experiments. One typical example is shown in Fig. 5e, from which SdH effects in MR data can be clearly seen above ~17 T at T = 0.5 K. Importantly, SdH oscillation is observed for any magnetic field orientations ranging from out-of-plane ( = 0°) to nearly in-plane ( = 82.6°) directions, which indicates a well-defined 3D Fermi surface and rule out the possibility of quantum oscillation from surface states in insulators35,36 The oscillation frequency does not vary strongly with field orientations, growing by only 16% from 500 T for  = 0° to 580 T for  = 82.6°, as shown in Fig. 5f. Since the oscillation frequency is proportional to the extremal cross-sectional area perpendicular to the magnetic field orientation, such angular dependence implies a nearly spherical Fermi surface, which is consistent with the 3D Fermi surface at Γ from the DFT band structure. As discussed above (Fig. 2f), in the polarized FM state, the strong d-f exchange splitting may lead to a Fermi surface at Γ from the spin polarized Gd d-band. Indeed, assuming a shifted EF by 0.245 eV attributed to self-doping from the P vacancies, the calculated angular dependence of oscillation frequency matches well with the experimental observations, as shown in Fig. 5f. Aside from the Gd d-band, other possible bands such as the P p-band are not resolvable by quantum oscillations.    13  The nearly spherical Fermi surface at Γ emerging with increasing magnetic fields is further supported by the surprising quantum oscillation anomalies. Figure 5g depicts the contour plot showing the temperature and field dependencies of the SdH effect under an in-plane magnetic field. Interestingly, the locations of the oscillation maxima and minima shift systematically with varying temperature, as illustrated by the bent stripes in Fig. 5g. Typical data taken at 2 K and 12 K are shown in the inset, where misaligned oscillations are clearly seen. Such behavior is observed in multiple samples, occurring in both SdH and dHvA effects (Extended Data Fig. 7), which is caused by the temperature dependent oscillation frequency that drops from 560 T at 1.5K to 450 T at 18K, as shown in Fig. 5h. This appears inconsistent with the standard quantum oscillation theory based on Landau quantization of electronic systems with unchanged charge densities, but agrees well with the scenario of exchange splitting-driven insulator-to-metal transition: The emergence of Fermi surface is due to the exchange-split Gd d-bands that crosses the EF, hence the size of the Fermi surface is determined by the strength of the sample magnetization which reduces upon warming. Similarly, quantum oscillation frequency also evolves with magnetic field in the same manner as magnetization. As shown Fig. 5i, the oscillation frequency at 1.5 K grows with field with a saturation behavior. At elevated temperatures (6 and 10 K), the frequency is lower but the field dependence becomes even stronger. All of these match well with the magnetization of GdPS (Fig. 3a). With the electronic structures revealed by ARPES and quantum oscillation, the much stronger MR in out-of-plane transport can now be understood. At zero magnetic field, transport is governed by P layers as revealed by ARPES observations, so that ρzz is much greater than ρxx in layered GdPS. Applying magnetic field induce a 3D spherical-like Fermi surface from Gd d-bands, so that ρzz drops strongly and eventually the out-of-plane transport becomes comparable with the in-plane transport.   14   The layered crystal structure of GdPS allows exfoliation down to thin flakes (Extended Data Fig. 8). Thus, this material offers an excellent platform for device integration, especially for omnidirectional sensing. The AFM ground state with vanishing stray fields further enables ultrafast and high-efficient device response9,10. In addition, in the field-induced metallic state, the Fermi surface formed from the exchange-split Gd d-band is spin polarized. Therefore, controlling EF by chemical or electrostatic doping to remove possible contribution from the phosphorus bands may lead to a fully spin polarized Fermi surface, which opens vast opportunities for spintronic applications. Our finding highlights the effectiveness of utilizing exchange splitting-driven insulator-to-metal transition to realize isotropic and gigantic modulation of transport by magnetic field, which is enabled by suppressed crystal field effect in f-electron systems and near-zero SOC in half-filled f-orbitals to create isotropic magnetism, as well as an AFM and semiconducting ground state. Such a strategy can be generalized to discover more candidate materials by computational high-throughput screening followed by experimental optimization of material parameters by doping or strain to ensure band edge states by d-orbitals experiencing strong d-f exchange, as well as appropriate spin polarization fields and magnetizations at elevated temperatures to extend the phenomena towards much higher temperatures and low fields for technology applications.       15  Methods Sample preparation Single crystals of GdPS were grown by a direct chemical vapor transport with a stoichiometric ratio of Gd, P, and S as source materials and TeCl4 as the transport agent. The growths were performed in a dual heating zone furnace with a temperature gradient from 1075 to 975 °C for 3 weeks. The produced crystals typically have either needle-like or plate-like structures (see Supplementary Note 1), both of which were confirmed to be the GdPS phase by the composition analysis using energy-dispersive x-ray spectroscopy and the structure analysis using x-ray diffraction. The structure of the GdPS single crystal was determined by single crystal XRD. The structure parameters obtained from single crystal XRD refinement were given in Supplementary Note 1. A careful structure investigation found that the atomic displacement parameter of P atoms (0.0116(10) Å2) is greater than other atoms including Gd1 (0.0065(5) Å2), Gd2 (0.0070(5) Å2), S1 (0.0060(12) Å2), and S2 (0.0050(12) Å2) under the same refinement conditions, which indicated the possible presence of vacancy at P atomic sites. The refinement of atomic occupancy of P atoms revealed the occupancy of 0.95, which indicates the presence of P vacancy that is consistent with the electron doping scenario revealed by ARPES.  Angle-resolved photoemission spectroscopy  ARPES measurements were conducted out at Beamline 10.0.1 of the Advanced Light Source (ALS) in Berkeley, California, USA. The energy resolution was better than 30 meV and the temperature   16  for measurement was 90 K. The incident photon energy was 55 eV. Samples were cleaved in situ and measured under vacuum better than 5  10-11 Torr.  Band structure calculations The band structure was calculated using density-functional theory (DFT) as implemented in VASP37,38 using the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional [ref] and projector augmented wave (PAW) pseudo-potentials. The cut-off for the kinetic energy of plane was to be 520 eV and a 4x4x1 k-point mesh was used in the Brillouin zone. We used the Hubbard U parameter for the Gd-4f electron (Ueff Gd,5f = 7 eV). The total energy was converged at least 10-6 eV (10-7 eV for MAE) during electron configuration relaxation. Spin-orbit coupling (SOC) was included in all calculations unless otherwise stated. To model the paramagnetic state for GdPS (Extended Data Fig. 1), the calculation was performed by using a non-magnetic state.  Magnetic anisotropy energy calculations The calculation of the variation of MEA with polar angle θ and azimuthal angle φ provides stable structures as references for MAE calculations. As shown in Supplementary Note 4, we observed that in the case of AFM GdPS, the configuration with the magnetic moment oriented at θ = 45° and φ = 45° is slightly more stable than other AFM configurations. Similarly, in the case of FM GdPS, the configuration with the magnetic moment oriented at θ = 0° and φ = 0° is slightly more stable than other FM configurations. We consider these stable structures as references for MAE calculations.    17  Extremal Fermi surface cross-section area and associated quantum oscillation frequency To calculate the Fermi surface with the interpolation of the energy bands, the maximally localized Wannier functions (MLWF) for the ferromagnetic state of GdPS were obtained using the Wannier90 package39. Moreover, the dHvA frequencies were calculated by the SKEAF code40 with the Fermi surface information.  Measurements The magnetization and magnetotransport up to 9 T were measured using a physical property measurement system (PPMS, Quantum Design). The standard 4-probe measurement was used for magnetotransport. The high field magnetization measured with VSM (up to 35 T), magnetotransport measurements (up to 31 T and 45 T), and TDO measurements (up to 45 T) were performed at the National High Magnetic Field Laboratory (NHMFL) at Tallahassee.  Extracting Temperature- and Field-dependent quantum oscillation frequency Because the Fermi surface of the polarized FM state is formed from the exchange-split Gd d-bands, it evolves with magnetization and thus temperature- and field-dependent. Therefore, the corresponding quantum oscillation frequency, which is proportional to the extremal Fermi Surface cross-section area perpendicular to magnetic field direction, also changes with temperature and magnetic field. For field-dependence presented in Fig. 5i, the oscillation frequency cannot be extracted from the conventional Fourier transform because the oscillation data is in the field-domain. Therefore, to determine oscillation frequency at various magnetic fields, we evaluate the   18  distance between neighboring oscillation peaks, which is the oscillation period. The inverse of the period yield frequency for that field range, as shown in Extended Data Fig. 6c. For temperature dependence, the oscillation frequency is obtained from the Fourier transform of the oscillation pattern, as shown in Extended Data Fig. 6b, which is the convention method for quantum oscillation analysis. For temperature-dependent Fermi surface in GdPS, this method bears small errors. That is because Fourier transform is applied to a certain field range containing multiple oscillation periods, but the frequency changes with field in GdPS. Nevertheless, given that the oscillation amplitude damps with reducing magnetic field, the peak position in Fourier spectrum is mainly determined by the oscillation in the high field range. Indeed, frequency extracted from Fourier transform in Fig. 5h agrees with the high field (29 – 31T) frequency obtained by the distance between neighboring oscillation peaks in Fig. 5i.  Data availability All data that support the findings of this study are available from the corresponding authors on request. Source data are provided with this paper.  Acknowledgements J.H acknowledges the support by the U.S. Department of Energy, Office of Science, Basic Energy Sciences program under Grant No. DE-SC0022006 for crystal growth, transport and magnetic property measurements. J. H. thanks Z. Q. Mao at PennState and X.G. Zhang at UF for informative discussions. B.N. and Y.W. acknowledge startup funds from the University of North Texas, and computational resources from the Texas Advanced Computing Center. Part of the modeling was   19  supported by computational resources from a user project at the Center for Nanophase Materials Sciences (CNMS), a US Department of Energy, Office of Science User Facility at Oak Ridge National Laboratory, and also partially by user project R0076 at the Pennsylvania State University Two-Dimensional Crystal Consortium – Materials Innovation Platform (2DCC-MIP) under NSF cooperative agreement DMR-2039351. A portion of this work (high field transport, TDO, and magnetization) was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR-2128556 and DMR-2128556 and the State of Florida. C.-H. H and G.C appreciates National Research Foundation, Singapore, under its Fellowship Award (NRF-NRFF13-2021-0010), the Agency for Science, Technology and Research (A*STAR) under its Manufacturing, Trade and Connectivity (MTC) Individual Research Grant (IRG) (Grant No.: M23M6c0100), and the Nanyang Assistant Professorship grant (NTU-SUG). Work at Princeton University was supported by the Gordon and Betty Moore Foundation (GBMF4547 and GBMF9461; M.Z.H.). The ARPES work was supported by the US DOE under the Basic Energy Sciences program (grant number DOE/BES DE-FG-02-05ER46200; M.Z.H.). This research used resources of the Advanced Light Source, which is a DOE Office of Science User Facility under contract no. DE-AC02-05CH11231. The authors want to thank S.-K. Mo at Beamline 10.0.1 of the ALS for support in getting the ARPES data. M.Z.H. acknowledges support from Lawrence Berkeley National Laboratory and the Miller Institute of Basic Research in Science at the University of California, Berkeley in the form of a Visiting Miller Professorship. M.Z.H. also acknowledges support from the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Science Center. Work at Wichita (single crystal XRD and structure refinement) is supported by NSF under award OSI-2328822.   20  Work at NIMS is supported by JSPS KAKENHI Grant Number JP21K14656, the Kurata Grants from The Hitachi Global Foundation and from The Iketani Science & Technology Foundation.   Author contributions G.A, B.N, C.-H.H, and X.P.Y contribute to this work equally. G.A, K.P., S.K.C, and R.B grew the crystals. G.A, R.B., S.R, H.C performed electrical transport and low field magnetotransport experiments. G.A, M.R.U.N, S.K.C, D.G., and E.S.C conducted high field magnetotransport, TDO, and magnetization measurements. J.W., B.D, and Z.H performed structure analysis. B.N, Y.W. C.-H.H, and G.C performed computations. X.P.Y. and M.Z.H. performed ARPES investigations. J.H. and Y.W. conceived the project.  Competing interests The authors decares no competing interests.           21   References 1. Pippard, A. B. Magnetoresistance in Metals. (Cambridge University Press, 1989). 2. Lin, J. J. & Bird, J. P. Recent experimental studies of electron dephasing in metal and semiconductor mesoscopic structures. J. Phys.: Condens. Matter 14, R501 (2002). 3. Shoenberg, D. Magnetic Oscillations in Metals. (Cambridge University Press, 2009). 4. Grünberg, P. A. Nobel Lecture: From spin waves to giant magnetoresistance and beyond. Rev. Mod. Phys. 80, 1531–1540 (2008). 5. Ramirez, A. P. Colossal magnetoresistance. J. Phys.: Condens. Matter 9, 8171 (1997). 6. Burkov, A. A. Chiral anomaly and transport in Weyl metals. J. Phys.: Condens. Matter 27, 113201 (2015). 7. Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod. Phys. 90, 015001 (2018). 8. Bernevig, A., Weng, H., Fang, Z. & Dai, X. Recent Progress in the Study of Topological Semimetals. J. Phys. Soc. Jpn. 87, 041001 (2018). 9. Baltz, V. et al. Antiferromagnetic spintronics. Rev. Mod. Phys. 90, 015005 (2018). 10. Jungwirth, T., Marti, X., Wadley, P. & Wunderlich, J. Antiferromagnetic spintronics. Nature Nanotech 11, 231–241 (2016). 11. Liang, T. et al. Ultrahigh mobility and giant magnetoresistance in the Dirac semimetal Cd3As2. Nature Mater 14, 280–284 (2015). 12. Hu, J., Xu, S.-Y., Ni, N. & Mao, Z. Transport of Topological Semimetals. Annual Review of Materials Research 49, 207–252 (2019). 13. Steren, L. B. et al. Angular dependence of the giant magnetoresistance effect. Phys. Rev. B 51, 292–296 (1995). 14. McGuire, T. & Potter, R. Anisotropic magnetoresistance in ferromagnetic 3d alloys. IEEE Trans. Magn. 11, 1018–1038 (1975). 15. O’Donnell, J., Eckstein, J. N. & Rzchowski, M. S. Temperature and magnetic field dependent transport anisotropies in La0.7Ca0.3MnO3 films. Applied Physics Letters 76, 218–220 (2000). 16. Li, R.-W. et al. Anomalously large anisotropic magnetoresistance in a perovskite manganite. Proceedings of the National Academy of Sciences 106, 14224–14229 (2009).   22  17. Sun, Z. L. et al. Field-induced metal-to-insulator transition and colossal anisotropic magnetoresistance in a nearly Dirac material EuMnSb2. npj Quantum Mater. 6, 1–8 (2021). 18. Shapira, Y., Foner, S. & Oliveira, N. F. Hall effect in EuTe. Physics Letters A 33, 323–324 (1970). 19. Ni, Y. et al. Colossal magnetoresistance via avoiding fully polarized magnetization in the ferrimagnetic insulator Mn3Si2Te6. Phys. Rev. B 103, L161105 (2021). 20. Seo, J. et al. Colossal angular magnetoresistance in ferrimagnetic nodal-line semiconductors. Nature 599, 576–581 (2021). 21. Wang, Z.-C. et al. Colossal Magnetoresistance without Mixed Valence in a Layered Phosphide Crystal. Advanced Materials 33, 2005755 (2021). 22. Skomski, R. & Sellmyer, D. J. Anisotropy of rare-earth magnets. Journal of Rare Earths 27, 675–679 (2009). 23. Riseborough, P. S. Magnetic anisotropy of transition-metal impurities in metallic hosts. Phys. Rev. B 20, 1362–1368 (1979). 24. Abdelouahed, S. & Alouani, M. Magnetic anisotropy in Gd, GdN, and GdFe2 tuned by the energy of gadolinium 4f states. Phys. Rev. B 79, 054406 (2009). 25. Yamada, H. & Takada, S. Magnetoresistance of Antiferromagnetic Metals Due to s-d Interaction. J. Phys. Soc. Jpn. 34, 51–57 (1973). 26. Yamada, H. & Takada, S. Negative Magnetoresistance of Ferromagnetic Metals due to Spin Fluctuations. Progress of Theoretical Physics 48, 1828–1848 (1972). 27. Larson, P. & Lambrecht, W. R. L. Electronic structure of Gd pnictides calculated within the LSDA+U approach. Phys. Rev. B 74, 085108 (2006). 28. Hulliger, F. GdPS and Related New Rare Earth Compounds. Nature 219, 373–374 (1968). 29. Tremel, W. & Hoffmann, R. Square nets of main-group elements in solid-state materials. J. Am. Chem. Soc. 109, 124–140 (1987). 30. Schoop, L. M. et al. Dirac cone protected by non-symmorphic symmetry and three-dimensional Dirac line node in ZrSiS. Nat Commun 7, 11696 (2016). 31. Xu, Q. et al. Two-dimensional oxide topological insulator with iron-pnictide superconductor LiFeAs structure. Phys. Rev. B 92, 205310 (2015). 32. Yin, J. et al. Large negative magnetoresistance in the antiferromagnetic rare-earth dichalcogenide EuTe2. Phys. Rev. Materials 4, 013405 (2020). 33. Rosa, P. et al. Colossal magnetoresistance in a nonsymmorphic antiferromagnetic insulator. npj Quantum Mater. 5, 1–6 (2020).   23  34. Breunig, O. et al. Gigantic negative magnetoresistance in the bulk of a disordered topological insulator. Nat Commun 8, 15545 (2017). 35. Li, L., Sun, K., Kurdak, C. & Allen, J. W. Emergent mystery in the Kondo insulator samarium hexaboride. Nat Rev Phys 2, 463–479 (2020). 36. Ando, Y. Topological Insulator Materials. J. Phys. Soc. Jpn. 82, 102001 (2013). 37. Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169–11186 (1996). 38. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 77, 3865–3868 (1996). 39. Marzari, N. & Vanderbilt, D. Maximally localized generalized Wannier functions for composite energy bands. Phys. Rev. B 56, 12847–12865 (1997). 40. Rourke, P. M. C. & Julian, S. R. Numerical extraction of de Haas–van Alphen frequencies from calculated band energies. Computer Physics Communications 183, 324–332 (2012).      24  Figures  Figure 1 | Strategy for realization isotropic magnetoresistance. a, Schematic band structure for the ground state (zero magnetic field) of an AFM semiconductor insulator, showing a gap opening at the Fermi energy EF. Exchange energy Ex is zero due to the vanished net magnetic moment in AFM state, as illustrated in the bottom. b, Applying magnetic field polarizes the moments to form a FM state, creating strong net moment and turn on exchange interaction that split the band. Sufficiently strong exchange splitting drives a band to cross the EF, resulting in a metallic state. c, Isotropic magnetization. The magnetization (M) of the sample is induced and parallel to the external magnetic field (H), rotating in-phase with the field rotation.     25      Figure 2 | Electronic structure of GdPS. a, the orthorhombic crystal structure of GdPS. b, arm-chair chain structure of the P layers. c, the tetragonal structure with square-net layer leads to Dirac crossings. d, the orthorhombic distorted arm-chair chain-type P layer opens a gap at the Dirac point (see Extended Data Fig. 2 for detailed band structures). e, Electronic band structure of GdPS in the AFM ground state. The right panel shows the checkerboard (Néel)-type AFM magnetic structure for the ground state determined by calculations (see Supplementary Note 2). While the magnetic structure is determined, the Gd moments’ orientations are schematic, which is difficult to determine owing to the very weak MAE. f, Electronic band structure of GdPS in the polarized FM state. Magnetic structure for the FM state shows the polarization of Gd moments along the direction of magnetic field. g, Exchange splitting between spin up and down bands for the Gd d-band near Γ, which can be seen by overlaying the electronic structures of the AFM state (solid line) and the FM state (dashed line). The comparison of the entire electronic band structure is shown in Supplementary Note 3.          26      Figure 3 | Magnetic properties of GdPS. a, Field-dependent magnetization of GdPS at (top) 1.5 K and (bottom) 10 K, measured with field along the c-axis (perpendicular to the P plane) and within the ab-plane (parallel to the P plane). Magnetization exhibit saturation behavior at high field, indicating a polarized FM state. b, TDO measured at various magnetic field orientations. The transition-like feature near 15 T (indicated by the black arrow) signatures the moment polarization to an FM state. At high field, dHvA oscillation in TDO can be observed. c, Critical field for moment polarization, as a function of the magnetic field orientation, extracted from TDO measurement. The polarization field Hp is essentially field-orientation-independent.     27     Figure 4 | Gigantic, isotropic negative MR in GdPS. a, Temperature dependence for in-plane resistivity ρxx, measured under various magnetic field from 0 to 9 T. Inset: measurement setup. b, Magnetic field dependence for ρxx, measured at various temperatures from 2 to 300 K. c, Angular dependence for MR at T = 5 K for in-plane resistivity ρxx, measured by rotating field perpendicular to (θ-dependence, left) and within (φ-dependence, right) the phosphorus plane. Inset: schematic of magnetic field rotation. d, Temperature dependence for out-of-plane resistivity ρzz, measured under various magnetic field from 0 to 9 T. Inset: measurement setup. e, Magnetic field dependence of ρzz, measured at various temperatures from 2 to 300 K. f, Angular dependence for MR at T = 5 K for out-of-plane resistivity ρ, measured by rotating field perpendicular to (θ-dependence, left) and within (φ-dependence, right) the phosphorus plane. Inset: schematic of magnetic field rotation.      28   Figure 5 | Electronic structure of GdPS. a-d, Electronic band structure at zero field. a-b, APRES observation of the electronic band dispersion at 90 K along (a) Γ-X and (b) Γ-Y directions. An electron band slightly crosses the EF at Y, leading to a small Fermi surface at Y point as shown in c. d, Brillouin zone for GdPS. The black dotted square in c marks the surface Brillouin zone. e, Angular dependence for MR for out-of-plane resistivity ρ, measured by rotating field perpendicular to (θ-dependence, left) and within (φ-dependence, right) the phosphorus plane. Inset: schematic of magnetic field rotation. e-i, Electronic band structure of the polarized FM state. e, High field MR at T = 0.5 K for GdPS, measured at various magnetic field orientations. Upper inset shows the measurement schematic. Lower inset shows the SdH oscillation at high field. f, Field-angular dependence for SdH oscillation frequency obtained from data in e (see Extended Data Fig. 7). The observed angular dependence matches well with the calculated Fermi surface formed from the exchange-split Gd d-band at Γ, assuming shifted EF by 0.245 eV (see Methods). g, Contour plot showing the dependence of the SdH oscillatory component ρxxosc with temperature and inverse magnetic field, constructed from SdH effect at various temperatures shown in Extended Data Fig. 8). Inset: Oscillation pattern at 2 and 12K. h-i, (h) Temperature- and (i) field-dependence for SdH oscillation frequency of the same sample (see Methods for extracting frequencies). Results in e-i are from the identical sample measured using (e-f) a 41 T magnet and (g-i) a 31 T magnet.     29  Extended Data  Extended Data Fig. 1 | Symmetry breaking gap in GdPS. Calculated electronic band structure in the non-magnetic state (see Methods) for (a) a tetragonal ZrSiS-type structure with square nets of P-layers, and (b) an orthorhombic structure with arm-chair chain of P-layers. The middle panels show the electronic band structure with the intensity indicating contribution from P. The right panels display zoom-in of the Γ-X cut to highlight the band opening at the Dirac crossings at the EF. SOC is included in the calculation. Contribution from the Gd-f orbital is not counted for modeling the non-magnetic state.         30      Extended Data Fig. 2 | Reproducible gigantic, isotropic negative MR in multiple samples. a-b, Temperature and magnetic field dependence for in-plane resistivity ρxx for another two samples. c-d, Temperature and magnetic field dependence for out-of-plane resistivity ρzz for another two samples.       31  Extended Data Table 1 | Negative MR of various samples  Samples Source Resistivity at 2K, 0T, ρ(H=0)  (mΩ cm) Resistivity at 2K, 9T, ρ(H=9T)  (mΩ cm) MR at 2K, normalized to ρ(H=0) 𝝆(𝑯 = 𝟎) − 𝝆(𝑯 = 𝟗𝐓)𝝆(𝑯 = 𝟎) MR at 2K, normalized to ρ(H=9T) 𝝆(𝑯 = 𝟎) − 𝝆(𝑯 = 𝟗𝐓)𝝆(𝑯 = 𝟗𝐓) ρxx, S#1 Fig. 3a 31.2 0.83 97.3% 3,659% ρxx, S#2 Extended Data Fig. 4 208.2 2.1 99.0% 9,814% ρxx, S#3 Extended Data Fig. 4 74.3 1.4 98.1% 5,207% ρzz, S#1 Fig. 3d 1,110.4 0.15 99.99% 740,167% ρzz , S#2 Extended Data Fig. 4 640 0.72 99.9% 88,788% ρzz , S#3 Extended Data Fig. 4 560 0.3 99.9% 185,667%     32    Extended Data Fig. 3 | Reproducible gigantic, isotropic negative MR at high temperatures for in-plane transport. a-b, Top: field dependence for in-plane resistivity ρxx at (a) T = 10 K and (b) T = 50 K, measured by applying magnetic field along different directions perpendicular to the phosphorus plane of GdPS (various θ angles, see inset). Bottom: corresponding angular dependence for MR at different fields at 10 and 50 K. c-d, similar measurements to a-b for the identical sample, but with various in-plane field orientations (φ angles, see inset). c-d, similar measurements to a-d, but for a different sample. The two samples display different amplitudes for MR, but both are gigantic and isotropic.     33     Extended Data Fig. 4 | Reproducible gigantic, isotropic negative MR at high temperatures for in-plane transport. a-b, Top: field dependence for in-plane resistivity ρzz at (a) T = 10 K and (b) T = 50 K, measured by applying magnetic field along different directions perpendicular to the phosphorus plane of GdPS (various θ angles, see inset). Bottom: corresponding angular dependence for MR at different fields at 10 and 50 K. c-d, similar measurements to a-b for the identical sample, but with various in-plane field orientations (φ angles, see inset). c-d, similar measurements to a-d, but for a different sample. The two samples display different amplitudes for MR, but both are gigantic and isotropic.     34     Extended Data Fig. 5 |Field-angle dependent quantum oscillation. a, The oscillatory components of SdH oscillations in ρxx under various magnetic field orientations presented in Fig. 4e, extracted by removing the non-oscillatory MR background. b, Fast Fourier transform (FFT) for the oscillatory components in a. Data are shifted for clarity.      35     Extended Data Fig. 6 | Temperature- and field-dependences for quantum oscillation. a, SdH oscillatory components at various temperatures, measured with field along the in-plane direction (θ = 90°), and used to construct the contour plot in Fig. 4g. Data are shifted for clarity. b, Fast Fourier transform (FFT) for the oscillatory components in a, from which the temperature dependent oscillation frequency shown in Fig. 4h is obtained. Data are shifted for clarity. c, Oscillation patterns at 1.5, 6, and 10 K extracted from a. The distance between the neighboring oscillation peaks, which is the SdH oscillation period, changes with magnetic field. The field-dependent oscillation frequency presented in Fig. 4i is obtained from inversing the period (see Methods).     36     Extended Data Fig. 7 | Temperature- and field-dependences for SdH and dHvA effects in additional samples. a, Field dependence for in-plane resistivity at various temperatures for a sample different from that in Fig. f-j, measured with field along the in-plane direction (θ = 90°). The corresponding SdH oscillation components are shown in b. c, Contour plot showing the temperature and field dependencies for SdH effect, constructed using data in d. Field dependence for TDO frequency at various temperatures, measured with field along the out-of-plane (θ = 0°) direction, The corresponding dHvA oscillation components are shown in e. f, Contour plot showing the temperature and field dependencies for dHvA effect, constructed using data in e.      37    Extended Data Fig. 8 | GdPS flakes. Exfoliated GdPS thin flakes on Si wafer. Scare bar: 10 μm.