We study the spin dynamics of a Heisenberg model at finite temperature in the presence of an external field or a uniaxial anisotropy. For the case of the uniaxial anisotropy our simulations show that the macro moment picture breaks down. An effect which we refer to as a spin-wave instability (SWI) results in a non-dissipative Bloch-Bloembergen type relaxation of the macro moment where the size of the macro moment changes, and can even be made to disappear. This relaxation mechanism is studied in detail by means of atomistic spin dynamics simulations.
Atomistic modelling of magnetic materials provides unprecedented detail about the underlying physical processes that govern their macroscopic properties, and allows the simulation of complex effects such as surface anisotropy, ultrafast laser-induced spin dynamics, exchange bias, and mi- crostructural effects. Here we present the key methods used in atomistic spin models which are then applied to a range of magnetic problems. We detail the parallelisation strategies used which enable the routine simulation of extended systems with full atomistic resolution.
We bridge an essential knowledge gap on the understanding of all-optical ultrafast switching in ferrimagnets; namely, the connection between atomistic spin dynamics methods and macroscopic phenomenological models. All-optical switching of the magnetization occurs after the application of a single femtosecond laser pulse to specific ferrimagnetic compounds. This strong excitation puts the involved degrees of freedom, electrons, lattice and spins out-of-equilibrium between each other. Atomistic spin models have quantitatively described all-optical switching in a wide range of experimental condit…
We investigate the magnetic properties of a range of low-dimensional ferromagnets using a combination of first-principles calculations and atomistic spin dynamics simulations. This approach allows us to evaluate the ground state and finite temperature properties of experimentally well characterized systems such as Co/Cu(111), Co/Cu(001), Fe/Cu(001) and Fe/W(110), for different thicknesses of the magnetic layer. We compare our calculated spin wave spectra with experimental data available in the literature, and find a good quantitative agreement. We also predict magnon spectra for systems for wh…
We propose a powerful extension to combined molecular and spin dynamics that fully captures the coupling between the atomic and spin subsystems via spin-orbit interactions. Its foundation is the inclusion of the local magnetic anisotropies that arise as a consequence of the lattice symmetry breaking due to phonons or defects. We demonstrate that our extension enables the exchange of angular momentum between the atomic and spin subsystems, which is critical to the challenges arising in the study of fluctuations and non-equilibrium processes in complex, natural, and engineered magnetic materials…
We present a method for performing atomistic spin dynamic simulations. A comprehensive summary of all pertinent details for performing the simulations such as equations of motions, models for including temperature, methods of extracting data and numerical schemes for performing the simulations is given. The method can be applied in a first principles mode, where all interatomic exchange is calculated self-consistently, or it can be applied with frozen parameters estimated from experiments or calculated for a fixed spin-configuration. Areas of potential applications to different magnetic questi…
We demonstrate the use of Langevin spin dynamics for studying dynamical properties of an archetypical spin glass system. Simulations are performed on CuMn (20% Mn) where we study the relaxation that follows a sudden quench of the system to the low temperature phase. The system is modeled by a Heisenberg Hamiltonian where the Heisenberg interaction parameters are calculated by means of first-principles density functional theory. Simulations are performed by numerically solving the Langevin equations of motion for the atomic spins. It is shown that dynamics is governed, to a large degree, by the…
The spin wave dispersions of the low temperature antiferromagnetic phase (AF1) MnWO$_4$ have been numerically calculated based on the recently reported non-collinear spin configuration with two different canting angles. A Heisenberg model with competing magnetic exchange couplings and single-ion anisotropy terms could properly describe the spin wave excitations, including the newly observed low-lying energy excitation mode $ω_2$=0.45 meV appearing at the magnetic zone centre. The spin wave dispersion and intensities are highly sensitive to two differently aligned spin-canting sublattices in th…
Using an atomistic model that simultaneously treats the dynamics of translational and spin degrees of freedom, we perform combined molecular and spin dynamics simulations to investigate the mutual influence of the phonons and magnons on their respective frequency spectra and lifetimes in ferromagnetic bcc iron. By calculating the Fourier transforms of the space- and time-displaced correlation functions, the characteristic frequencies and the linewidths of the vibrational and magnetic excitation modes were determined. Comparison of the results with that of the stand-alone molecular dynamics and…
We present a computationally efficient general first-principles based method for spin-lattice simulations for solids. Our method is based on a combination of atomistic spin dynamics and molecular dynamics, expressed through a spin-lattice Hamiltonian where the bilinear magnetic term is expanded to second order in displacement, and all parameters are computed using density functional theory. The effect of first-order spin-lattice coupling on the magnon and phonon dispersion in bcc Fe is reported as an example, and is seen to be in good agreement with previous simulations performed with an empir…
Temperature-dependent magnetic properties of Nd$_2$Fe$_{14}$B permanent magnets, i.e., saturation magnetization $M_\text{s}(T)$, effective magnetic anisotropy constants $K_i^\text{eff}(T)$ ($i=1,2,3$), domain wall width $δ_w(T)$, and exchange stiffness constant $A_\text{e}(T)$, are calculated by using \textit{ab-initio} informed atomistic spin model simulations. We construct the atomistic spin model Hamiltonian for Nd$_2$Fe$_{14}$B by using the Heisenberg exchange of Fe$-$Fe and Fe$-$Nd atomic pairs, the uniaxial single-ion anisotropy of Fe atoms, and the crystal-field energy of Nd ions which …
The Landau-Lifshitz (LL) equation, originally proposed at the macrospin level, is increasingly used in Atomistic Spin Dynamic (ASD) models. The models are based on a spin Hamiltonian featuring atomic spins of fixed length, with the exchange introduced using the Heisenberg formalism. ASD models are proving a powerful approach to the fundamental understanding of ultrafast magnetisation dynamics, including the prediction of the thermally induced magnetisation switching phenomenon in which the magnetisation is reversed using an ultrafast laser pulse in the absence of an externally applied field. T…
Single femtosecond pulse toggle switching of ferrimagnetic alloys is an essential building block for ultrafast spintronics. Very different element-specific demagnetization dynamics is believed to be a hard limit for switching in ferrimagnets. This suggests that ferrimagnets composed of two ions of different nature, such as rare earth transition metal alloys, are necessary for switching. However, experimental observation of toggle switching in Mn$_2$Ru$_x$Ga Heusler alloys, has contested this limit since Mn ions are of the same nature. To shed some light into this question, we present an atomis…
We analyze the spontaneous magnetization reversal of supported monoatomic chains of finite length due to thermal fluctuations via atomistic spin-dynamics simulations. Our approach is based on the integration of the Landau-Lifshitz equation of motion of a classical spin Hamiltonian at the presence of stochastic forces. The associated magnetization lifetime is found to obey an Arrhenius law with an activation barrier equal to the domain wall energy in the chain. For chains longer than one domain-wall width, the reversal is initiated by nucleation of a reversed magnetization domain primarily at t…
Anti-ferromagnetic materials have the possibility to offer ultra fast, high data density spintronic devices. A significant challenge is the reliable detection of the state of the antiferromagnet, which can be achieved using exchange bias. Here we develop an atomistic spin model of the athermal training effect, a well known phenomenon in exchange biased systems where the bias is significantly reduced after the first hysteresis cycle. We find that the setting process in granular thin films relies on the presence of interfacial mixing between the ferromagnetic and antiferromagnetic layers. We sys…
As impermeable to gas molecules and at the same time transparent to high-energy ions, graphene has been suggested as a window material for separating a high-vacuum ion beam system from targets kept at ambient conditions. However, accumulation of irradiation-induced damage in the graphene membrane may give rise to its mechanical failure. Using atomistic simulations, we demonstrate that irradiated graphene even with a high vacancy concentration does not show signs of such instability, indicating a considerable robustness of graphene windows. We further show that upper and lower estimates for the…
We present full atomistic calculations of the spin-flip time (T$_{1}$) of electrons and holes mediated by acoustic phonons in self-assembled In$_{1-x}$Ga$_x$As/GaAs quantum dots at zero magnetic field. At low magnetic field, the first-order process is suppressed, and the second-order process becomes dominant. We find that the spin-phonon-interaction induced spin relaxation time is 40 - 80 s for electrons, and 1 - 20 ms for holes at 4.2 K. The calculated hole-spin relaxation times are in good agreement with recent experiments, which suggests that the two-phonon process is the main relaxation me…
In this work, we leverage atomistic spin-lattice simulations to examine how magnetic interactions impact the propagation of sound waves through a ferromagnetic material. To achieve this, we characterize the sound wave velocity in BCC iron, a prototypical ferromagnetic material, using three different approaches that are based on the oscillations of kinetic energy, finite-displacement derived forces, and corrections to the elastic constants, respectively. Successfully applying these methods within the spin-lattice framework, we find good agreement with the Simon effect including high order terms…
Ultrafast magnetization dynamics probes the most fundamental properties of magnetic materials, exploring questions about the fundamental interactions responsible for magnetic phenomena. Thermal effects are known to be extremely important for laser-induced dynamics in metallic systems, but the dynamics of topological magnetic structures are little understood. Here we apply a massively parallel atomistic spin dynamics simulation to study the response of a permalloy vortex to a 50 fs laser pulse. We find that macroscopically the short timescale dynamics are indistinguishable from the bulk, but th…
The exchange interaction determines the ferromagnetic (FM) or antiferromagnetic (AFM) ordering of atomic spins. When ferromagnets and antiferromagnets are coupled together, they often exhibit the exchange bias effect, a unidirectional interface exchange field causing a shift of the magnetic hysteresis loop. The effective magnitude of this interface exchange field is at most a few percent of the bulk exchange, arising from pinned interfacial spins in the antiferromagnet. The pinned spins are known to comprise a small fraction of the total number of interface spins, yet their exact nature and ph…
Hybrid micromagnetic and atomistic modeling of magnetization dynamics induced by engineered defects
Department of Physics and Astronomy, Uppsala University（乌普萨拉大学物理与天文系）；WISE - Wallenberg Initiative Materials Science for Sustainability, Department of Physics and Astronomy, Uppsala University（乌普萨拉大学物理与天文系瓦伦堡可持续性材料科学倡议）；PDC Center for High Performance Computing, KTH Royal Institute of Technology（瑞典皇家理工学院PDC高性能计算中心）
The dynamical behavior of the magnetism of diluted magnetic semiconductors (DMS) has been investigated by means of atomistic spin dynamics simulations. The conclusions drawn from the study are argued to be general for DMS systems in the low concentration limit, although all simulations are done for 5% Mn-doped GaAs with various concentrations of As antisite defects. The magnetization curve, $M(T)$, and the Curie temperature $T_C$ have been calculated, and are found to be in good correspondence to results from Monte Carlo simulations and experiments. Furthermore, equilibrium and non-equilibrium…
We present a general approach to the derivation of the effective anisotropy field which determines the dynamical behaviour of magnetic spins according to the Landau-Lifshitz-Gilbert equation. The approach is based on the gradient in spherical polar coordinates with the final results being expressed in Cartesian coordinates as usually applied in atomistic and micromagnetic model calculations. The approach is generally valid for all orders of anisotropies including higher order combinations of azimuthal and rotational anisotropies often found in functional magnetic materials such as permanent ma…
Signature of surface anisotropy in the spin-flip neutron scattering cross section of spherical nanoparticles: atomistic simulations and analytical theory
An extended atomistic spin model allowing for studies of the finite temperature magnetic properties of alloys is proposed. The model is obtained by extending the Heisenberg Hamiltonian via a parameterization from a first principles basis, interpolating from both the low temperature ferromagnetic and the high temperature paramagnetic reference states. This allows us to treat magnetic systems with varying degree of itinerant character within the model. Satisfactory agreement with both previous theoretical studies and experiments are obtained in terms of Curie temperatures and paramagnetic proper…
While the ground state of magnetic materials is in general well described on the basis of spin density functional theory (SDFT), the theoretical description of finite-temperature and non-equilibrium properties require an extension beyond the standard SDFT. Time-dependent SDFT (TD-SDFT), which give for example access to dynamical properties are computationally very demanding and can currently be hardly applied to complex solids. Here we focus on the alternative approach based on the combination of a parameterized phenomenological spin Hamiltonian and SDFT-based electronic structure calculations…
Three-temperature atomistic spin-lattice dynamics in LAMMPS: a moment-consistent, fluctuation-dissipation-correct extension and its validation on ultrafast demagnetization and all-optical switching of GdFeCo
DGIST (Daegu Gyeongbuk Institute of Science and Technology)（大邱庆北科学技术院）；Kunsan National University（群山国立大学）；Chungbuk National University（忠北国立大学）
A coupled atomistic spin and lattice dynamics approach is developed which merges the dynamics of these two degrees of freedom into a single set of coupled equations of motion. The underlying microscopic model comprises local exchange interactions between the electron spin and magnetic moment and the local couplings between the electronic charge and lattice displacements. An effective action for the spin and lattice variables is constructed in which the interactions among the spin and lattice components are determined by the underlying electronic structure. In this way, expressions are obtained…
Temperature dependence of spin-model parameters in ferrimagnets
Instituto de Ciencia de Materiales de Madrid, CSIC（马德里材料科学研究所，西班牙国家科学研究委员会）；Hitachi Cambridge Laboratory（日立剑桥实验室）；HUN-REN Wigner Research Centre for Physics（匈牙利研究网络维格纳物理研究中心）；Budapest University of Technology and Economics（布达佩斯技术与经济大学）
Uncovering multi-channel magnetic hopfion annihilation via a single-node, billion-spin-scale atomistic framework
KTH Royal Institute of Technology（皇家理工学院）；Swedish e-Science Research Center（瑞典电子科学研究中心）；Wallenberg Initiative Materials Science for Sustainability (WISE)（瓦伦贝格材料科学可持续发展倡议）
We study the Fe/W(110) monolayer system through a combination of first principles calculations and atomistic spin dynamics simulations. We focus on the dispersion of the spin waves parallel to the [001] direction. Our results compare favorably with the experimental data of Prokop et al. [Phys. Rev. Lett. 102, 177206], and correctly capture a drastic softening of the magnon spectrum, with respect to bulk bcc Fe. The suggested shortcoming of the itinerant electron model, in particular that given by density functional theory, is refuted. We also demonstrate that finite temperature effects are sig…
Extreme-Scale Atomistic Simulation of Real-Temperature Magnetic Skyrmion Dynamics by Coupled Spin-Lattice Modeling
Sun Yat-sen University（中山大学）；Graduate School of China Academy of Engineering Physics（中国工程物理研究院研究生院）；Southeast University（东南大学）；Suzhou Laboratory（苏州实验室）；Central South University（中南大学）
Halide perovskites make efficient solar cells due to their exceptional optoelectronic properties, but suffer from several stability issues. The characterization of the degradation processes is challenging because of the limitations in the spatio-temporal resolution in experiments and the absence of efficient computational methods to study the reactive processes. Here, we present the first effort in developing reactive force fields for large scale molecular dynamics simulations of the phase instability and the defect-induced degradation reactions in inorganic CsPbI$_{3}$. We find that the phase…
The full realization of spin qubits for quantum technologies relies on the ability to control and design the formation processes of spin defects in semiconductors and insulators. We present a computational protocol to investigate the synthesis of point-defects at the atomistic level, and we apply it to the study of a promising spin-qubit in silicon carbide, the divacancy (VV). Our strategy combines electronic structure calculations based on density functional theory and enhanced sampling techniques coupled with first principles molecular dynamics. We predict the optimal annealing temperatures …
The path probability method (PPM), which is a natural extension of the cluster variation method (CVM) to a time domain, has been employed in a relaxation process of atomic configurations in alloy systems. Although the vacancy mechanism is the main atomic migration process in an alloy system, most studies of PPM have used the spin flipping mechanism (or the direct exchange mechanism) because of the huge computational burden imposed by the vacancy mechanism. In this paper the computational problem is circumvented by treating various path variables in the PPM as cluster probabilities in the CVM, …
Controlling magnetism in low dimensional materials is essential for designing devices that have feature sizes comparable to several critical length scales that exploit functional spin textures, allowing the realization of low-power spintronic and magneto-electric hardware. [1] Unlike conventional covalently-bonded bulk materials, van der Waals (vdW)-bonded layered magnets [2-4] offer exceptional degrees of freedom for engineering spin textures. [5] However, their structural instability has hindered microscopic studies and manipulations. Here, we demonstrate nanoscale structural control in the …
We studied the effect of strong electric current on spin waves interacting relativistically with the current. The spin-wave spectrum is calculated at arbitrary direction of the wave vector. It is shown that the alternating Hall current generated by the alternating magnetic moment of the spin waves, reduces the spin-wave damping. At strong enough unpolarized dc current the damping changes sign, and the spin-wave amplitude starts to increase exponentially fast with time. The critical current for the spin-wave instability is determined mainly by the anomalous Hall effect, and can be much smaller …
It is well known that conventional harmonic lattice dynamics cannot be applied to energetically unstable crystals at 0 K, such as high temperature body centered cubic (BCC) phase of crystalline Zr. Predicting phonon spectra at finite temperature requires the calculation of force constants to the third, fourth and even higher orders, however, it remains challenging to determine to which order the Taylor expansion of the potential energy surface for different materials should be cut off. Molecular dynamics, on the other hand, intrinsically includes arbitrary orders of phonon anharmonicity, howev…
The magnetization of body-centered cubic iron at low temperatures is calculated with the steepest-entropy-ascent quantum thermodynamics (SEAQT) framework. This framework assumes that a thermodynamic property in an isolated system traces the path through state space with the greatest entropy production. Magnetization is calculated from the expected value of a thermodynamic ensemble of quantized spin waves based on the Heisenberg spin model applied to an ensemble of coupled harmonic oscillators. A realistic energy landscape is obtained from a magnon dispersion relation calculated using spin-dens…
The stability is analyzed of the magnetic junction collinear configurations against small fluctuations under amplitude-modulated current with CPP mode. High spin injection is assumed. Under parametric resonance conditions, with the modulation frequency twice the precession frequency, instability is possible of one, or another, or both the collinear configurations. When the dc component of the current density exceeds the instability threshold of the antiparallel configuration, the parametric instability is suppressed by nonparametric one which is induced by the dc current. The parametric instab…
It is well established that it is possible to switch certain antiferromagnets electrically, yet the interplay of Néel-spin-orbit torques and thermal activation is only poorly understood. Combining ab initio calculations and atomistic spin dynamics simulations we develop a multiscale model to study the current induced switching in Mn2Au. We compute from first principles the strength and direction of the electrically induced magnetic moments, caused by the Rashba--Edelstein effect, and take these into account in atomistic spin dynamics simulations. Our simulations reveal the switching paths as w…
The calculation of free energies from first principles in materials is a formidable task which enables the prediction of phase stability with high accuracy; these calculations are complicated in magnetic materials by the interplay of electronic, magnetic, and vibrational degrees of freedom. In this work, we show the feasibility and accuracy of the calculation of phase stability in magnetic systems with ab initio methods and thermodynamic integration by sampling the magnetic and vibrational phase space with coupled atomistic spin dynamics-ab initio molecular dynamics (ASD-AIMD) simulations [Sto…
In the framework of phenomenological model we consider dynamics of a compensated collinear antiferromagnet (AFM) in the presence of spin-polarised current. The model is based on the assumption that AFM spins are localised and spin torque is transferred to each magnetic sublattice independently. It is shown that in AFM spin current i) can be a source of the "negative friction"; and ii) modifies spin-wave frequencies. Equilibrium state of AFM can be destabilized by the current polarized in parallel to AFM vector. Threshold current at which the loss of stability takes place depends upon the magne…
A simulation framework that couples atomistic electronic structures to Boltzmann transport formalism is developed and applied to calculate the transport characteristics of thin silicon nanowires (NWs) up to 12nm in diameter. The sp3d5s*-spin-orbit-coupled atomistic tight-binding (TB) model is used for the electronic structure calculation. Linearized Boltzmann transport theory is applied, including carrier scattering by phonons, surface roughness (SRS), and impurities. We present a comprehensive investigation of the low-field mobility in silicon NWs considering: i) n- and p-type NWs, ii) [100],…
A temperature-dependent approach involving Green-Kubo equilibrium atomic and spin dynamics (GKEASD) is reported to assess phonon and magnon thermal transport processes accounting for phonon-magnon interactions. Using body-center cubic (BCC) iron as a case study, GKEASD successfully reproduces its characteristic temperature-dependent spiral and lattice thermal conductivities. The non-electronic thermal conductivity, i.e., the sum of phonon and magnon thermal conductivities, calculated using GKEASD for BCC Fe agrees well with experimental measurements. Spectral energy analysis reveals that high-…
We have developed a method to simulate behavior of nanoporous materials in a molecular dynamics code. The nanoporous solid is produced via a spinodal decomposition of a material brought from a supercritical fluid into the two phase (liquid-vapor) region and then quenching and freezing the liquid into an interconnected nanoporous solid. We have simulated, at the atomic level, compression in crystal/nanoporous configurations, demonstrating that this is a powerful technique for studying the equation-of-state of cold and warm dense matter. By performing compression simulations relevant to high ene…
This work reports the measurement of electron $g$ factor anisotropy ($| Δg |$ = $| g_{001} - g_{1 \bar 1 0} |$) for phosphorous donor qubits in strained silicon (sSi = Si/Si$_{1-x}$Ge$_x$) environments. Multi-million-atom tight-binding simulations are performed to understand the measured decrease in $| Δg |$ as a function of $x$, which is attributed to a reduction in the interface-related anisotropy. For $x <$7\%, the variation in $| Δg |$ is linear and can be described by $η_x x$, where $η_x \approx$1.62$\times$ 10$^{-3}$. At $x$=20\%, the measured $| Δg |$ is 1.2 $\pm$ 0.04 $\times$ 10$^{-3}…
We show that an unpolarized electric current incident perpendicular to the plane of a thin ferromagnet can excite a spin-wave instability transverse to the current direction if source and drain contacts are not symmetric. The instability, which is driven by the current-induced ``spin-transfer torque'', exists for one current direction only.
A sufficiently large unpolarized current can cause a spin-wave instability in thin nanomagnets with asymmetric contacts. The dynamics beyond the instability is understood in the perturbative regime of small spin-wave amplitudes, as well as by numerically solving a discretized model. In the absence of an applied magnetic field, our numerical simulations reveal a hierarchy of instabilities, leading to chaotic magnetization dynamics for the largest current densities we consider.
The localized magnon modes of isolated $kπ$ skyrmions on a field-polarized background are analyzed based on the Landau-Lifshitz-Gilbert equation within the terms of an atomistic classical spin model, with system parameters based on the Pd/Fe biatomic layer on Ir(111). For increasing skyrmion order $k$ a higher number of excitation modes are found, including modes with nodes in the radial eigenfunctions. It is shown that at low fields $2π$ and $3π$ skyrmions are destroyed via a burst instability connected to a breathing mode, while $1π$ skyrmions undergo an elliptic instability. At high fields …
