Quantum oscillations of the quasiparticle lifetime in a metal.

Following nearly a century of research, it remains a puzzle that the low-lying excitations of metals are remarkably well explained by effective single-particle theories of non-interacting bands1-4. The abundance of interactions in real materials raises the question of direct spectroscopic signatures of phenomena beyond effective single-particle, single-band behaviour. Here we report the identification of quantum oscillations (QOs) in the three-dimensional topological semimetal CoSi, which defy the standard description in two fundamental aspects. First, the oscillation frequency corresponds to the difference of semiclassical quasiparticle (QP) orbits of two bands, which are forbidden as half of the trajectory would oppose the Lorentz force. Second, the oscillations exist up to above 50 K, in strong contrast to all other oscillatory components, which vanish below a few kelvin. Our findings are in excellent agreement with generic model calculations of QOs of the QP lifetime (QPL). Because the only precondition for their existence is a nonlinear coupling of at least two electronic orbits, for example, owing to QP scattering on defects or collective excitations, such QOs of the QPL are generic for any metal featuring Landau quantization with several orbits. They are consistent with certain frequencies in topological semimetals5-9, unconventional superconductors10,11, rare-earth compounds12-14 and Rashba systems15, and permit to identify and gauge correlation phenomena, for example, in two-dimensional materials16,17 and multiband metals18.

Resonant Elastic X-Ray Scattering of Antiferromagnetic Superstructures in EuPtSi_{3}.

We report resonant elastic x-ray scattering of long-range magnetic order in EuPtSi_{3}, combining different scattering geometries with full linear polarization analysis to unambiguously identify magnetic scattering contributions. At low temperatures, EuPtSi_{3} stabilizes type A antiferromagnetism featuring various long-wavelength modulations. For magnetic fields applied in the hard magnetic basal plane, well-defined regimes of cycloidal, conical, and fanlike superstructures may be distinguished that encompass a pocket of commensurate type A order without superstructure. For magnetic field applied along the easy axis, the phase diagram comprises the cycloidal and conical superstructures only. Highlighting the power of polarized resonant elastic x-ray scattering, our results reveal a combination of magnetic phases that suggest a highly unusual competition between antiferromagnetic exchange interactions with Dzyaloshinsky-Moriya spin-orbit coupling of similar strength.

Small-angle neutron scattering of long-wavelength magnetic modulations in reduced sample dimensions.

Magnetic small-angle neutron scattering (SANS) is ideally suited to providing direct reciprocal-space information on long-wavelength magnetic modulations, such as helicoids, solitons, merons or skyrmions. SANS of such structures in thin films or micro-structured bulk materials is strongly limited by the tiny scattering volume vis a vis the prohibitively high background scattering by the substrate and support structures. Considering near-surface scattering just above the critical angle of reflection, where unwanted signal contributions due to substrate or support structures become very small, it is established that the scattering patterns of the helical, conical, skyrmion lattice and fluctuation-disordered phases in a polished bulk sample of MnSi are equivalent for conventional transmission and near-surface SANS geometries. This motivates the prediction of a complete repository of scattering patterns expected for thin films in the near-surface SANS geometry for each orientation of the magnetic order with respect to the scattering plane.

Extending MIEZE spectroscopy towards thermal wavelengths.

A modulation of intensity with zero effort (MIEZE) setup is proposed for high-resolution neutron spectroscopy at momentum transfers up to 3 Å-1, energy transfers up to 20 meV and an energy resolution in the microelectronvolt range using both thermal and cold neutrons. MIEZE has two prominent advantages compared with classical neutron spin echo. The first is the possibility to investigate spin-depolarizing samples or samples in strong magnetic fields without loss of signal amplitude and intensity. This allows for the study of spin fluctuations in ferromagnets, and facilitates the study of samples with strong spin-incoherent scattering. The second advantage is that multi-analyzer setups can be implemented with comparatively little effort. The use of thermal neutrons increases the range of validity of the spin-echo approximation towards shorter spin-echo times. In turn, the thermal MIEZE option for greater ranges (TIGER) closes the gap between classical neutron spin-echo spectroscopy and conventional high-resolution neutron spectroscopy techniques such as triple-axis, time-of-flight and back-scattering. To illustrate the feasibility of TIGER, this paper presents the details of its implementation at the RESEDA beamline at FRM II by means of an additional velocity selector, polarizer and analyzer.

Optimization strategies and artifacts of time-involved small-angle neutron scattering experiments.

Kinetic small-angle neutron scattering provides access to the microscopic properties of mesoscale systems under slow, periodic perturbations. By interlocking the phases of neutron pulse, sample modulation and detector signal, time-involved small-angle neutron scattering experiments (TISANE) allow one to exploit the neutron velocity spread and record data without major sacrifice in intensity at timescales down to microseconds. This article reviews the optimization strategies of TISANE that arise from specific aspects of the process of data acquisition and data analysis starting from the basic principles of operation. Typical artifacts of data recorded in TISANE due to the choice of time binning and neutron chopper pulse width are illustrated by virtue of the response of the skyrmion lattice in MnSi under periodic changes of the direction of the stabilizing magnetic field.

Emergence of mesoscale quantum phase transitions in a ferromagnet.

Mesoscale patterns as observed in, for example, ferromagnets, ferroelectrics, superconductors, monomolecular films or block copolymers1,2 reflect spatial variations of a pertinent order parameter at length scales and time scales that may be described classically. This raises the question for the relevance of mesoscale patterns near zero-temperature phase transitions, also known as quantum phase transitions. Here we report the magnetic susceptibility of LiHoF4-a dipolar Ising ferromagnet-near a well-understood transverse-field quantum critical point (TF-QCP)3,4. When tilting the magnetic field away from the hard axis such that the Ising symmetry is always broken, a line of well-defined phase transitions emerges from the TF-QCP, characteristic of further symmetry breaking, in stark contrast to a crossover expected microscopically. We show that the scenario of a continuous suppression of ferromagnetic domains, representing a breaking of translation symmetry on mesoscopic scales in an environment of broken magnetic Ising symmetry on microscopic scales, is in excellent qualitative and quantitative agreement with the field and temperature dependence of the susceptibility and the magnetic phase diagram of LiHoF4 under tilted field. This identifies a new type of phase transition that may be referred to as mesoscale quantum criticality, which emanates from the textbook example of a microscopic ferromagnetic TF-QCP. Our results establish the surroundings of quantum phase transitions as a regime of mesoscale pattern formation, in which non-analytical quantum dynamics and materials properties without classical analogue may be expected.

Network of Topological Nodal Planes, Multifold Degeneracies, and Weyl Points in CoSi.

We showcase the importance of global band topology in a study of the Weyl semimetal CoSi as a representative of chiral space group (SG) 198. We identify a network of band crossings comprising topological nodal planes, multifold degeneracies, and Weyl points consistent with the fermion doubling theorem. To confirm these findings, we combined the general analysis of the band topology of SG 198 with Shubnikov-de Haas oscillations and material-specific calculations of the electronic structure and Berry curvature. The observation of two nearly dispersionless Shubnikov-de Haas frequency branches provides unambiguous evidence of four Fermi surface sheets at the R point that reflect the symmetry-enforced orthogonality of the underlying wave functions at the intersections with the nodal planes. Hence, irrespective of the spin-orbit coupling strength, SG 198 features always six- and fourfold degenerate crossings at R and Γ that are intimately connected to the topological charges distributed across the network.

Symmetry-enforced topological nodal planes at the Fermi surface of a chiral magnet.

Despite recent efforts to advance spintronics devices and quantum information technology using materials with non-trivial topological properties, three key challenges are still unresolved1-9. First, the identification of topological band degeneracies that are generically rather than accidentally located at the Fermi level. Second, the ability to easily control such topological degeneracies. And third, the identification of generic topological degeneracies in large, multisheeted Fermi surfaces. By combining de Haas-van Alphen spectroscopy with density functional theory and band-topology calculations, here we show that the non-symmorphic symmetries10-17 in chiral, ferromagnetic manganese silicide (MnSi) generate nodal planes (NPs)11,12, which enforce topological protectorates (TPs) with substantial Berry curvatures at the intersection of the NPs with the Fermi surface (FS) regardless of the complexity of the FS. We predict that these TPs will be accompanied by sizeable Fermi arcs subject to the direction of the magnetization. Deriving the symmetry conditions underlying topological NPs, we show that the 1,651 magnetic space groups comprise 7 grey groups and 26 black-and-white groups with topological NPs, including the space group of ferromagnetic MnSi. Thus, the identification of symmetry-enforced TPs, which can be controlled with a magnetic field, on the FS of MnSi suggests the existence of similar properties-amenable for technological exploitation-in a large number of materials.

Atomistic investigation of surface characteristics and electronic features at high-purity FeSi(110) presenting interfacial metallicity.

Iron silicide (FeSi) is a fascinating material that has attracted extensive research efforts for decades, notably revealing unusual temperature-dependent electronic and magnetic characteristics, as well as a close resemblance to the Kondo insulators whereby a coherent picture of intrinsic properties and underlying physics remains to be fully developed. For a better understanding of this narrow-gap semiconductor, we prepared and examined FeSi(110) single-crystal surfaces of high quality. Combined insights from low-temperature scanning tunneling microscopy and density functional theory calculations (DFT) indicate an unreconstructed surface termination presenting rows of Fe-Si pairs. Using high-resolution tunneling spectroscopy (STS), we identify a distinct asymmetric electronic gap in the sub-10 K regime on defect-free terraces. Moreover, the STS data reveal a residual density of states in the gap regime whereby two in-gap states are recognized. The principal origin of these features is rationalized with the help of the DFT-calculated band structure. The computational modeling of a (110)-oriented slab notably evidences the existence of interfacial intragap bands accounting for a markedly increased density of states around the Fermi level. These findings support and provide further insight into the emergence of surface metallicity in the low-temperature regime.

Microwave Spectroscopy of the Low-Temperature Skyrmion State in Cu_{2}OSeO_{3}.

In the cubic chiral magnet Cu_{2}OSeO_{3} a low-temperature skyrmion state (LTS) and a concomitant tilted conical state are observed for magnetic fields parallel to ⟨100⟩. Here, we report on the dynamic resonances of these novel magnetic states. After promoting the nucleation of the LTS by means of field cycling, we apply broadband microwave spectroscopy in two experimental geometries that provide either predominantly in-plane or out-of-plane excitation. By comparing the results to linear spin-wave theory, we clearly identify resonant modes associated with the tilted conical state, the gyrational and breathing modes associated with the LTS, as well as the hybridization of the breathing mode with a dark octupole gyration mode mediated by the magnetocrystalline anisotropies. Most intriguingly, our findings suggest that under decreasing fields the hexagonal skyrmion lattice becomes unstable with respect to an oblique deformation, reflected in the formation of elongated skyrmions.

Magnetoelastic hybrid excitations in CeAuAl3.

Nearly a century of research has established the Born-Oppenheimer approximation as a cornerstone of condensed-matter systems, stating that the motion of the atomic nuclei and electrons may be treated separately. Interactions beyond the Born-Oppenheimer approximation are at the heart of magneto-elastic functionalities and instabilities. We report comprehensive neutron spectroscopy and ab initio phonon calculations of the coupling between phonons, CEF-split localized 4f electron states, and conduction electrons in the paramagnetic regime of [Formula: see text], an archetypal Kondo lattice compound. We identify two distinct magneto-elastic hybrid excitations that form even though all coupling constants are small. First, we find a CEF-phonon bound state reminiscent of the vibronic bound state (VBS) observed in other materials. However, in contrast to an abundance of optical phonons, so far believed to be essential for a VBS, the VBS in [Formula: see text] arises from a comparatively low density of states of acoustic phonons. Second, we find a pronounced anticrossing of the CEF excitations with acoustic phonons at zero magnetic field not observed before. Remarkably, both magneto-elastic excitations are well developed despite considerable damping of the CEFs that arises dominantly by the conduction electrons. Taking together the weak coupling with the simultaneous existence of a distinct VBS and anticrossing in the same material in the presence of damping suggests strongly that similarly well-developed magneto-elastic hybrid excitations must be abundant in a wide range of materials. In turn, our study of the excitation spectra of [Formula: see text] identifies a tractable point of reference in the search for magneto-elastic functionalities and instabilities.

High-resolution neutron depolarization microscopy of the ferromagnetic transitions in Ni3Al and HgCr2Se4 under pressure.

We performed neutron imaging of ferromagnetic transitions in Ni3Al and HgCr2Se4 crystals. These neutron depolarization measurements revealed bulk magnetic inhomogeneities in the ferromagnetic transition temperature with spatial resolution of about 100 µm. To obtain such spatial resolution, we employed a novel neutron microscope equipped with Wolter mirrors as a neutron image-forming lens and a focusing neutron guide as a neutron condenser lens. The images of Ni3Al show that the sample does not homogeneously go through the ferromagnetic transition; the improved resolution allowed us to identify a distribution of small grains with slightly off-stoichiometric composition. Additionally, neutron depolarization imaging experiments on the chrome spinel, HgCr2Se4, under pressures up to 15 kbar highlight the advantages of the new technique especially for small samples or sample environments with restricted sample space. The improved spatial resolution enables one to observe domain formation in the sample while decreasing the acquisition time despite having a bulky pressure cell in the beam.

Reciprocal space tomography of 3D skyrmion lattice order in a chiral magnet.

It is commonly assumed that surfaces modify the properties of stable materials within the top few atomic layers of a bulk specimen only. Exploiting the polarization dependence of resonant elastic X-ray scattering to go beyond conventional diffraction and imaging techniques, we have determined the depth dependence of the full 3D spin structure of skyrmions-that is, topologically nontrivial whirls of the magnetization-below the surface of a bulk sample of Cu2OSeO3 We found that the skyrmions change exponentially from pure Néel- to pure Bloch-twisting over a distance of several hundred nanometers between the surface and the bulk, respectively. Though qualitatively consistent with theory, the strength of the Néel-twisting at the surface and the length scale of the variation observed experimentally exceed material-specific modeling substantially. In view of the exceptionally complete quantitative theoretical account of the magnetic rigidities and associated static and dynamic properties of skyrmions in Cu2OSeO3 and related materials, we conclude that subtle changes of the materials properties must exist at distances up to several hundred atomic layers into the bulk, which originate in the presence of the surface. This has far-reaching implications for the creation of skyrmions in surface-dominated systems and identifies, more generally, surface-induced gradual variations deep within a bulk material and their impact on tailored functionalities as an unchartered scientific territory.

Entropy-limited topological protection of skyrmions.

Magnetic skyrmions are topologically protected whirls that decay through singular magnetic configurations known as Bloch points. We used Lorentz transmission electron microscopy to infer the energetics associated with the topological decay of magnetic skyrmions far from equilibrium in the chiral magnet Fe1-x Co x Si. We observed that the lifetime τ of the skyrmions depends exponentially on temperature, [Formula: see text]. The prefactor τ0 of this Arrhenius law changes by more than 30 orders of magnitude for small changes of the magnetic field, reflecting a substantial reduction of the lifetime of skyrmions by entropic effects and, thus, an extreme case of enthalpy-entropy compensation. Such compensation effects, being well known across many different scientific disciplines, affect topological transitions and, thus, topological protection on an unprecedented level.

Heuristic Description of Magnetoelectricity of Cu2OSeO3.

CuO2SeO3 is an insulating material that hosts topologically nontrivial spin whirls, so-called skyrmions, and exhibits magnetoelectric coupling allowing to manipulate these skyrmions by means of electric fields. We report magnetic force microscopy imaging of the real-space spin structure on the surface of a bulk single crystal of CuO2SeO3. Based on measurements of the electric polarization using Kelvin-probe force microscopy, we develop a heuristic description of the magnetoelectric properties in CuO2SeO3. The model successfully describes the dependency of the electric polarization on the magnetization in all magnetically modulated phases.

Positron spectroscopy of point defects in the skyrmion-lattice compound MnSi.

Outstanding crystalline perfection is a key requirement for the formation of new forms of electronic order in a vast number of widely different materials. Whereas excellent sample quality represents a standard claim in the literature, there are, quite generally, no reliable microscopic probes to establish the nature and concentration of lattice defects such as voids, dislocations and different species of point defects on the level relevant to the length and energy scales inherent to these new forms of order. Here we report an experimental study of the archetypical skyrmion-lattice compound MnSi, where we relate the characteristic types of point defects and their concentration to the magnetic properties by combining different types of positron spectroscopy with ab-initio calculations and bulk measurements. We find that Mn antisite disorder broadens the magnetic phase transitions and lowers their critical temperatures, whereas the skyrmion lattice phase forms for all samples studied underlining the robustness of this topologically non-trivial state. Taken together, this demonstrates the unprecedented sensitivity of positron spectroscopy in studies of new forms of electronic order.

Spin-Resolved Fermi Surface of the Localized Ferromagnetic Heusler Compound Cu2MnAl Measured with Spin-Polarized Positron Annihilation.

We determined the bulk electronic structure of the prototypical Heusler compound Cu(2)MnAl by measuring the angular correlation of annihilation radiation using spin-polarized positrons. To this end, a new algorithm for reconstructing 3D densities from projections is introduced that allows us to corroborate the excellent agreement between our electronic structure calculations and the experimental data. The contribution of each individual Fermi surface sheet to the magnetization was identified, and summed to a total spin magnetic moment of 3.6±0.5 µ(B)/f.u..