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Meguro-ku, Japan

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Recently intensive efforts have been devoted to the emerging field of antiferromagnetic (AFM) spintronics, where ferromagnetic electrodes are substituted by antiferromagnets. This study investigates the anisotropic magnetoresistance (AMR) of epitaxial tetragonal antiferromagnetic bimetallic films: Mn Au and Mn Au/Fe bilayers. An anomalous AMR effect with additional peaks is observed. This study theoretically and experimentally demonstrates that the AFM spins of Mn Au can be viewed and controlled at room temperature, and this is achievable with a notably relatively small magnetic field of 200 mT. Strong hybridization between Au and Mn, and strong modification of the intrinsic quadratic anisotropy of Mn Au from interfacial biquadratic anisotropy result in an additional anomalous AMR component of 1%. The findings suggest that Mn Au films can be used in room temperature antiferromagnetic spintronics.

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We start by performing a dynamical space-time ideal MHD simulation with adaptive mesh refinement (AMR) of the 25M (at zero-age-main-sequence) presupernova model E25 (ref. 31), with initial conditions for differential rotation as in ref. 9 (initial central angular velocity of the iron core is 2.8 rad s−1; x  = 500 km; z  = 2,000 km; M , mass of the Sun). This model could be considered as a type Ic-bl/hypernova and long γ-ray burst progenitor22. At the onset of collapse, we set up a modified dipolar magnetic field structure from a vector potential A with components A  = A  = 0 and  A  =  , where r is the radius, r  = 1,000 km (as in ref. 9) is a parameter controlling the fall-off of the magnetic field, and B  = 1010 G sets the initial strength of the magnetic field. This progenitor seed field is not unreasonable for γ-ray-burst supernova progenitor cores22, 32. With the grid set-up (nine levels of box-in-box AMR; finest resolution dx = 375 m) and methods being identical to those in refs 9 and 33, we follow this simulation until t  = 20 ms after core bounce. At this time, the initial supernova shockwave has stalled at a radius of about 130 km. Extended Data Figs 5 and 6 show the radial profiles of important state variables (density, entropy, angular velocity and fast magnetosonic speed) of the simulation at the time of mapping. Both the protoneutron star and the postshock region have reached a quasi-equilibrium state, and the underlying space-time changes only very slowly and secularly, allowing us to carry out subsequent high-resolution GRMHD simulations that assume a fixed background space-time for about 10–20 ms. The resolution of the AMR box covering the shear layer of the protoneutron star in this initial simulation is dx = 750 m, but to resolve the FGM of the MRI for the chosen initial magnetic field of 1010 G, a linear resolution of at least dx = 100 m is required34. This is why the common method of obtaining the field strength necessary to power a magnetorotational explosion (≥1015 G) has been by flux compression (B ∝ ρ2/3; amplification by a factor of about 103; ρ is the density of the gas in the collapsing core) from unrealistically high seed fields (B ≥ 1012 G precollapse)8, 10, 35, 36. A magnetized fluid is unstable to weak-field shearing modes in the presence of a negative angular velocity gradient that is not compensated for by compositional or entropy gradients of the fluid2. At the time of mapping of the initial AMR simulation to the high-resolution domain, the plasma in the shocked region around the protoneutron star is locally unstable to weak-field shearing modes, as given by (refs 2, 34, 37). Here, C is the stability criterion of the MRI; ω is the Brunt–Väisälä frequency indicating convective stability/instability; characterizes the rotational shear; and Ω is the angular velocity. We follow refs 11 and 37 and calculate the stability criterion C , as well as the wavelength (λ ) and growth time (τ ) of the FGM of the MRI, in two-dimensional x–y and x–z slices through our three-dimensional domain. To better approximate the background flow in our three-dimensional AMR stellar collapse simulation, we average in space and time. We first carry out a spatial averaging step and calculate averaged versions of the state variables of our simulation (for example, the spatially averaged density ρ ) at every time step. For that, we choose a centred stencil that takes into account three points in each direction (this is the maximum number of points that we have available at AMR component boundaries). Because this is insufficient to get a large enough sample of points for the averaging procedure, we also calculate a moving time average of the form ρ  = αρ  + (1 − α)ρ , where i denotes the current time step and i −1 the previous one. We choose a weight function for each data set in the moving average as α = 2(n ∆t /∆t  + 1)−1, where ∆t is the time step on the current refinement level, and ∆t is the time step of the coarsest level. This choice of weight function guarantees that 86% of the data in the average comprise the last n time-step data sets. The time-step size in our AMR simulation on the refinement level that contains the shear layer around the protoneutron star is ∆t = 5 × 10−4 ms, and we choose n such that α = 2,000, ensuring temporal averaging over a timescale of about 1 ms. We calculate C , λ and τ from the space and time averages of the state variables in our simulation (Extended Data Fig. 2). Next, we map the configuration to a three-dimensional domain with uniform spacing of the form x, y, z = [−66.5 km, 66.5 km] for four resolutions, h = [500 m, 200 m, 100 m, 50 m]. To guarantee divergence-free initial data for the magnetic field, we carry out a constraint projection step after we have interpolated the magnetic field to the new domain. This is technically challenging as we have to make sure that all operators used in the projection are consistent in their definition with the discrete form of the divergence operator maintained in our specific implementation of constrained transport33. We use a discrete analogue of the Helmholtz decomposition36 to decompose the magnetic field into a discrete curl, ∇  ×, and a discrete gradient, ∇ : where Φ is a discrete scalar field. The discrete divergence, ∇  ⋅, of equation (1) leads to a discrete Poisson equation: where ∆ is the discrete Laplace operator. We solve equation (2) augmented with homogeneous Dirichlet boundary conditions to machine precision for Φ using the conjugate gradient solver provided by the PETSc2 library in combination with the parallel algebraic multigrid preconditioner HYPRE37. We then obtain a divergence-free field, B′, from the projection B′ = B − ∇ Φ. Finally, we recompute ∇ ⋅ B′ to check that it is zero to floating-point precision. We perform ideal, fixed background space-time, GRMHD simulations using the open-source Einstein Toolkit33, 38 with WENO5 reconstruction39, 40, the HLLE Riemann solver41 and constrained transport42 for maintaining ∇ ⋅ B = 0. We use the K  = 220 MeV variant of the finite-temperature nuclear equation of state of ref. 43, and the neutrino leakage/heating approximations described in refs 44, 45, with a heating scale factor f  = 1.0. We perform simulations on a domain with uniform spacing of the form x, y = [0 km, 66.5 km] and z = [−66.5 km, 66.5 km] for four resolutions, h = [500 m, 200 m, 100 m, 50 m], in quadrant symmetry three dimensions (90° rotational symmetry in the x–y plane). We keep all variables at the boundary fixed in time. This is justifiable for several reasons. First, the accretion boundary flow itself only changes on timescales longer than those simulated. Second, the fast magnetosonic speed (Extended Data Figs 5d and 6c) is of the order of a few per cent of the speed of light throughout the high-resolution computational domain. This implies a boundary crossing time for the simulation box of about 20 ms. This leaves the results in the shear layer unaltered by boundary effects for the simulated times of 10 ms. Additionally, as the cylindrically rotating flow in the shocked region is rotating in and out of the purely Cartesian boundary zones, sound waves can be reflected at the boundaries. Although these reflections are not necessarily unrealistic, as there will be perturbations in the shocked region of any rotating iron core, they pose an additional complication for the numerical stability of the simulations46. We find these reflections to be minimal in the hydrodynamical variables themselves, but they do cause spurious oscillations in the magnetic field towards the boundary zones. To prevent these oscillations at the outer boundary, without affecting the solution in the shear layer around the protoneutron star, we apply diffusivity at the level of the induction equation for the magnetic field via a modified Ohm’s law. We choose E = −v × B + ηJ, where J = ∇ × B is the three-current density; we set η = η (0.5 + 0.5tanh((r − r )b−1)) with η  = 10−2, r  = 40 km and b = 3 km. That is, we apply diffusivity only in a region outside of radius r and transition smoothly over a blending zone with width b to no diffusivity inside r . We compute spectra of the turbulent kinetic and magnetic energy as instantaneous snapshots using the discrete Fourier transform (ref. 47), where u is a vector field, L is the extent of the computational box, and N the number of grid points in the computational box. The spectra shown in Fig. 3 are densitized to better reflect the overall energy contained in the turbulent kinetic motion and the magnetic field. We show the spectra of the non-densitized turbulent velocity in Extended Data Fig. 7a, and the non-densitized magnetic field in Extended Data Fig. 7b, and also window the data to account for the non-periodicity at the boundaries of our computational domain. For that, we use a mollifier of the form and respectively for y and z. This effectively blends the data to zero over a stencil width d at the outer boundary. We choose d = 3, but note that other choices yield similar results. These non-densitized and windowed spectra illustrate that the lack of an exponential turnoff at large k in the turbulent kinetic energy in Fig. 3 is due to the inclusion of the nearly discontinuous density fall-off at the edge of the protoneutron star core (at r ≈ 12 km) in the calculation of the spectrum for Fig. 3 and the non-periodicity of our computational domain. The non-densitized and windowed turbulent kinetic energy spectrum in Extended Data Fig. 7 is compensated for k−5/3 scaling (as expected according to Kolmogorov theory48). We observe a slightly steeper scaling between k−5/3 and k−2. Within the first 3 ms, there is a rapid transition into a fully turbulent state at large k (Fig. 3b and Extended Data Fig. 7a). Afterwards, the turbulent kinetic energy decreases at large k and the spectrum gradually evolves towards a steeper fall-off. There is no increase in the turbulent kinetic energy at small values of k at late times. The magnetic energy, similarly to the turbulent kinetic energy, peaks at large k at t − t  ≈ 3 ms, which correlates well with the observed saturation of the maximum toroidal field shown in Fig. 1. Subsequently, the magnetic energy at small k grows first exponentially and then linearly with time. This picture is consistent with energy being extracted from the turbulent kinetic motion at large k and being pumped into an inverse cascade that leads to growth of magnetic-field energy at small values of k. As the kinematic phase ends and transitions into saturation, magnetic fields and numerical resistivity become important for the evolution49. This may explain the transition to linear growth. We also observe a superposed 2-ms modulation on top of the k = 4 exponential growth that corresponds roughly to the Alfven crossing time across the shear layer (t  ≈ 2 ms). We compute the two-dimensional angle-averaged (in φ) magnetic flux and poloidal current to determine which magnetic-field structures are global in φ (Extended Data Figs 3 and 4). The magnetic flux is computed as and the current as J = ∇ × B. The isocontours of the magnetic flux represent the poloidal field lines, while the poloidal current approximates the toroidal magnetic field. We find that the shear layer of the protoneutron star distorts the initial poloidal magnetic field of the iron core, but we find no emerging global poloidal field created from turbulence. The toroidal field (poloidal current), however, does show a global structure that roughly fills the width of the shear layer in the polar region of our simulation, supporting the idea that the toroidal magnetar-strength field in our simulations (see also Fig. 4) truly is global in φ. The limitations of this study are finite resolution of the simulations (most visible in the not-fully-converged saturation magnetic field), and the sensitivity of the detailed turbulent state to the numerical methods. Also, the impact of the imposed 90° rotational symmetry has to be investigated. Ultimately, high-resolution simulations such as these have to be embedded back into a full-star simulation to determine the detailed shock revival and explosion geometry. All computer code used here that is not already freely available, and the initial data, are available at http://stellarcollapse.org.

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Mn Sn is a hexagonal antiferromagnet (AFM) that exhibits non-collinear ordering of Mn magnetic moments at the Néel temperature of T ≈ 420 K (refs 10, 11, 13). The system has a hexagonal Ni Sn-type structure with space group P6 /mmc (Fig. 1a). The structure is stable only in the presence of excess Mn, which randomly occupies the Sn site13. The basal plane projection of the Mn sublattice can be viewed as a triangular lattice arrangement of a twisted triangular tube made of face-sharing octahedra (Fig. 1a, b). Each a–b plane consists of a slightly distorted kagome lattice of Mn moments each of ~3 μ (where μ is the Bohr magneton), and the associated geometrical frustration manifests itself as an inverse triangular spin structure that carries a very small net ferromagnetic moment of ~0.002 μ per Mn atom (Fig. 1c)10, 11. All Mn moments lie in the a–b plane and form a chiral spin texture with an opposite vector chirality to the usual 120° structure (Extended Data Fig. 1). This inverse triangular configuration has an orthorhombic symmetry, and only one of the three moments in each Mn triangle is parallel to the local easy-axis10, 11, 12 (Fig. 1c). Thus, the canting of the other two spins towards the local easy-axis is considered to be the origin of the weak ferromagnetic moment10, 11, 12. It is known that as-grown crystals retain the inverse triangular spin state over a wide temperature (T) range between T and ~50 K (ref. 14). At low temperatures, a cluster glass phase appears with a large c-axis ferromagnetic component due to spin canting towards the c axis11, 15, 16. In this work, we used as-grown single crystals that have the composition Mn Sn and confirmed no transition except the one at 50 K (Methods). As the detailed spin structure is unknown for the low temperature phase, here we focus on the phase stable above 50 K, and use ‘Mn Sn’ to refer to our crystals for clarity. We first show our main experimental evidence for the large anomalous Hall effect (AHE) at room temperature. Figure 2a presents the field dependence of the Hall resistivity, ρ (B), obtained at 300 K for the field along (a axis). ρ (B) exhibits a clear hysteresis loop with a sizable jump of |Δρ | ≈ 6 μΩ cm. This is strikingly large for an AFM, and is larger than those found in elemental transition metal ferromagnets (FMs) such as Fe, Co and Ni (refs 2, 3, 17). Notably, the sign change occurs at a small field of ~300 Oe. Furthermore, the hysteresis remains sharp and narrow in all the temperature range between 100 K and 400 K (Fig. 2b). In this temperature region, a large anomaly as a function of field has been seen only in the Hall resistivity. The longitudinal resistivity ρ(B) remains constant except for spikes at the critical fields where the Hall resistivity jumps (Fig. 2a). Correspondingly, the Hall conductivity, σ = −ρ /ρ2, for in-plane fields along both and shows a large jump and narrow hysteresis (Fig. 2c, d). For instance, with B || , σ has large values near zero field, ~20 Ω−1 cm−1 at 300 K and nearly 100 Ω−1 cm−1 at 100 K. This is again quite large for an AFM and comparable to those values found in ferromagnetic metals3, 17. On the other hand, the Hall conductivity for B || [0001] (c axis) shows no hysteresis but only a linear field dependence. The magnetization curve M(B) shows anisotropic hysteresis similar to that found for the Hall effect. For example, M versus B || at temperatures between 100 K and 400 K shows a clear hysteresis, indicating that a weak ferromagnetic moment (4–7 mμ per formula unit (f.u.)) changes its direction with coercivity of only a few hundred oersted (Fig. 3a). Whereas the in-plane M is almost isotropic and has a narrow hysteresis, the magnetization shows only a linear dependence on B for B || [0001] at all the temperatures measured between 100 K and 450 K (Fig. 3b). The similar anisotropic and hysteretic behaviours found in both ρ (B) and M(B) indicate that the existence of the small and soft ferromagnetic component allows us to switch the sign of the Hall effect. Indeed, previous neutron diffraction measurements and theoretical analyses clarified that the inverse triangular spin structure has no in-plane anisotropy energy up to the fourth-order term10, 12, which is consistent with the observed small coercivity. This further indicates that by rotating the net ferromagnetic moment, one may switch the staggered moment direction of the triangular spin structure10, 12. This switch should be the origin of the sign change of the Hall effect, as we discuss below. On heating, this ferromagnetic component vanishes at the Néel temperature of 430 K, above which the hysteresis disappears in both the T and B dependence of the magnetization (Fig. 3a and its inset). To reveal the temperature evolution of the spontaneous component of the AHE, both the zero-field Hall resistivity ρ (B = 0) and the zero-field longitudinal resistivity ρ(B = 0) were measured after cooling samples in a magnetic field of B  = 7 T from 400 K down to 5 K and subsequently setting B to 0 at 5 K (Methods). Figure 4a shows the temperature dependence of the zero-field Hall conductivity σ (B = 0) = −ρ (B = 0)/ρ2(B = 0) obtained after the above field-cooling (FC) procedure using three different configurations of the magnetic field (B ) and electric current (I) directions. Here, σ stands for the Hall conductivity obtained after the FC procedure in B || with I || [0001], and σ for B || and I || . Both show large values at low temperatures, and in particular, |σ | exceeds 100 Ω−1 cm−1 at T < 80 K. Both |σ | and |σ | decrease on heating but still retain values of ~10 Ω−1 cm−1 at 400 K, which is the highest temperature of our measurements. On the other hand, σ obtained after the FC procedure in B || [0001] with I || is zero within our experimental accuracy at T > 50 K. In the low temperature phase below 50 K, |σ | increases on cooling and reaches 140 Ω−1 cm−1 at 5 K, the lowest temperature of our measurements. In the three FC procedures described above, the temperature dependence of the longitudinal resistivity ρ(B = 0) was also concomitantly obtained (Fig. 4a, inset). Both in-plane and out-of-plane components show saturation at T > 300 K, indicating the presence of strong inelastic scattering at high temperatures. Conventionally, the Hall resistivity is described as ρ  = R B + R μ M. Here, R and R are the ordinary and anomalous Hall coefficients, and μ is the permeability. To further examine the field and magnetization dependence of the AHE, we estimated the ordinary Hall contribution R B by using the temperature dependent ρ and M/B for B || c (Extended Data Fig. 2, Methods). The obtained R  = 3.0 × 10−4 cm3 C−1 indicates that R B is negligibly small compared to the observed ρ . Plotting ρ versus M in Fig. 2e, we note that ρ for B || c has a normal M-linear AHE. Likewise, ρ for B || a–b also shows an M-linear AHE in field, Δρ  = R μ M (broken lines). Clearly, however, the large hysteresis with a sharp sign change in ρ cannot be described by the simple linear term, indicating that there is another dominant contribution to the AHE. If we label this additional term as , the Hall resistivity in Mn Sn can be described by By subtracting R B and R μ M from ρ , we find that is nearly independent of B or M, unlike what is found in FMs (Fig. 2f, Extended Data Fig. 3). With the reversal of a small applied field, changes sign, corresponding to the rotation of the staggered moments of the non-collinear spin structure10, 12. Thus, the large AHE, , must have a distinct AF-driven origin. In a magnetic conductor with relatively high resistivity, the AHE is dominated by contributions ∝ρ2. Thus, it is useful to compare S  = μ R /ρ2 for Mn Sn with those for various magnets (Extended Data Table 1, Methods)3. Normally for FMs such as Fe, Ni and MnSi, S is known to be field-independent, and takes values of the order of 0.01–0.1 V−1 (refs 3, 18, 19). Indeed, the field-induced M-linear contribution of the AHE has a field-independent S , which has the positive sign and the same order of magnitude as in FMs. On the other hand, one can also define S for the spontaneous component at zero field as  = ρ (B = 0)/[ρ2(B = 0)M(B = 0)] =  (B = 0)/[ρ2(B = 0)M(B = 0)] + S . We find significantly large | | |S |, reaching 14 V−1 at 100 K and with a different sign from S (Fig. 4b, Extended Data Table 1). This indicates that , which is the dominant part of the spontaneous component, has a different origin from the conventional AHE (Methods). A large AHE in a non-collinear AFM was first theoretically predicted for Mn Ir, which has a stacked kagome lattice of Mn atoms, similarly to Mn Sn (ref. 7). Chen et al.7 considered that an AHE may be induced by breaking a symmetry of a single layer kagome lattice that has a triangular magnetic order, and confirmed the large AHE by numerical calculations. Similar symmetry arguments apply to Mn Sn. In this case, the inverse triangular magnetic order breaks the in-plane hexagonal symmetry of the lattice, and thus may induce an AHE in the a–b plane. Indeed Kübler et al.8 have theoretically found a large AHE in Mn Sn, calculating the anomalous Hall conductivity through the Brillouin zone integration of the Berry curvature3, 20. Interestingly, they found a significant enhancement of the Berry curvature, particularly around band crossing points called Weyl points21, 22 near the Fermi energy. Experimental confirmation of the existence of the Weyl points (for example by ARPES measurements) is awaited. On the other hand, the observed σ  = 0 at T > 50 K is consistent with the in-plane coplanar spin structure and with the absence of a topological Hall effect due to spin chirality4, 5, 6, 8. The large enhancement in |σ | at T < 50 K, however, may contain the topological Hall contribution as spins cant towards the c axis in the low-T phase. It would be interesting to verify this possibility in future studies. Various applications are conceivable for the observed large AHE and its soft response to applied field. One possibility is for non-volatile memory. To date, FMs have been used as the main active materials for memory devices23. However, AFMs have recently attracted attention, because the small magnitude of their stray fields provides stability against magnetic field perturbations, opening new avenues to achieving high data retention and high-density memory integration24, 25, 26, 27, 28, 29. In addition, AFMs have much faster spin dynamics than FMs, which may lead to ultrafast data processing26, 29. To develop useful magnetic devices, one needs to find detectable macroscopic effects that are induced by the rotation of the order parameter. Such effects commonly used for FMs are often unavailable in AFMs as they have zero or vanishingly small magnetization. However, recent theoretical studies have proposed that such spin-axis change can be observed in AFMs24, 25, 26, 29—for instance, through anisotropic magnetoresistance (AMR) effects25. Experimental demonstrations have been performed at room temperature in which rotation of AF moments was detected as the AMR changed by a few tens of milliohms, which is of the order of 0.1% of the total resistance28. The AHE provides another useful probe for the spin-axis switch, and therefore may serve as an electrical means for reading magnetically stored information. In fact, in Mn Sn the Hall voltage can be readily detected as it generates a sizeable resistance jump at room temperature (for example, >500 mΩ for a 100-nm-thick thin film and |Δρ |/ρ > 1%) and this material has no magnetoresistance up to several tesla. Thus, the sharp Hall resistance change can be easily tuned to be more than 10% of the total resistance by reducing the misalignment of Hall voltage contacts. Whereas the remanent magnetization and thus stray fields of Mn Sn should be two to three orders of magnitude smaller than in ordinary FMs, its coercivity of a few hundred oersted is close to that of FMs used in magnetic devices. It will therefore be interesting to explore the possibility of using electrical means not only for reading but also for writing information—for example, by spin transfer torque24, 26, 29. Finally, we note that the present exceptionally large AHE found in an AFM with vanishingly small magnetization indicates that a large fictitious field due to Berry phase must exist in momentum space, and is expected to generate various effects including orbital ferromagnetism4, 20 and the spin Hall effect30. Exploration of such effects and their external-field control are suitable subjects for future studies.

Henriques V.A.R.,AMR | De Oliveira J.L.,AMR | Diniz E.F.,AMR | Dutra A.C.S.M.,ETEP Faculdades
SAE Technical Papers | Year: 2011

Gamma-TiAl alloys are potential replacements for nickel alloys and conventional titanium alloys in hot sections of turbine engines, as well as in orbital platform vehicles. The combination of high specific stiffness and good oxidation resistance at intermediate temperatures can provide significant weight savings. However, they have a limited plasticity at room temperature and the tendency to brittle fracture. Powder metallurgy is a near net shape process that allows the parts production with complex geometry at low costs. An improved plasticity of the Ti-Al alloys is received by adding alloying elements and by microstructure modification. An alloy of two-phase structure Ti-48Al-2Cr-2Nb (at.%) was investigated using the blended elemental technique. Samples were produced by mixing of initial metallic powders followed by uniaxial and cold isostatic pressing with subsequent densification by sintering between 1100-1400°C, in vacuum. It was shown that the samples presented a two-phase structure consisting of lamellar colonies of alternating layers of gamma and alpha-2 phase, with high porosity. Copyright © 2011 SAE International. Source

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A highly sensitive flexible magnetic sensor based on the anisotropic magnetoresistance effect has been fabricated. A limit of detection of 150 nT has been observed and excellent deformation stability was achieved after wrapping the flexible sensor with bending radii down to 5 mm. The flexible AMR sensor has been used to read the magnetic pattern with a thickness of 10 μm formed by ferrite magnetic inks.

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