Showing papers on "Spin wave published in 2015"

Linear spin wave theory for single-Q incommensurate magnetic structures

Abstract: Linear spin wave theory provides the leading term in the calculation of the excitation spectra of long-range ordered magnetic systems as a function of 1/root S. This term is acquired using the Holstein-Primakoff approximation of the spin operator and valid for small dS fluctuations of the ordered moment. We propose an algorithm that allows magnetic ground states with general moment directions and single-Q incommensurate ordering wave vector using a local coordinate transformation for every spin and a rotating coordinate transformation for the incommensurability. Finally we show, how our model can determine the spin wave spectrum of the magnetic C-site langasites with incommensurate order.

Topological Magnon Bands in a Kagome Lattice Ferromagnet

Abstract: There is great interest in finding materials possessing quasiparticles with topological properties. Such materials may have novel excitations that exist on their boundaries which are protected against disorder. We report experimental evidence that magnons in an insulating kagome ferromagnet can have a topological band structure. Our neutron scattering measurements further reveal that one of the bands is flat due to the unique geometry of the kagome lattice. Spin wave calculations show that the measured band structure follows from a simple Heisenberg Hamiltonian with a Dzyaloshinkii-Moriya interaction. This serves as the first realization of an effectively two-dimensional topological magnon insulator--a new class of magnetic material that should display both a magnon Hall effect and protected chiral edge modes.

Giant Spin Pumping and Inverse Spin Hall Effect in the Presence of Surface and Bulk Spin−Orbit Coupling of Topological Insulator Bi2Se3

Abstract: Three-dimensional (3D) topological insulators are known for their strong spin-orbit coupling (SOC) and the existence of spin-textured surface states that might be potentially exploited for "topological spintronics." Here, we use spin pumping and the inverse spin Hall effect to demonstrate successful spin injection at room temperature from a metallic ferromagnet (CoFeB) into the prototypical 3D topological insulator Bi2Se3. The spin pumping process, driven by the magnetization dynamics of the metallic ferromagnet, introduces a spin current into the topological insulator layer, resulting in a broadening of the ferromagnetic resonance (FMR) line width. Theoretical modeling of spin pumping through the surface of Bi2Se3, as well as of the measured angular dependence of spin-charge conversion signal, suggests that pumped spin current is first greatly enhanced by the surface SOC and then converted into a dc-voltage signal primarily by the inverse spin Hall effect due to SOC of the bulk of Bi2Se3. We find that the FMR line width broadens significantly (more than a factor of 5) and we deduce a spin Hall angle as large as 0.43 in the Bi2Se3 layer.

Thickness dependence of the interfacial Dzyaloshinskii–Moriya interaction in inversion symmetry broken systems

Abstract: In magnetic multilayer systems, a large spin-orbit coupling at the interface between heavy metals and ferromagnets can lead to intriguing phenomena such as the perpendicular magnetic anisotropy, the spin Hall effect, the Rashba effect, and especially the interfacial Dzyaloshinskii–Moriya (IDM) interaction. This interfacial nature of the IDM interaction has been recently revisited because of its scientific and technological potential. Here we demonstrate an experimental technique to straightforwardly observe the IDM interaction, namely Brillouin light scattering. The non-reciprocal spin wave dispersions, systematically measured by Brillouin light scattering, allow not only the determination of the IDM energy densities beyond the regime of perpendicular magnetization but also the revelation of the inverse proportionality with the thickness of the magnetic layer, which is a clear signature of the interfacial nature. Altogether, our experimental and theoretical approaches involving double time Green's function methods open up possibilities for exploring magnetic hybrid structures for engineering the IDM interaction.

Micro-focused Brillouin light scattering: imaging spin waves at the nanoscale

Abstract: Spin waves constitute an important part of research in the field of magnetization dynamics. Spin waves are the elementary excitations of the spin system in a magnetically ordered material state and magnons are their quasi particles. In the following article, we will discuss the optical method of Brillouin light scattering (BLS) spectroscopy which is a now a well established tool for the characterization of spin waves. BLS is the inelastic scattering of light from spin waves and confers several benefits: the ability to map the spin wave intensity distribution with spatial resolution and high sensitivity as well as the potential to simultaneously measure the frequency and the wave vector and, therefore, the dispersion properties. For several decades, the field of spin waves gained huge interest by the scientific community due to its relevance regarding fundamental issues of spindynamics in the field of solid states physics. The ongoing research in recent years has put emphasis on the high potential of spin waves regarding information technology. In the emerging field of \textit{magnonics}, several concepts for a spin-wave based logic have been proposed and realized. Opposed to charge-based schemes in conventional electronics and spintronics, magnons are charge-free currents of angular momentum, and, therefore, less subject to scattering processes that lead to heating and dissipation. This fact is highlighted by the possibility to utilize spin waves as information carriers in electrically insulating materials. These developments have propelled the quest for ways and mechanisms to guide and manipulate spin-wave transport. In particular, a lot of effort is put into the miniaturization of spin-wave waveguides and the excitation of spin waves in structures with sub-micrometer dimensions. For the further development of potential spin-wave-based devices, the ability to directly observe spin-wave propagation with spatial resolution is crucial. As an optical technique BLS do

Thermal Generation of Spin Current in an Antiferromagnet.

Abstract: The longitudinal spin Seebeck effect has been investigated for a uniaxial antiferromagnetic insulator Cr(2)O(3), characterized by a spin-flop transition under magnetic field along the c axis. We have found that a temperature gradient applied normal to the Cr(2)O(3)/Pt interface induces inverse spin Hall voltage of spin-current origin in Pt, whose magnitude turns out to be always proportional to magnetization in Cr(2)O(3). The possible contribution of the anomalous Nernst effect is confirmed to be negligibly small. The above results establish that an antiferromagnetic spin wave can be an effective carrier of spin current.

Thermal Hall Effect of Spin Excitations in a Kagome Magnet

Abstract: At low temperatures, the thermal conductivity of spin excitations in a magnetic insulator can exceed that of phonons. However, because they are charge neutral, the spin waves are not expected to display a thermal Hall effect. However, in the kagome lattice, theory predicts that the Berry curvature leads to a thermal Hall conductivity κ(xy). Here we report observation of a large κ(xy) in the kagome magnet Cu(1-3, bdc) which orders magnetically at 1.8 K. The observed κ(xy) undergoes a remarkable sign reversal with changes in temperature or magnetic field, associated with sign alternation of the Chern flux between magnon bands. The close correlation between κ(xy) and κ(xx) firmly precludes a phonon origin for the thermal Hall effect.

Magnetic thin-film insulator with ultra-low spin wave damping for coherent nanomagnonics

Abstract: Wave control in the solid state has opened new avenues in modern information technology. Surface-acoustic-wave-based devices are found as mass market products in 100 millions of cellular phones. Spin waves (magnons) would offer a boost in today's data handling and security implementations, i.e., image processing and speech recognition. However, nanomagnonic devices realized so far suffer from the relatively short damping length in the metallic ferromagnets amounting to a few 10 micrometers typically. Here we demonstrate that nm-thick YIG films overcome the damping chasm. Using a conventional coplanar waveguide we excite a large series of short-wavelength spin waves (SWs). From the data we estimate a macroscopic of damping length of about 600 micrometers. The intrinsic damping parameter suggests even a record value about 1 mm allowing for magnonics-based nanotechnology with ultra-low damping. In addition, SWs at large wave vector are found to exhibit the non-reciprocal properties relevant for new concepts in nanoscale SW-based logics. We expect our results to provide the basis for coherent data processing with SWs at GHz rates and in large arrays of cellular magnetic arrays, thereby boosting the envisioned image processing and speech recognition.

Temperature dependence of spin diffusion length and spin Hall angle in Au and Pt

Abstract: We have studied the spin transport and the spin Hall effect as a function of temperature for platinum (Pt) and gold (Au) in lateral spin valve structures. First, by using the spin absorption technique, we extract the spin diffusion length of Pt and Au. Secondly, using the same devices, we have measured the spin Hall conductivity and analyzed its evolution with temperature to identify the dominant scattering mechanisms behind the spin Hall effect. This analysis confirms that the intrinsic mechanism dominates in Pt whereas extrinsic effects are more relevant in Au. Moreover, we identify and quantify the phonon-induced skew scattering. We show that this contribution to skew scattering becomes relevant in metals such as Au, with a low residual resistivity.

Exchange magnon-polaritons in microwave cavities

Abstract: We formulate a scattering theory to study magnetic films in microwave cavities beyond the independent-spin and rotating-wave approximations of the Tavis-Cummings model. We demonstrate that strong coupling can be realized not only for the ferromagnetic resonance mode, but also for spin-wave resonances; the coupling strengths are mode dependent and decrease with increasing mode index. The strong-coupling regime can also be accessed electrically by spin pumping into a metal contact.

Length Scale of the Spin Seebeck Effect.

Abstract: We investigate the origin of the spin Seebeck effect in yttrium iron garnet (YIG) samples for film thicknesses from 20 nm to 50 μm at room temperature and 50 K. Our results reveal a characteristic increase of the longitudinal spin Seebeck effect amplitude with the thickness of the insulating ferrimagnetic YIG, which levels off at a critical thickness that increases with decreasing temperature. The observed behavior cannot be explained as an interface effect or by variations of the material parameters. Comparison to numerical simulations of thermal magnonic spin currents yields qualitative agreement for the thickness dependence resulting from the finite magnon propagation length. This allows us to trace the origin of the observed signals to genuine bulk magnonic spin currents due to the spin Seebeck effect ruling out an interface origin and allowing us to gauge the reach of thermally excited magnons in this system for different temperatures. At low temperature, even quantitative agreement with the simulations is found.

Narrow Magnonic Waveguides Based on Domain Walls.

Abstract: The channeling of spin waves with domain walls in ultrathin ferromagnetic films is demonstrated theoretically and through micromagnetics simulations. It is shown that propagating excitations localized to the wall, which appear in the frequency gap of bulk spin wave modes, can be guided in curved geometries and propagate in close proximity to other channels. For Neel-type walls arising from a Dzyaloshinskii-Moriya interaction, the channeling is strongly nonreciprocal and group velocities can exceed 1 km/s in the long wavelength limit for certain propagation directions. The channeled modes represent an unusual analogy of whispering gallery waves that are one dimensional and nonreciprocal with this interaction. Moreover, a sufficiently strong Dzyaloshinskii-Moriya interaction can create a degeneracy of channeled and propagating modes at a critical wave vector.

Room Temperature Electrical Detection of Spin Polarized Currents in Topological Insulators.

Abstract: Topological insulators (TIs) are a new class of quantum materials that exhibit a current-induced spin polarization due to spin-momentum locking of massless Dirac Fermions in their surface states. This helical spin polarization in three-dimensional (3D) TIs has been observed using photoemission spectroscopy up to room temperatures. Recently, spin polarized surface currents in 3D TIs were detected electrically by potentiometric measurements using ferromagnetic detector contacts. However, these electric measurements are so far limited to cryogenic temperatures. Here we report the room temperature electrical detection of the spin polarization on the surface of Bi2Se3 by employing spin sensitive ferromagnetic tunnel contacts. The current-induced spin polarization on the Bi2Se3 surface is probed by measuring the magnetoresistance while switching the magnetization direction of the ferromagnetic detector. A spin resistance of up to 70 mΩ is measured at room temperature, which increases linearly with current bias, reverses sign with current direction, and decreases with higher TI thickness. The magnitude of the spin signal, its sign, and control experiments, using different measurement geometries and interface conditions, rule out other known physical effects. These findings provide further information about the electrical detection of current-induced spin polarizations in 3D TIs at ambient temperatures and could lead to innovative spin-based technologies.

Spin-Wave Diode

Abstract: Conventional electric circuits use electrons as information carriers, a process that dissipates vast quantities of waste heat. A new design for a spin-wave diode, which produces no Joule heating, is presented.

Optically reconfigurable magnetic materials

Abstract: The periodic modulation of the magnetic properties of magnonic crystals controls the flow of spin waves. An optical method is now shown that can produce such modulations by heating, which can be reprogrammed during operation. Structuring of materials is the most general approach for controlling waves in solids. As spin waves—eigen-excitations of the electrons’ spin system—are free from Joule heating, they are of interest for a range of applications, such as processing1,2,3,4,5, filtering6,7,8 and short-time data storage9. Whereas all these applications rely on predefined constant structures, a dynamic variation of the structures would provide additional, novel applications. Here, we present an approach for producing fully tunable, two-dimensionally structured magnetic materials. Using a laser, we create thermal landscapes in a magnetic medium that result in modulations of the saturation magnetization and in the control of spin-wave characteristics. This method is demonstrated by the realization of fully reconfigurable one- and two-dimensional magnonic crystals—artificial periodic magnetic lattices.

Competing spin-liquid states in the spin- 1 2 Heisenberg model on the triangular lattice

Abstract: We study the spin-$\frac{1}{2}$ Heisenberg model on the triangular lattice with antiferromagnetic first- $({J}_{1})$ and second- $({J}_{2})$ nearest-neighbor interactions using density matrix renormalization group. By studying the spin correlation function, we find a ${120}^{\ensuremath{\circ}}$ magnetic order phase for ${J}_{2}\ensuremath{\lesssim}0.07{J}_{1}$ and a stripe antiferromagnetic phase for ${J}_{2}\ensuremath{\gtrsim}0.15{J}_{1}$. Between these two phases, we identify a spin-liquid region characterized by exponential decaying spin and dimer correlations, as well as large spin singlet and triplet excitation gaps on finite-size systems. We find two near degenerating ground states with distinct properties in two sectors, which indicates more than one spin-liquid candidate in this region. While the sector with spinons is found to respect time reversal symmetry, the even sector without spinons breaks such a symmetry for finite-size systems. Furthermore, we detect the signature of the fractionalization by following the evolution of different ground states with inserting spin flux into the cylinder system. Moreover, by tuning the anisotropic bond coupling, we explore the nature of the spin-liquid phase and find the optimal parameter region for gapped ${Z}_{2}$ spin liquids.

Magnetic correlations in the quasi-two-dimensional semiconducting ferromagnet CrSiTe 3

Abstract: Intrinsic, two-dimensional ferromagnetic semiconductors are an important class of materials for overcoming the limitations of dilute magnetic semiconductors for spintronics applications. ${\text{CrSiTe}}_{3}$ is a particularly interesting member of this class, since it can likely be exfoliated down to single layers, where ${T}_{C}$ is predicted to increase dramatically. Establishing the nature of the magnetism in the bulk is a necessary precursor to understanding the magnetic behavior in thin-film samples and the possible applications of this material. In this work, we use elastic and inelastic neutron scattering to measure the magnetic properties of single-crystalline ${\text{CrSiTe}}_{3}$. We find that there is a very small single ion anisotropy favoring magnetic ordering along the $c$ axis and that the measured spin waves fit well to a model where the moments are only weakly coupled along that direction. Finally, we find that both static and dynamic correlations persist within the $ab$ plane up to at least 300 K, strong evidence of this material's two-dimensional characteristics that are relevant for future studies on thin-film and monolayer samples.

Nanometre-scale probing of spin waves using single electron spins

Abstract: Pushing the frontiers of condensed-matter magnetism requires the development of tools that provide real-space, few-nanometre-scale probing of correlated-electron magnetic excitations under ambient conditions. Here we present a practical approach to meet this challenge, using magnetometry based on single nitrogen-vacancy centres in diamond. We focus on spin-wave excitations in a ferromagnetic microdisc, and demonstrate local, quantitative and phase-sensitive detection of the spin-wave magnetic field at ∼50 nm from the disc. We map the magnetic-field dependence of spin-wave excitations by detecting the associated local reduction in the disc's longitudinal magnetization. In addition, we characterize the spin-noise spectrum by nitrogen-vacancy spin relaxometry, finding excellent agreement with a general analytical description of the stray fields produced by spin-spin correlations in a 2D magnetic system. These complementary measurement modalities pave the way towards imaging the local excitations of systems such as ferromagnets and antiferromagnets, skyrmions, atomically assembled quantum magnets, and spin ice.

Skyrmion-Based Dynamic Magnonic Crystal

Abstract: A linear array of periodically spaced and individually controllable skyrmions is introduced as a magnonic crystal. It is numerically demonstrated that skyrmion nucleation and annihilation can be accurately controlled by a nanosecond spin polarized current pulse through a nano- contact. Arranged in a periodic array, such nanocontacts allow the creation of a skyrmion lattice that causes a periodic modulation of the waveguide's magnetization, which can be dynamically controlled by changing either the strength of an applied external magnetic field or the density of the injected spin current through the nanocontacts. The skyrmion diameter is highly dependent on both the applied field and the injected current. This implies tunability of the lowest band gap as the skyrmion diameter directly affects the strength of the pinning potential. The calculated magnonic spectra thus exhibit tunable allowed frequency bands and forbidden frequency bandgaps analogous to that of conventional magnonic crystals where, in contrast, the periodicity is structurally induced and static. In the dynamic magnetic crystal studied here, it is possible to dynamically turn on and off the artificial periodic structure, which allows switching between full rejection and full transmission of spin waves in the waveguide. These findings should stimulate further research activities on multiple functionalities offered by magnonic crystals based on periodic skyrmion lattices.

Towards graded-index magnonics: Steering spin waves in magnonic networks

Abstract: Magnonics explores precessional excitations of ordered spins in magnetic materials---so-called spin waves---and their use as information and signal carriers within networks of magnonic waveguides. Here, we demonstrate that the nonuniformity of the internal magnetic field and magnetization inherent to magnetic structures creates a medium of graded refractive index for propagating magnetostatic waves and can be used to steer their propagation. The character of the nonuniformity can be tuned and potentially programmed using the applied magnetic field, which opens exciting prospects for the field of graded-index magnonics.

Magnonic Waveguides Studied by Microfocus Brillouin Light Scattering

Abstract: The paradigm of magnonics is based on utilization of propagating spin waves (or their quanta—magnons) for signal transmission and processing in magnetic-field-controlled devices. Implementation of magnonic devices for future-generation microelectronics requires the use of spin-wave guiding structures with micrometer- to nanometer-sized dimensions. Therefore, the deep understanding of propagation, excitation, and control of spin waves in microscopic waveguides is an absolute prerequisite for further developments in the field. Here we review recent experiments on spin-wave propagation in microscopic magnonic waveguides utilizing high-resolution Brillouin light-scattering spectroscopy enabling 2-D visualization of spin waves on the nanoscale.

Quantitative simulation of temperature-dependent magnetization dynamics and equilibrium properties of elemental ferromagnets

Abstract: Atomistic spin model simulations are immensely useful in determining temperature dependent magnetic prop- erties, but are known to give the incorrect dependence of the magnetization on temperature compared to exper- iment owing to their classical origin. We find a single parameter rescaling of thermal fluctuations which gives quantitative agreement of the temperature dependent magnetization between atomistic simulations and experi- ment for the elemental ferromagnets Ni, Fe, Co and Gd. Simulating the sub-picosecond magnetization dynam- ics of Ni under the action of a laser pulse we also find quantitative agreement with experiment in the ultrafast regime. This enables the quantitative determination of temperature dependent magnetic properties allowing for accurate simulations of magnetic materials at all temperatures.

Spin-wave logic devices based on isotropic forward volume magnetostatic waves

Abstract: We propose the utilization of isotropic forward volume magnetostatic spin waves in modern wave-based logic devices and suggest a concrete design for a spin-wave majority gate operating with these waves. We demonstrate by numerical simulations that the proposed out-of-plane magnetized majority gate overcomes the limitations of anisotropic in-plane magnetized majority gates due to the high spin-wave transmission through the gate, which enables a reduced energy consumption of these devices. Moreover, the functionality of the out-of-plane majority gate is increased due to the lack of parasitic generation of short-wavelength exchange spin waves.

Dynamical skyrmion state in a spin current nano-oscillator with perpendicular magnetic anisotropy.

Abstract: We study the spectral characteristics of spin current nano-oscillators based on the $\mathrm{Pt}/[\mathrm{Co}/\mathrm{Ni}]$ magnetic multilayer with perpendicular magnetic anisotropy. By varying the applied magnetic field and current, both localized and propagating spin wave modes of the oscillation are achieved. At small fields, we observe an abrupt onset of the modulation sidebands. We use micromagnetic simulations to identify this state as a dynamical magnetic skyrmion stabilized in the active device region by spin current injection, whose current-induced dynamics is accompanied by the gyrotropic motion of the core due to the skew deflection. Our results demonstrate a practical route for controllable skyrmion manipulation by spin current in magnetic thin films.

All-magnetic control of skyrmions in nanowires by a spin wave.

Abstract: Magnetic skyrmions are topologically protected nanoscale objects, which are promising building blocks for novel magnetic and spintronic devices. Here, we investigate the dynamics of a skyrmion driven by a spin wave in a magnetic nanowire. It is found that (i) the skyrmion is first accelerated and then decelerated exponentially; (ii) it can turn L-corners with both right and left turns; and (iii) it always turns left (right) when the skyrmion number is positive (negative) in the T- and Y-junctions. Our results will be the basis of skyrmionic devices driven by a spin wave.

Spin-wave logic devices based on isotropic forward volume magneto-static waves

Abstract: We propose the utilization of isotropic forward volume magneto-static spin waves in modern wave-based logic devices and suggest a concrete design for a spin-wave majority gate operating with these waves. We demonstrate by numerical simulations that the proposed out-of-plane magnetized majority gate overcomes the limitations of anisotropic in-plane magnetized majority gates due to the high spin-wave transmission through the gate, which enables a reduced energy consumption of these devices. Moreover, the functionality of the out-of-plane majority gate is increased due to the lack of parasitic generation of short-wavelength exchange spin waves.

Photodrive of magnetic bubbles via magnetoelastic waves

Abstract: Precise control of magnetic domain walls continues to be a central topic in the field of spintronics to boost infotech, logic, and memory applications. One way is to drive the domain wall by current in metals. In insulators, the incoherent flow of phonons and magnons induced by the temperature gradient can carry the spins, i.e., spin Seebeck effect, but the spatial and time dependence is difficult to control. Here, we report that coherent phonons hybridized with spin waves, magnetoelastic waves, can drive magnetic bubble domains, or curved domain walls, in an iron garnet, which are excited by ultrafast laser pulses at a nonabsorbing photon energy. These magnetoelastic waves were imaged by time-resolved Faraday microscopy, and the resultant spin transfer force was evaluated to be larger for domain walls with steeper curvature. This will pave a path for the rapid spatiotemporal control of magnetic textures in insulating magnets.

Radiative damping in waveguide-based ferromagnetic resonance measured via analysis of perpendicular standing spin waves in sputtered permalloy films

Abstract: The damping $\alpha$ of the spinwave resonances in 75 nm, 120 nm, and 200nm -thick Permalloy films is measured via vector-network-analyzer ferromagnetic-resonance (VNA-FMR) in the out-of-plane geometry. Inductive coupling between the sample and the waveguide leads to an additional radiative damping term. The radiative contribution to the over-all damping is determined by measuring perpendicular standing spin waves (PSSWs) in the Permalloy films, and the results are compared to a simple analytical model. The damping of the PSSWs can be fully explained by three contributions to the damping: The intrinsic damping, the eddy-current damping, and the radiative damping. No other contributions were observed. Furthermore, a method to determine the radiative damping in FMR measurements with a single resonance is suggested.

All-electrical manipulation of magnetization dynamics in a ferromagnet by antiferromagnets with anisotropic spin Hall effects

Abstract: We investigate spin-orbit torques of metallic CuAu-I-type antiferromagnets using spin-torque ferromagnetic resonance tuned by a dc-bias current. The observed spin torques predominantly arise from diffusive transport of spin current generated by the spin Hall effect. We find a growth-orientation dependence of the spin torques by studying epitaxial samples, which may be correlated to the anisotropy of the spin Hall effect. The observed anisotropy is consistent with first-principles calculations on the intrinsic spin Hall effect. Our work demonstrates large tunable spin-orbit effects in magnetically ordered materials.

Nanometer-scale probing of spin waves using single electron spins

Abstract: Pushing the frontiers of condensed-matter magnetism requires the development of tools that provide real-space, few-nanometre-scale probing of correlated-electron magnetic excitations under ambient conditions. Here we present a practical approach to meet this challenge, using magnetometry based on single nitrogen-vacancy centres in diamond. We focus on spin-wave excitations in a ferromagnetic microdisc, and demonstrate local, quantitative and phase-sensitive detection of the spin-wave magnetic field at ∼50 nm from the disc. We map the magnetic-field dependence of spin-wave excitations by detecting the associated local reduction in the disc's longitudinal magnetization. In addition, we characterize the spin–noise spectrum by nitrogen-vacancy spin relaxometry, finding excellent agreement with a general analytical description of the stray fields produced by spin–spin correlations in a 2D magnetic system. These complementary measurement modalities pave the way towards imaging the local excitations of systems such as ferromagnets and antiferromagnets, skyrmions, atomically assembled quantum magnets, and spin ice.