Strain-tunable Berry curvature in quasi-two-dimensional chromium telluride

Magnetic transition metal chalcogenides form an emerging platform for exploring spin-orbit driven Berry phase phenomena owing to the nontrivial interplay between topology and magnetism. Here we show that the anomalous Hall effect in pristine Cr2Te3 thin films manifests a unique temperature-dependent sign reversal at nonzero magnetization, resulting from the momentum-space Berry curvature as established by first-principles simulations. The sign change is strain tunable, enabled by the sharp and well-defined substrate/film interface in the quasi-two-dimensional Cr2Te3 epitaxial films, revealed by scanning transmission electron microscopy and depth-sensitive polarized neutron reflectometry. This Berry phase effect further introduces hump-shaped Hall peaks in pristine Cr2Te3 near the coercive field during the magnetization switching process, owing to the presence of strain-modulated magnetic layers/domains. The versatile interface tunability of Berry curvature in Cr2Te3 thin films offers new opportunities for topological electronics.

An important consequence of the band topology in Cr 2 Te 3 is the Berry curvature [17,18] underlying the anomalous Hall effect (AHE) [19].The intrinsic AHE is topological in nature and a hallmark of itinerant ferromagnets, which has also been observed in more exotic systems even without a net magnetization, such as spin liquids [20], antiferromagnets [21] and Weyl semimetals [22].When SOC coexists with long-range magnetic order, the Berry curvature can be significantly influenced near avoided band crossings, rendering the system an incredibly rich playground combining topology and magnetism [23,24].
Here, we report the unique magnetotransport signatures of high-quality quasi-2D Cr 2 Te 3 MBE grown thin films governed by non-trivial band topologies.Via synergetic structural, magnetic and transport measurements, together with first-principles simulations, we have uncovered novel Berry-curvature-induced magnetism featuring an extraordinary sign reversal of the AHE, as we modulate the temperature and the strain for the thin films containing 3 to 24 unit cells (u.c.) on Al 2 O 3 (0001) or SrTiO 3 (111) substrates.Moreover, a hump-shaped Hall feature emerges, most likely due to the presence of multiple magnetic domains under different levels of inter- facial strain.This work identifies pristine ferromagnetic Cr 2 Te 3 thin films as a fascinating platform for further engineering topological effects given their nontrivial Berry curvature physics.Atomic structure, interfaces and strain.
The crystalline structure of Cr 2 Te 3 thin films is described first, followed by the development of strain at the substrate/film interface by the epitaxy.Bulk Cr 2 Te 3 crystalizes in the structure with space group P 31c (D 2 3d , No. 163), as shown in Figs.1a-c, where each unit cell contains four vertically stacked hexagonal layers of Cr [25].There are three symmetrically unique sites for Cr, labeled Cr1, Cr2 and Cr3, respectively: The Cr1 atoms are sparsely arranged in a weakly antiferromagnetic sublattice [26], while the Cr2/Cr3 atoms form ferromagnetic layers similar to those in CrTe 2 [27].Since the Cr1 sites are often only partially filled (Figs.1i-m), Cr 2 Te 3 behaves essentially as a quasi-2D magnet [28][29][30].This quasi-2D nature of Cr 2 Te 3 allows for high-quality, layer-by-layer epitaxial growth of c-oriented films on a variety of substrates.The hexagonal c axis is the easy magnetic axis, leading to PMA for the films.
The six-fold in-plane (IP) symmetry is seen in the honeycombs visualized by atomic resolution scanning tunnel-ing microscopy (STM, Fig. 1e) and scanning transmission electron microscopy (STEM, Fig. 1g) high-angle annular dark-field (HAADF) imaging, as well as in the reflection high-energy electron diffraction (RHEED, Supplementary Fig. 1) and X-ray diffraction (XRD, Supplementary Fig. 2) patterns.The sharp substrate/film interface is confirmed by the cross sectional HAADF (Fig. 1i) and the corresponding integrated differential phase contrast (iDPC, Fig. 1j) images.The intrinsic random distribution of Cr atoms on the Cr1 sites is resolved in the enlarged view of the atoms in Figs.1k-m, shown overlaid with red circles, while the overall chemical composition of the thin film is uniform within the resolution of energy dispersive X-ray spectroscopy (EDS, see Supplementary Fig. 3).
Figure 1f illustrates the basic sample architecture, where the strain in the Cr 2 Te 3 thin films is governed by the interface with the substrate.Upon reducing the thickness (t), films grown on Al 2 O 3 (0001) can develop an IP compressive strain up to −0.15%, as determined by XRD and summarized in Fig. 1d.A higher strain level can be sustained using SrTiO 3 (111) substrates.Such control of strain is well suited for exploring interfacesensitive properties in Cr 2 Te 3 thin films.Magnetism, domains and PNR.
The magnetic properties of Cr 2 Te 3 thin films with selected thicknesses were assessed using vibrating sample magnetometry (VSM).Figure 2a shows the temperature dependence of the magnetization M (T ) for a t = 24 u.c.film on Al 2 O 3 (0001) substrate with an out-of-plane (OOP) applied magnetic field µ 0 H = 0.1 T.Under the field-cool (FC) condition, M (T ) rises below the Curie temperature T C ∼ 180 K, reaching M ∼ 2.50 µ B (Bohr magneton) per Cr at 2 K in the 0.1 T field.The zerofield-cool (ZFC) scan on the other hand deviates from the FC curve below the blocking temperature T b , signaling the freezing out of domains in random direction in the absence of an aligning field H.
As illustrated in Fig. 2b, Cr 2 Te 3 favors PMA with coercive field µ 0 H c = 0.76 T and saturation magnetization M s ∼ 2.83 µ B per Cr at 2 K for t = 24 u.c., whereas the IP measurements have weaker ferromagnetic hysteresis loops.The low-T zero-field kink in the OOP M (H) becomes more prominent at reduced t (for additional data on t = 6 u.c., see Supplementary Fig. 4), indicating the presence of interfacial strain-induced multiple magnetic domains with continuously varying spin canting [31][32][33].
Depth-sensitive polarized neutron reflectometry (PNR) measurements, responsive to the IP magnetization, were carried out at chosen T and H on samples with t = 24 and 6 u.c., in order to uncover the impact of interfacial strain.The PNR spin asymmetry ratio SA = (R + − R − )/(R + + R − ), measured as a function of the wave vector transfer Q = 4π sin(θ)/λ with R + and R − being the reflectivity for the neutron spin parallel (+) or antiparallel (−) to the external field, evidently confirms the magnetization (Supplementary Fig. 5).By simultaneously refining PNR and X-ray reflectivity (XRR, Supplementary Fig. 5) data, the depth profiles of nuclear (NSLD) and magnetic (MSLD) scattering length densities (SLD) at µ 0 H = 1 T, 0.8 T and 0.05 T for t = 24 u.c. were obtained and shown in Fig. 2c.The uniform MSLD profile at the IP saturation field µ 0 H = 1 T attests to the high quality of the magnetic Cr 2 Te 3 layer with well-defined interfaces of 0.5 nm.Remarkably, at a reduced IP field µ 0 H = 0.8 T and 0.05 T, M develops a non-uniform depth-dependent magnetization profile with two components, revealing a lower (higher) value close to (away from) the substrate.Since PNR is sensitive to IP magnetization, these results collectively suggest that more pronounced strain at the interface leads to a higher OOP magnetic anisotropy and hence a lower measured IP MSLD.This scenario is further substantiated by the lower M observed for t = 6 u.c. with stronger strain measured at 5 K and 60 K under 1 T IP magnetic field (Supplementary Fig. 4d).The salient structural and magnetic features pave the way for an in-depth investigation of the magneto-transport responses in Cr 2 Te 3 thin films.

Strain-tunable AHE and sign reversal.
The unusual Hall effects are the most outstanding properties of the Cr 2 Te 3 thin films.The development of long-range magnetic ordering is manifested in the AHEinduced hysteresis in the Hall resistivity in Fig. 3a (for more details on the transport parameters, see Supplementary Fig. 6).Here R H characterizes the linear-in-H ordinary Hall effect (OHE) that dominates at high H and R S is the AHE coefficient denoting contribution from the underlying magnetic order.
) By removing the linear OHE background in Fig. 3a, we now turn to the rich T and H dependences of the Hall traces ∆ρ yx (H) and the unconventional AHE in the ferromagnetic regime in Fig. 3b.For t = 6 u.c., at T 30 K, when fully magnetized under a positive H, the system produces a negative AHE signal ρ AHE , i.e., ∆ρ yx (H) loops around the origin in the opposite direction of that for the M (H) hysteresis (see Supplementary Fig. 4b).The T dependence of the corresponding anomalous Hall conductivity σ AHE = ρ AHE /(ρ 2 AHE + ρ 2 xx ), with ρ xx being the longitudinal electrical resistivity, is summarized in Fig. 3c.Upon rising T , ρ AHE surprisingly changes sign at a transition temperature T S ∼ 40 K for t = 6 u.c..Note that the sign change signifies a compensation point at T S where ρ AHE or the anomalous Hall conductivity σ AHE traverses through zero while M remains finite (see Supplementary Fig. 7b).This is a highly unusual behavior, although rather similar to the transport anomaly present in SrRuO 3 as a result of the nontrivial band topology [23].Furthermore, this unique sign reversal behavior of the AHE is sensitively governed by the interfacial strain (Fig. 3d).As evident in Fig. 3e, T S largely decreases upon increasing compressive strain at reduced t (Fig. 1d).At t = 3 u.c., the strain is found to be sufficient in driving σ AHE > 0 in the ground state, leading to the absence of a temperature-induced sign flipping at finite T .
To elucidate the physical origin of the AHE sign reversal of Cr 2 Te 3 , we examined the Berry curvature Ω z (k) = n f n Ω z n (k) (Fig. 4a, summed over the occupied bands with f n the equilibrium Fermi-Dirac distribution function) based on the electronic band structure (Fig. 4b) obtained using density functional theory (DFT).As exemplified by the left inset of Fig. 4a, a significant spike feature develops in Ω z (k), originating from the nearly degenerate SOC anti-crossing bands along the A-L k-path.The intrinsic AHE conductivity is evaluated by integrating over the Brillouin zone (BZ) where e is the electron charge and is the reduced Planck's constant.As shown in Fig. 4c, the calculated σ AHE = −12.7 Ω −1 cm −1 at the Fermi level ε F for Cr 2 Te 3 under equilibrium state (the black curve in Fig. 4c, see also Supplementary Fig. 8 for the convergence test under different k-mesh), which is in excellent agreement with the experimental value of −11.5 Ω −1 cm −1 for t = 24 u.c..It attests to the dominance of the intrinsic Berry phase mechanism, rather than the extrinsic side jump or skew scattering [19], as the primary origin of the observed AHE in Cr 2 Te 3 .The calculation also reveals a sensitive energy dependence of σ AHE -not only the magnitude but also the sign changes near ε F .At finite T , due to the thermal broadening in f n , the slight asymmetry of σ AHE above and below ε F , naturally explains the experimentally observed AHE sign anomaly.Modeling of strained cases in Fig. 4c further reveals that σ AHE at ε F changes sign under −1% compressive strain, substantiating that Berry physics underlies the observed strain-driven AHE sign reversal in Fig. 3d.This unique capability of achieving zero σ AHE or ρ AHE while maintaining nonzero M in Cr 2 Te 3 thin films, deviating from the classic Eq. ( 1), offers direct insight into the intrinsic AHE solely owing to the Berry curvature [19,34].Left inset in a, nearly degenerate SOC anti-crossing bands contributing to the sharp peak in Ω z (k) along A-L.c, Anomalous Hall conductivity σAHE near the Fermi level εF, in equilibrium state (black), under compressive (blue) or tensile (red) strain conditions, respectively.The shades in c are guide for the eye showing the slight asymmetry of the energy dependence of σAHE above and below εF which at finite T may lead to a sign reversal in σAHE owing to thermal broadening.

Hump-shaped Hall peaks at the coercive field.
Figure 3b also shows additional hump-shaped peaks on top of the otherwise square AHE hysteresis loop.The peaks are centered at the characteristic fields H peak that tracks well with the coercive fields H c determined from the magnetic measurements (Supplementary Fig. 7).These hump-shaped Hall peaks in our pristine Cr 2 Te 3 are related to the presence of strain-modulated magnetic multidomain structures with opposite signs of AHE (Fig. 5a), rather than the skyrmion-induced topological Hall effect as postulated in various related phases and heterostructures [35][36][37][38][39][40][41].To better understand the mechanism(s) underlying the Hall peaks observed in ∆ρ yx (H), minor loop experiments were carried out for t = 6 u.c. at T = 30 K and are shown in Fig. 5b.For each scan, the loop starts from a well-defined initial state that is fully magnetized under a positive H, which is then swept towards a negative H min around −H peak and scanned back to the initial positive H.The minor loops are hysteretic, where the emergence of the Hall peak with positive H depends on whether H min surpasses −H peak .The two-component origin of the Hall anomaly peaks can be quite well explained by a distribution of domains having T -dependent H c using [42] ∆ρ (3) Here ρ AHE (T ) and H c (T ) are functionals based on experimental ρ AHE and H c (Fig. 3c and Supplementary Fig. 7), H Heav (x) is the Heaviside function approximat-ing the switching of M , and the Gaussian distribution characterizes the strain-driven distribution of domains with varying T S by assuming an effective temperature spreading factor T σ .As compared in Fig. 5c, the numerical simulation indeed reproduces qualitatively well the behavior of the minor loops.The observed AHE sign change and the emergence of hump-shpaed Hall features are also present in films grown on SrTiO 3 (111) (Supplementary Fig. 9).The quality of the substrate/film interface plays a pivotal role in materializing this exquisite tunability of the Berry curvature in Cr 2 Te 3 films.In summary, we have discovered several unusual Berry curvature driven effects in the anomalous Hall transport of Cr 2 Te 3 thin films.We report on the growth, detailed magnetic and transport properties of pristine Cr 2 Te 3 MBE thin films deposited on Al 2 O 3 (0001) and SrTiO 3 (111) substrates.A striking sign reversal in the anomalous Hall resistivity, accompanied by a finite magnetization, has been observed and theoretically modeled, revealing the relevance of the nontrivial Berry curvature physics.This unique sign reversal, coupled with the intrinsic strain-modulated magnetic domains in the material, induces a hump-shaped Hall feature in Cr 2 Te 3 thin films.The Berry curvature effect is observed in this case due to the high quality of the substrate/film interface, which is further tunable via different level of  strain given by varying film thickness and/or choice of substrates.Our comprehensive experimental and theoretical investigations establish Cr 2 Te 3 to host tunable topological effects related to the intrinsic Berry curvature, thereby providing new perspectives in the field of topological electronics.

METHODS
Sample growth.The growth of Cr 2 Te 3 thin films, with nominal t ranging from 3 -24 u.c. were carried out in a MBE system under an ultrahigh-vacuum (UHV) environment of 10 −10 − 10 −9 Torr.Insulating Al 2 O 3 (0001) was primarily used as substrate, whose surface quality was insured by ex situ chemical and thermal cleaning and in situ outgassing at 800 • C for 30 min.When using SrTiO 3 (111), the insulating substrates were first annealed at 930 • C for 3 hours in a tube furnace under flowing oxygen environment to achieve passivated surface with atomic flatness and then in situ outgassed at 580 • C for 30 min.After surface preparation, the substrate temperature was lowered to 230 • C for film growth, allowing enough surface mobility for the epitaxial crystallization of the desired phase of Cr 2 Te 3 .High-purity (5N) Cr was evaporated from an electron-beam source, while Te was thermally co-evaporated from a Knudsen effusion cell ad-justed to maintain a typical Cr:Te flux ratio of 1:10 and a growth rate of approximately 0.005 nm s −1 .The epitaxial growth process was monitored by in situ RHEED (see Supplementary Fig. 1) operated at 15 kV.The as-grown films were in situ annealed at the growth temperature for 30 min, and naturally cooled to room temperature.For ex situ characterizations, films were protected by in situ capping with Te (2 nm) and AlO x (10 nm) or Se (20 nm) for later removal for STM measurements.The schematic of film stack is illustrated in Fig. 1f.
Structural characterizations.The XRD patterns were obtained using a parallel beam of Cu K α1 radiation with wavelength λ = 0.15406 nm in a Rigaku Smart-Lab system.The 2θ (for OOP measurement) and/or 2θ χ (for IP configuration) scan angles were between 10 • and 120 • with a typical step size of 0.05 • .XRR measurements were performed at the Center for Nanophase Materials Sciences (CNMS), Oak Ridge National Laboratory, on a PANalytical X'Pert Pro MRD equipped with hybrid monochromator and Xe proportional counter.For the XRR measurements, the X-ray beam was generated at 45 kV/40 mA, and the X-ray beam wavelength after the hybrid mirror was λ = 0.15406 nm (Cu K α1 radiation).To facilitate electron microscopy, planner samples were deposited on Si 3 N 4 TEM grids with thin Sb 2 Te 3 buffer while cross-sectional samples were prepared using focused ion beam (FIB) lift-out method on a Thermo Scientific FEI Quanta 3D dual beam system.STEM imaging was carried out on a Thermo Scientific FEI Titan aberrationcorrected system operated at 200 kV.A semi-convergence angle of 17.9 mrad was used.DPC and iDPC images were recorded using a segmented detector.For the 3 u.c.sample, STEM images were acquired with a Themis Z G3 instrument provided by Thermo Fischer Scientific at 200 kV with a beam current of 40 pA and a convergence angle of 25 mrad.
Scanning tunneling microscopy.STM experiments were performed at the Laboratory for Physical Sciences using a home-built low-temperature scanning tunneling microscope [43] controlled by a Topometrix digital feedback electronic control unit.Samples were loaded into an UHV environment with a base pressure of 5 × 10 −10 Torr and heated in front of a residual gas analyzer to verify the removal of the Se capping layer before being transferred to the microscope at 77 K. Scans were performed with an electrochemically-etched tungsten tip and differential spectroscopy data were extracted via a Stanford Research Systems SR830 lock-in amplifier.
Polarized neutron reflectometry.PNR is a highly penetrating depth-sensitive technique to probe the chemical and magnetic depth profiles with a resolution of 0.5 nm.The depth profiles of the NSLD and MSLD correspond to the depth profile of the chemical and IP magnetization vector distributions on the atomic scale, respectively [44][45][46].Based on these neutron scattering merits, PNR serves as the powerful technique to simulta-neously and nondestructively characterize chemical and magnetic nature of buried interfaces [47].PNR experiments were performed on the Magnetism Reflectometer at the Spallation Neutron Source at Oak Ridge National Laboratory [48][49][50], using neutrons with wavelengths λ in a band of 0.2 -0.8 nm and a high polarization of 98.5-99%.Measurements were conducted in a closed cycle refrigerator (Advanced Research System) equipped with a 1.15 T Bruker electromagnet.Using the time-of-flight method, a collimated polychromatic beam of polarized neutrons with the wavelength band δλ impinges on the film at a grazing angle θ, interacting with atomic nuclei and the spins of unpaired electrons.The reflected intensity R + and R − are measured as a function of momentum transfer, Q = 4π sin(θ)/λ, with the neutron spin parallel (+) or antiparallel (−), respectively, to the applied field.To separate the nuclear from the magnetic scattering, the spin asymmetry ratio SA = (R + − R − )/(R + + R − ) is calculated, for which SA = 0 designating no magnetic moment in the system.Being electrically neutral, spinpolarized neutrons penetrate the entire multilayer structures and probe magnetic and structural composition of the film and buried interfaces down to the substrate.
Transport and magnetic measurements.Electrical transport measurements as a function of of temperature, field and angle were performed in the temperature range of 2 − 300 K in a Quantum Design Physical Property Measurement System (PPMS) equipped with a 9 T superconducting magnet.A typical ac current (I x ) of 5 µA was injected into the Hall bar (∼ 0.3 × 1.0 mm 2 for hand-scratched or 10 × 30 µm 2 for e-beam patterned) residing in the crystallographic a-b plane, while longitudinal (V x ) and transverse (V y ) voltages were simultaneously monitored using a lock-in technique.For aligning the magnetic field H, a horizontal rotator was used with an angular resolution of ∼ 0.1 • .VSM was used to characterize the magnetization, where linear diamagnetic backgrounds from sample holders/substrates were subtracted to obtain M (H) and M (T ).
Theoretical simulations.First-principles calculations were performed using the Quantum Espresso packages [51].The generalized gradient approximation with the Perdew-Burke-Ernzerhof parameterization (GGA-PBE) was used as the exchange-correlation functional [52].An energy cutoff of 40 Ry and a 6 × 6 × 4 Γ-centered k-mesh were applied for the relaxation calculation.The crystal structure of Cr 2 Te 3 was fully optimized until the force on each atom is smaller than 0.05 eV nm −1 .The optimized lattice constants of bulk Cr 2 Te 3 are a = b = 0.6799 nm and c = 1.2022 nm.For the self-consistent field calculation, SOC was included and a higher 12×12×8 kmesh was used.The magnetization was set along the −z axis.The resulting absolute magnetic moments of the Cr atoms are 3.08, 2.99 and 3.06 µ B for Cr1, Cr2 and Cr3, respectively.For the Berry curvature and anomalous Hall conductivity calculations, the Wannier90 packages were used [53].Maximally localized Wannier functions (MLWF) including both Cr d-orbitals and Te p-orbitals were employed to reproduce the DFT-calculated band structure with SOC.

Figure 1 |
Figure 1 | Crystal structure of Cr2Te3 thin films.a, Atomistic structure of Cr2Te3 viewed along the crystallographic [210] direction.b, Among the three Cr species, Cr1 (red) form sparse honeycombs that are stacked between those of Cr2/Cr3 (purple/blue) with six-fold in-plane symmetry (c).d, Enhanced in-plane compressive strain at reduced thickness t, quantified by the relative change of the a lattice parameter via XRD for Cr2Te3 grown on Al2O3(0001) (solid) or SrTiO3(111) (open).f, Schematic of the film stacks, where the interfacial strain plays a pivotal role in inducing extraordinary magnetic and transport phenomena.Atomically resolved STM morphology of a 13 × 13 nm 2 surface after removing Se capping (e) and planar HAADF STEM image (g) of Cr2Te3 confirm the honeycomb-like Te lattice, where the HAADF intensity line scan (h) reveals the Cr sites.i-m, Cross-sectional images of Cr2Te3 films grown on SrTiO3(111).The HAADF (i) and iDPC (j) imaging along the [210] zone axis of Cr2Te3 illustrates the dominating Te-Cr2/Cr3-Te layers.The enlarged view (dashed box region in i) of HAADF (k), DPC (l), and iDPC (m) images identify the random distribution of the interlayer Cr1 (circles), which deviates from the ideal Cr2Te3 structure with full occupancy.The color wheel in the DPC image indicates the projected electric field direction.

Figure 2 |
Figure 2 | Magnetic properties of Cr2Te3 thin films.a, Temperature dependence of the magnetization M of a typical 24 u.c.Cr2Te3 film under the zero-field-cool (ZFC) and field-cool (FC) conditions with an out-of-plane (OOP) external magnetic field µ0H = 0.1 T. The Curie (TC) and blocking (T b ) temperatures are labeled by the arrows.b, Field dependence of M under OOP and in-plane (IP) configurations for t = 24 u.c. at selected temperatures (top three, black, green and orange) and OOP M (H) for t = 6 u.c. and 3 u.c. at 2 K (bottom two, red and blue).For clarity, the curves are vertically shifted and the IP data are magnified by a factor of 3. c, Depth profiles of PNR nuclear (NSLD), magnetic (MSLD, at IP fields of 1 T, 0.8 T and 0.05 T, respectively) and X-ray scattering length densities (SLD) of 24 u.c.Cr2Te3 on Al2O3(0001) with Te/AlOx capping.

Figure 3 |
Figure 3 | The unconventional Hall effects in Cr2Te3 thin films.a, Magnetic field dependence of the Hall resistivity ρyx(H) at selected temperatures of 6 u.c.Cr2Te3 on Al2O3(0001).b, Hall traces ∆ρyx after removing the high-field ordinary Hall backgrounds.At T S ∼ 40 K, a sign change occurs in the anomalous Hall resistivity ρAHE, defined as the value of ∆ρyx when the system is fully magnetized under a positive H. Apart from the AHE hysteresis loop, additional hump-shaped features develop.c, Temperature dependence of the anomalous Hall conductivity σAHE for t = 3 -24 u.c.(symbols, where solid lines are guide for the eye and dashed lines are linear fit to the low T data).d-e, Thickness dependence of σAHE at 2 K (d), AHE sign reversal temperature T S (e) and T S , the T -intercept of the linear AHE component at low T .

Figure 4 |
Figure 4 | Berry curvature and anomalous Hall conductivity in Cr2Te3.a-b, Calculated Berry curvature Ω z (k) (a) along the high symmetry k-paths in the Brillouin zone (right inset in a) and the corresponding electronic band structure (b).Left inset in a, nearly degenerate SOC anti-crossing bands contributing to the sharp peak in Ω z (k) along A-L.c, Anomalous Hall conductivity σAHE near the Fermi level εF, in equilibrium state (black), under compressive (blue) or tensile (red) strain conditions, respectively.The shades in c are guide for the eye showing the slight asymmetry of the energy dependence of σAHE above and below εF which at finite T may lead to a sign reversal in σAHE owing to thermal broadening.

Figure 5 |
Figure 5 | Characteristics of hump-shaped Hall peaks.a, Simplified superposition of two AHE components with opposite sign and different coercive fields.b-c, Minor loop experiments on a 6 u.c.Cr2Te3 on Al2O3(0001) at T = 30 K, first fully magnetized at µ0H = +3 T (complete loop shown in grey as guide for the eye) and then swept back and forth between +2 T and selected µ0Hmin.The experimental minor loops in b are qualitatively reproduced in c using simulations that underscore the significance of strain-driven multidomain features and the sign reversal in ρAHE.