Search

CrX$_3$ (X = Cl, Br, I) have the same crystal structure and Hamiltonian but different ligand spin-orbit coupling (SOC) constant $\lambda_X$, providing excellent material platform exploring for exotic two-dimensional (2D) spin orders. Their microscopic mechanism underlying 2D spin physics and Hamiltonian remain unestablished, along with experimental corroboration of Kitaev exchange interaction, central to realizing topological quantum spin liquids. We report Kitaev interaction signature in magnetic anisotropy measured by ferromagnetic resonance (FMR) spectroscopy. We present measured values of Heisenberg J, Kitaev K, and off-diagonal symmetric $\Gamma$ exchange interactions in CrX$_3$ determined using FMR and exact diagonalization. K and $\Gamma$ exhibit dominant quadratic dependencies on $\lambda_X$, indicating its central role in 2D magnetism. Our study provides foundation for exploring exotic 2D magnetic topologies by tuning intrinsic material parameters such as SOC., Comment: 18 pages, 4 figures

The field of nanoscale magnetic resonance imaging (NanoMRI) was started 30 years ago. It was motivated by the desire to image single molecules and molecular assemblies, such as proteins and virus particles, with near-atomic spatial resolution and on a length scale of 100 nm. Over the years, the NanoMRI field has also expanded to include the goal of useful high-resolution nuclear magnetic resonance (NMR) spectroscopy of molecules under ambient conditions, including samples up to the micron-scale. The realization of these goals requires the development of spin detection techniques that are many orders of magnitude more sensitive than conventional NMR and MRI, capable of detecting and controlling nanoscale ensembles of spins. Over the years, a number of different technical approaches to NanoMRI have emerged, each possessing a distinct set of capabilities for basic and applied areas of science. The goal of this roadmap article is to report the current state of the art in NanoMRI technologies, outline the areas where they are poised to have impact, identify the challenges that lie ahead, and propose methods to meet these challenges. This roadmap also shows how developments in NanoMRI techniques can lead to breakthroughs in emerging quantum science and technology applications.

Creating materials that do not exist in nature can lead to breakthroughs in science and technology. Magnetic skyrmions are topological excitations that have attracted great attention recently for their potential applications in low power, ultrahigh density memory. A major challenge has been to find materials that meet the dual requirement of small skyrmions stable at room temperature. Here we meet both these goals by developing epitaxial FeGe films with excess Fe using atomic layer molecular beam epitaxy (MBE) far from thermal equilibrium. Our novel atomic layer design permits the incorporation of 20% excess Fe while maintaining a non-centrosymmetric crystal structure supported by theoretical calculations and necessary for stabilizing skyrmions. We show that the Curie temperature is well above room temperature, and that the skyrmions probed by topological Hall effect have sizes down to 15 nm as imaged by Lorentz transmission electron microscopy (LTEM) and magnetic force microscopy (MFM). Our results illustrate new avenues for creating artificial materials tailored at the atomic scale that can impact nanotechnology., Comment: 22 figures, 34 pages

Magnetic skyrmions are promising for next-generation information storage and processing owing to their potential advantages in data storage density, robustness, and energy efficiency. The magnetic multilayers consisting of Pt, Co, and a third metal element $X$ provide an ideal platform to study the skyrmions due to their highly tunable magnetic properties. Here, we report the observation of room-temperature bubble-like N\'eel skyrmions in epitaxial Pt/Co/Cu multilayers in samples with multidomain states in zero field. The magneto-optic Kerr effect (MOKE) and superconducting quantum interference device (SQUID) magnetometry are applied to investigate the shapes of the hysteresis loops, the magnetic anisotropy, and the saturation magnetization. By tuning the Co thickness and the number of periods, we achieve perpendicular and in-plane magnetized states and multidomain states that are identified by a wasp-waisted hysteresis loop. Skyrmions are directly imaged by magnetic force microscopy (MFM) and Lorentz transmission electron microscopy (LTEM). The development of room-temperature skyrmions in Pt/Co/Cu multilayers may lead to advances in skyrmion-related research and applications., Comment: 11 pages, 6 Figures

Y3Fe5O12 is arguably the best magnetic material for magnonic quantum information science (QIS) because of its extremely low damping. We report ultralow damping at 2 K in epitaxial Y3Fe5O12 thin films grown on a diamagnetic Y3Sc2Ga3O12 substrate that contains no rare-earth elements. Using these ultralow damping YIG films, we demonstrate for the first time strong coupling between magnons in patterned YIG thin films and microwave photons in a superconducting Nb resonator. This result paves the road towards scalable hybrid quantum systems that integrate superconducting microwave resonators, YIG film magnon conduits, and superconducting qubits into on-chip QIS devices.

Engineering magnetic anisotropy in a ferro- or ferrimagnetic (FM) thin film is crucial in spintronic device. One way to modify the magnetic anisotropy is through the surface of the FM thin film. Here, we report the emergence of a perpendicular magnetic anisotropy (PMA) induced by interfacial interactions in a heterostructure comprised of a garnet ferrimagnet, Y3Fe5O12 (YIG), and the low-symmetry, high spin orbit coupling (SOC) transition metal dichalcogenide, WTe2. At the same time, we also observed an enhancement in Gilbert damping in the WTe2 covered YIG area. Both the magnitude of interface-induced PMA and the Gilbert damping enhancement have no observable WTe2 thickness dependence down to single quadruple-layer, indicating that the interfacial interaction plays a critical role. The ability of WTe2 to enhance the PMA in FM thin film, combined with its previously reported capability to generate out-of-plane damping like spin torque, makes it desirable for magnetic memory applications., Comment: Nano Letters, in press

We investigate the physical origin of nonlinear damping due to mode coupling between several auto-oscillatory modes driven by spin-orbit torque in constricted Py/Pt heterostructures by examining the dependence of auto-oscillation on temperature and applied field orientation. We observe a transition in the nonlinear damping of the auto-oscillation modes extracted from the total oscillation power as a function of drive current, which coincides with the onset of power redistribution amongst several modes and the crossover from linewidth narrowing to linewidth broadening in all individual modes. This indicates the activation of another relaxation process by nonlinear magnon-magnon scattering within the modes. We also find that both nonlinear damping and threshold current in the mode-interaction damping regime at high drive current after transition are temperature independent, suggesting that the mode coupling occurs dominantly through a non-thermal magnon scattering process via a dipole or exchange interaction rather than thermally excited magnon-mediated scattering. This finding presents a promising pathway to overcome the current limitations of efficiently controlling the interaction between two highly nonlinear magnetic oscillators to prevent mode crosstalk or inter-mode energy transfer and deepens understanding of complex nonlinear spin dynamics in multimode spin wave systems., Comment: 13 pages, 15 figures

We demonstrate a high-quality spin orbit torque nano-oscillator comprised of spin wave modes confined by the magnetic field by the strongly inhomogeneous dipole field of a nearby micromagnet. This approach enables variable spatial confinement and systematic tuning of magnon spectrum and spectral separations for studying the impact of multi-mode interactions on auto-oscillations. We find these dipole field-localized spin wave modes exhibit good characteristic properties as auto-oscillators--narrow linewidth and large amplitude--while persisting up to room temperature. We find that the linewidth of the lowest-lying localized mode is approximately proportional to temperature in good agreement with theoretical analysis of the impact of thermal fluctuations. This demonstration of a clean oscillator with tunable properties provides a powerful tool for understanding the fundamental limitations and linewidth contributions to improve future spin-Hall oscillators., Comment: corrected funding number, added Supplementary Information, revised manuscript

The NV center in diamond has proven to be a powerful tool for locally characterizing the magnetic response of microwave excited ferromagnets. To date, this has been limited by the requirement that the FMR excitation frequency be less than the NV spin resonance frequency. Here we report NV relaxometry based on a two-magnon Raman-like process, enabling detection of FMR at frequencies higher than the NV frequency. For high microwave drive powers, we observe an unexpected field-shift of the NV response relative to a simultaneous microwave absorption signal from a low damping ferrite film. We show that the field-shifted NV response is due to a second order Suhl instability. The instability creates a large population of non-equilibrium magnons which relax the NV spin, even when the uniform mode FMR frequency exceeds that of the NV spin resonance frequency, hence ruling out the possibility that the NV is relaxed by a single NV-resonant magnon. We argue that at high frequencies the NV response is due to a two-magnon relaxation process in which the difference frequency of two magnons matches the NV frequency, and at low frequencies we evaluate the lineshape of the one-magnon NV relaxometry response using spinwave instability theory.

Electrical detection of topological magnetic textures such as skyrmions is currently limited to conducting materials. While magnetic insulators offer key advantages for skyrmion technologies with high speed and low loss, they have not yet been explored electrically. Here, we report a prominent topological Hall effect in Pt/Tm$_3$Fe$_5$O$_{12}$ bilayers, where the pristine Tm$_3$Fe$_5$O$_{12}$ epitaxial films down to 1.25 unit cell thickness allow for tuning of topological Hall stability over a broad range from 200 to 465 K through atomic-scale thickness control. Although Tm$_3$Fe$_5$O$_{12}$ is insulating, we demonstrate the detection of topological magnetic textures through a novel phenomenon: 'spin-Hall topological Hall effect' (SH-THE), where the interfacial spin-orbit torques allow spin-Hall-effect generated spins in Pt to experience the unique topology of the underlying skyrmions in Tm$_3$Fe$_5$O$_{12}$. This novel electrical detection phenomenon paves a new path for utilizing a large family of magnetic insulators in future skyrmion technologies.

We lay the foundation for determining the microscopic spin interactions in two-dimensional (2D) ferromagnets by combining angle-dependent ferromagnetic resonance (FMR) experiments on high quality CrI$_3$ single crystals with theoretical modeling based on symmetries. We discover that the Kitaev interaction is the strongest in this material with $K \sim -5.2$ meV, 25 times larger than the Heisenberg exchange $J \sim -0.2$ meV, and responsible for opening the $\sim$5 meV gap at the Dirac points in the spin-wave dispersion. Furthermore, we find that the symmetric off-diagonal anisotropy $\Gamma \sim -67.5$ $\mu$eV, though small, is crucial for opening a $\sim$0.3 meV gap in the magnon spectrum at the zone center and stabilizing ferromagnetism in the 2D limit. The high resolution of the FMR data further reveals a $\mu$eV-scale quadrupolar contribution to the $S=3/2$ magnetism. Our identification of the underlying exchange anisotropies opens paths toward 2D ferromagnets with higher $T_\text{C}$ as well as magnetically frustrated quantum spin liquids based on Kitaev physics., Comment: 5 pages, 4 figures

Magnetic sensing technology has found widespread application in industries as diverse as transportation, medicine, and resource exploration. Such use cases often require highly sensitive instruments to measure the extremely small magnetic fields involved, relying on difficult to integrate Superconducting Quantum Interference Device (SQUID) and Spin-Exchange Relaxation Free (SERF) magnetometers. A potential alternative, nitrogen vacancy (NV) centers in diamond, has shown great potential as a high sensitivity and high resolution magnetic sensor capable of operating in an unshielded, room-temperature environment. Transitioning NV center based sensors into practical devices, however, is impeded by the need for high power RF excitation to manipulate them. Here we report an advance that combines two different physical phenomena to enable a highly efficient excitation of the NV centers: magnetoelastic drive of ferromagnetic resonance (FMR) and NV-magnon coupling. Our work demonstrates a new pathway to combine acoustics and magnonics that enables highly energy efficient and local excitation of NV centers without the need for any external RF excitation, and thus could lead to completely integrated, on-chip, atomic sensors., Comment: Fixed an issue with the display of figure 1

Spin currents are generated within the bulk of magnetic materials due to heat flow, an effect called intrinsic spin-Seebeck. This bulk bosonic spin current consists of a diffusing thermal magnon cloud, parametrized by the magnon chemical potential ($\mu_{m}$), with a diffusion length of several microns in yttrium iron garnet (YIG). Transient opto-thermal measurements of the spin-Seebeck effect (SSE) as a function of temperature reveal the time evolution of $\mu_{m}$ due to intrinsic SSE in YIG. The interface SSE develops at times < 2 ns while the intrinsic SSE signal continues to evolve at times > 500 $\mu$s, dominating the temperature dependence of SSE in bulk YIG. Time-dependent SSE data are fit to a multi-temperature model of coupled spin/heat transport using finite element method (FEM), where the magnon spin lifetime ($\tau$) and magnon-phonon thermalization time ($\tau_{mp}$) are used as fit parameters. From 300 K to 4 K, $\tau_{mp}$ varies from 1 to 10 ns, whereas $\tau$ varies from 2 to 60 $\mu$s with the spin lifetime peaking at 90 K. At low temperature, a reduction in $\tau$ is observed consistent with impurity relaxation reported in ferromagnetic resonance measurements. These results demonstrate that the thermal magnon cloud in YIG contains extremely low frequency magnons (~10 GHz) providing spectral insight to the microscopic scattering processes involved in magnon spin/heat diffusion., Comment: 35 pages and 17 figures

We report quantitative measurements of optically detected ferromagnetic resonance (ODFMR) of ferromagnetic thin films that use nitrogen-vacancy (NV) centers in diamonds to transduce FMR into a fluorescence intensity variation. To uncover the mechanism responsible for these signals, we study ODFMR as we 1) vary the separation of the NV centers from the ferromagnet (FM), 2) record the NV center longitudinal relaxation time $T_1$ during FMR, and 3) vary the material properties of the FM. Based on the results, we propose the following mechanism for ODFMR. Decay and scattering of the driven, uniform FMR mode results in spinwaves that produce fluctuating dipolar fields in a spectrum of frequencies. When the spinwave spectrum overlaps the NV center ground-state spin resonance frequencies, the dipolar fields from these resonant spinwaves relax the NV center spins, resulting in an ODFMR signal. These results lay the foundation for an approach to NV center spin relaxometry to study FM dynamics without the constraint of directly matching the NV center spin-transition frequency to the magnetic system of interest, thus enabling an alternate modality for scanned-probe magnetic microscopy that can sense ferromagnetic resonance with nanoscale resolution.

The availability of compact, low-cost magnetic resonance imaging instruments would further broaden the substantial impact of this technology. We report highly sensitive detection of magnetic resonance using low-stress silicon nitride (SiN$_x$) membranes. We use these membranes as low-loss, high-frequency mechanical oscillators and find they are able to mechanically detect spin-dependent forces with high sensitivity enabling ultrasensitive magnetic resonance detection. The high force detection sensitivity stems from their high mechanical quality factor $Q\sim10^6$ combined with the low mass of the resonator. We use this excellent mechanical force sensitivity to detect the electron spin magnetic resonance using a SiN$_x$ membrane as a force detector. The demonstrated force sensitivity at 300 K is 4 fN/$\sqrt{\mathrm{Hz}}$, indicating a potential low temperature (4 K) sensitivity of 25 aN/$\sqrt{\mathrm{Hz}}$. Given their sensitivity, robust construction, large surface area and low cost, SiN$_x$ membranes can potentially serve as the central component of a compact room-temperature ESR and NMR instrument that has superior spatial resolution to conventional approaches.

We demonstrate optical detection of a broad spectrum of ferromagnetic excitations using nitrogen-vacancy (NV) centers in an ensemble of nanodiamonds. Our recently developed approach exploits a straightforward CW detection scheme using readily available diamond detectors, making it easily implementable. The NV center is a local detector, giving the technique spatial resolution, which here is defined by our laser spot, but in principle can be extended far into the nanoscale. Among the excitations we observe are propagating dipolar and dipolar-exchange spinwaves, as well as dynamics associated with the multi-domain state of the ferromagnet at low fields. These results offer an approach, distinct from commonly used ODMR techniques, for spatially resolved spectroscopic study of magnetization dynamics at the nanoscale., Comment: 6 pages and 1 page supplmentary information

A key limitation of electron paramagnetic resonance (EPR), an established and powerful tool for studying atomic-scale biomolecular structure and dynamics is its poor sensitivity, samples containing in excess of 10^12 labeled biomolecules are required in typical experiments. In contrast, single molecule measurements provide improved insights into heterogeneous behaviors that can be masked by ensemble measurements and are often essential for illuminating the molecular mechanisms behind the function of a biomolecule. We report EPR measurements of a single labeled biomolecule that merge these two powerful techniques. We selectively label an individual double-stranded DNA molecule with a single nanodiamond containing nitrogen-vacancy (NV) centers, and optically detect the paramagnetic resonance of NV spins in the nanodiamond probe. Analysis of the spectrum reveals that the nanodiamond probe has complete rotational freedom and that the characteristic time scale for reorientation of the nanodiamond probe is slow compared to the transverse spin relaxation time. This demonstration of EPR spectroscopy of a single nanodiamond labeled DNA provides the foundation for the development of single molecule magnetic resonance studies of complex biomolecular systems., Comment: 24 pages, 5 figures

We report a systematic study of spin transport in antiferromagnetic (AF) insulators having a wide range of ordering temperatures. Spin current is dynamically injected from Y3Fe5O12 (YIG) into various AF insulators in Pt/insulator/YIG trilayers. Robust, long-distance spin transport in the AF insulators is observed, which shows strong correlation with the AF ordering temperatures. We find a striking linear relationship between the spin decay length in the AFs and the damping enhancement in YIG, suggesting the critical role of magnetic correlations in the AF insulators as well as at the AF/YIG interfaces for spin transport in magnetic insulators., Comment: 16 pages, 4 figures

We observe highly efficient dynamic spin injection from Y$_3$Fe$_5$O$_{12}$ (YIG) into NiO, an antiferromagnetic (AF) insulator, via strong coupling, and robust spin propagation in NiO up to 100-nm thickness mediated by its AF spin correlations. Strikingly, the insertion of a thin NiO layer between YIG and Pt significantly enhances the spin currents driven into Pt, suggesting exceptionally high spin transfer efficiency at both YIG/NiO and NiO/Pt interfaces. This offers a powerful platform for studying AF spin pumping and AF dynamics as well as for exploration of spin manipulation in tailored structures comprising metallic and insulating ferromagnets, antiferromagnets and nonmagnetic materials., Comment: 17 pages, 4 figures

Using simultaneous magnetic force microscopy (MFM) and transport measurements of a graphene spin valve, we correlate the non-local spin signal with the magnetization of the device electrodes. The imaged magnetization states corroborate the influence of each electrode within a one-dimensional spin transport model and provide evidence linking domain wall pinning to additional features in the transport signal., Comment: The following article has been submitted to Applied Physics Letters. If it is published, it will be found online at http://apl.aip.org

Spin wave modes confined in a ferromagnetic film by the spatially inhomogeneous magnetic field generated by a scanned micromagnetic tip of a ferromagnetic resonance force microscope (FMRFM) enable microscopic imaging of the internal fields and spin dynamics in nanoscale magnetic devices. Here we report a detailed study of spin wave modes in a thin ferromagnetic film localized by magnetic field configurations frequently encountered in FMRFM experiments, including geometries in which the probe magnetic moment is both parallel and antiparallel to the applied uniform magnetic field. We demonstrate that characteristics of the localized modes, such as resonance field and confinement radius, can be broadly tuned by controlling the orientation of the applied field relative to the film plane. Micromagnetic simulations accurately reproduce our FMRFM spectra allowing quantitative understanding of the localized modes. Our results reveal a general method of generating tightly confined spin wave modes in various geometries with excellent spatial resolution that significantly facilitates the broad application of FMRFM. This paves the way to imaging of magnetic properties and spin wave dynamics in a variety of contexts for uncovering new physics of nanoscale spin excitations., Comment: 35pages, 6figures

Using Y3Fe5O12 (YIG) thin films grown by our sputtering technique, we study dynamic spin transport in nonmagnetic (NM), ferromagnetic (FM) and antiferromagnetic (AF) materials by ferromagnetic resonance (FMR) spin pumping. From both inverse spin Hall effect (ISHE) and damping enhancement, we determine the spin mixing conductance and spin Hall angle in many metals. Surprisingly, we observe robust spin conduction in AF insulators excited by an adjacent YIG at resonance. This demonstrates that YIG spin pumping is a powerful and versatile tool for understanding spin Hall physics, spin-orbit coupling (SOC), and magnetization dynamics in a broad range of materials., Comment: 24 pages, 8 figures, 1 table

It is generally believed that spin-orbit coupling (SOC) follows Z4 (atomic number) dependence and becomes significant only in heavy elements. Consequently, SOC in 3d transition metals should be negligible given their small Z. Using dynamic spin pumping of Y3Fe5O12-based structures, we uncover a systematic evolution of spin Hall angle with d-orbital filling in a series of 3d metals, reminiscent of behavior observed in 5d metals. In particular, Cr and Ni show very large spin Hall angle (half of that for Pt), indicating that d-orbital filling rather than Z plays a dominant role in spin Hall effect (SHE) in 3d metals. This result enriches our understanding of SHE and broadens the scope of materials available for exploring the rich phenomena enabled by SOC as well as presenting a guidepost for testing theoretical models of spin-orbit coupling in transition metals., Comment: 17 pages, 4 figures, 1 table

Understanding the complex properties of electronic and spintronic devices at the micro- and nano-scale is a topic of intense current interest as it becomes increasingly important for scientific progress and technological applications. In-operando characterization of such devices by scanned probe techniques is particularly well-suited for the microscopic study of these properties. We have developed a scanned probe microscope (SPM) which is capable of both standard force imaging (atomic, magnetic, electrostatic) and simultaneous electrical transport measurements. We utilize flexible and inexpensive FPGA (field programmable gate array) hardware and a custom software framework developed in National Instrument's LabVIEW environment to perform the various aspects of microscope operation and device measurement. The FPGA-based approach enables sensitive, real-time cantilever frequency-shift detection. Using this system, we demonstrate electrostatic force microscopy of an electrically-biased graphene FET device. The combination of SPM and electrical transport also enables imaging of the transport response to a localized perturbation provided by the scanned cantilever tip. Facilitated by the broad presence of LabVIEW in the experimental sciences and the openness of our software solution, our system permits a wide variety of combined scanning and transport measurements by providing standardized interfaces and flexible access to all aspects of a measurement (input and output signals, and processed data). Our system also enables precise control of timing (synchronization of scanning and transport operations) and implementation of sophisticated feedback protocols, and thus should be broadly interesting and useful to practitioners in the field.

Spin transport efficiency in heterostructures depends on the spin conductances of each constituent and their interfaces. We report a comparative study of spin pumping in Y3Fe5O12/Cu/Pt and Y3Fe5O12/Cu/W trilayers. Surprisingly, the insertion of a Cu interlayer between Y3Fe5O12 and W substantially improves (over a factor of 4) the spin current injection into W while similar insertion between Y3Fe5O12 and Pt degrades the spin current. This is a consequence of a much improved Y3Fe5O12/Cu spin mixing conductance relative to that for Y3Fe5O12/W. This result implies the possibility of engineered heterostructures with matching spin conductances to enable optimal spin transport efficiency., Comment: 18 pages, 5 figures

We observe a dependence of the damping of a confined mode of precessing ferromagnetic magnetization on the size of the mode. The micron-scale mode is created within an extended, unpatterned YIG film by means of the intense local dipolar field of a micromagnetic tip. We find that damping of the confined mode scales like the surface-to-volume ratio of the mode, indicating an interfacial damping effect (similar to spin pumping) due to the transfer of angular momentum from the confined mode to the spin sink of ferromagnetic material in the surrounding film. Though unexpected for insulating systems, the measured intralayer spin-mixing conductance $g_{\uparrow \downarrow} = 5.3 \times 10^{19} {\rm m}^{-2}$ demonstrates efficient intralayer angular momentum transfer., Comment: 4 pages, 5 figures

We report the manipulation of nitrogen vacancy (NV) spins in diamond when nearby ferrimagnetic insulator, yttrium iron garnet, is driven into precession. The change in NV spin polarization, as measured by changes in photoluminescence, is comparable in magnitude to that from conventional optically detected magnetic resonance, but relies on a distinct mechanism as it occurs at a microwave frequency far removed from the magnetic resonance frequency of the NV spin. This observation presents a new approach to transferring ferromagnetic spin information into a paramagnet and then transducing the response into a robust optical signal. It also opens new avenues for studying ferromagnetism and spin transport at the nanoscale., Comment: 3 figures

We investigate the local symmetry of the tilting of the CuO$_6$ octahedra in La$_{\rm 1.65}$Eu$_{\rm 0.2}$Sr$_{\rm 0.15}$CuO$_{4}$ by means of $^{63}$Cu NMR spectroscopy and the Van Vleck susceptibility of the $\rm Eu^{3+}$ ions. The Cu NMR central line lineshape is sensitive to local structure through the coupling of the $^{63}\rm Cu$ nuclear quadrupole moment to the local electric field gradient.The $\rm Eu^{3+}$ Van Vleck susceptibility, as a single ion effect, locally probes the symmetry of the crystal field at the Eu site. Both techniques independently provide clear evidence for a change of the local tilt symmetry at the first order structural transition from the orthorhombic to the low temperature tetragonal phase, in excellent agreement with the average structure obtained by diffraction techniques. We conclude that the symmetry of the average crystal structure accurately represents the symmetry of the octahedral tilt pattern on a local scale., Comment: 5 pages, 5 figures. Submitted