Landau Lifshitz Gilbert Slonczewski Equation Research Articles

Non-uniform superlattice magnetic tunnel junctions

We propose a new class of non-uniform superlattice magnetic tunnel junctions (Nu-SLTJs) with the linear, Gaussian, Lorentzian, and Pöschl–Teller width and height based profiles manifesting a sizable enhancement in the TMR (≈104 − 106%) with a significant suppression in the switching bias (≈9 folds) owing to the physics of broad-band spin filtering. By exploring the negative differential resistance region in the current–voltage characteristics of the various Nu-SLTJs, we predict the Nu-SLTJs offer fastest spin transfer torque switching in the order of a few hundred picoseconds. We self-consistently employ the atomistic non-equilibrium Green’s function formalism coupled with the Landau–Lifshitz–Gilbert–Slonczewski equation to evaluate the device performance of the various Nu-SLTJs. We also present the design of minimal three-barrier Nu-SLTJs having significant TMR (≈104%) and large spin current for the ease of device fabrication. We hope that the class of Nu-SLTJs proposed in this work may lay the bedrock to embark on the exhilarating voyage of exploring various non-uniform superlattices for the next generation of spintronic devices.

Field-free switching of magnetization induced by spin–orbit torque in Pt/CoGd/Pt thin film

All-electric control of magnetism is essential for the future ultralow-power memory and logic spintronic applications. Here, we report a highly efficient way to realize field-free current-induced switching of magnetization in ferrimagnetic CoGd, which is deposited with composite target. The critical switching current density is as low as ∼1 × 107 A/cm2. Without any gradient designs, our CoGd film intrinsically exhibits extraordinary titled magnetic anisotropy and bias-fields. The field-free switching of magnetizations can be achieved by applying current pulses in x or y directions. We further establish a coupled ferrimagnetic macrospin model subjected to the Landau–Lifshitz–Gilbert–Slonczewski equation, and the theoretical results agree with experiments well. All these results suggest that deposition with composite target is the optimal route to fabricate a high performance spin–orbit torque device, which provides multiple routes to achieve field-free, deterministic, and low-consumption magnetization switching.

Spin torque oscillations triggered by in-plane field

We study the dynamics of a spin torque nano oscillator that consists of parallelly magnetized free and pinned layers by numerically solving the associated Landau–Lifshitz–Gilbert–Slonczewski equation in the presence of a field-like torque. We observe that an in-plane magnetic field which is applied for a short interval of time (1 ns) triggers the magnetization to exhibit self-oscillations from low energy initial magnetization state. Also, we confirm that the frequency of oscillations can be tuned over the range ∼25–∼72 GHz by current, even in the absence of field-like torque. We find the frequency enhancement up to 10 GHz by the presence of field-like torque. We determine the Q-factor for different frequencies and show that it increases with frequency. Our analysis with thermal noise confirms that the system is stable against thermal noise and the dynamics is not altered appreciably by it.

Emergence of Nontrivial Spin Textures in Frustrated Van Der Waals Ferromagnets.

In this work, first principles ground state calculations are combined with the dynamic evolution of a classical spin Hamiltonian to study the metamagnetic transitions associated with the field dependence of magnetic properties in frustrated van der Waals ferromagnets. Dynamically stabilized spin textures are obtained relative to the direction of spin quantization as stochastic solutions of the Landau–Lifshitz–Gilbert–Slonczewski equation under the flow of the spin current. By explicitly considering the spin signatures that arise from geometrical frustrations at interfaces, we may observe the emergence of a magnetic skyrmion spin texture and characterize the formation under competing internal fields. The analysis of coercivity and magnetic hysteresis reveals a dynamic switch from a soft to hard magnetic configuration when considering the spin Hall effect on the skyrmion. It is found that heavy metals in capped multilayer heterostructure stacks host field-tunable spiral skyrmions that could serve as unique channels for carrier transport. The results are discussed to show the possibility of using dynamically switchable magnetic bits to read and write data without the need for a spin transfer torque. These results offer insight to the spin transport signatures that dynamically arise from metamagnetic transitions in spintronic devices.

Proposal for energy efficient spin transfer torque-magnetoresistive random access memory device

Utilizing the electronic analogs of optical phenomena such as anti-reflection coating and resonance for spintronic devices, we propose and theoretically analyze the design of a spin transfer torque-magnetoresistive random access memory (STT-MRAM) device. The proposed device consists of a superlattice heterostructure terminated with the anti-reflective regions sandwiched between the fixed and free ferromagnetic layers. Employing Green’s function spin-transport formalism coupled self-consistently with the stochastic Landau–Lifshitz–Gilbert–Slonczewski equation, we design an STT-MRAM based on the anti-reflective superlattice magnetic tunnel junction (AR-SLMTJ) device having an ultrahigh tunnel magnetoresistance (≈3.5×104%) and large spin current. We demonstrate that the STT-MRAM based on the AR-SLMTJ structure owing to the physics of bandpass spin filtering is nearly 1100% more energy efficient than trilayer magnetic tunnel junction (MTJ) based STT-MRAM. We also present detailed probabilistic switching and energy analysis to find out the optimal point of operation of a trilayer MTJ and AR-SLMTJ based STT-MRAM. Our predictions serve as a template to consider the heterostructures for next-generation spintronic device applications.

Study of Spin Pumping through α‐Sn Thin Films

Elemental tin in the α‐phase is an intriguing member of the family of topological quantum materials. In thin films, with decreasing thickness, α‐Sn transforms from a 3D topological Dirac semimetal (TDS) to a 2D topological insulator (TI). Getting access to and making use of its topological surface states is challenging and requires interfacing to a magnetically ordered material. Herein, the successful epitaxial growth of α‐Sn thin films on Co, forming the core of a spin‐valve structure, is reported. Time‐ and element‐selective ferromagnetic resonance experiments are conducted to investigate the presence of spin pumping through the spin‐valve structure. A rigorous statistical analysis of the experimental data using a model based on the Landau–Lifshitz–Gilbert–Slonczewski equation is applied. A strong exchange coupling contribution is found, however no unambiguous proof for spin pumping. Nevertheless, the incorporation of α‐Sn into a spin valve remains a promising approach given its simplicity as an elemental TI and its room‐temperature application potential.

Self-organization in the one-dimensional Landau–Lifshitz–Gilbert–Slonczewski equation with non-uniform anisotropy fields

In magnetic films driven by spin-polarized currents, the perpendicular-to-plane anisotropy is equivalent to breaking the time translation symmetry, i.e., to a parametric pumping. In this work, we numerically study those current-driven magnets via the Landau–Lifshitz–Gilbert–Slonczewski equation in one spatial dimension. We consider a space-dependent anisotropy field in the parametric-like regime. The anisotropy profile is antisymmetric to the middle point of the system. We find several dissipative states and dynamical behavior and focus on localized patterns that undergo oscillatory and phase instabilities. Using numerical simulations, we characterize the localized states’ bifurcations and present the corresponding diagram of phases.

Frequency enhancement and power tunability in tilted polarizer spin-torque nano-oscillator

In the absence of an applied magnetic field, a spin-torque nano-oscillator with a tilted polarizer is studied using numerical simulations of the associated Landau–Lifshitz–Gilbert–Slonczewski equation. We find considerable enhancement of frequencies by tilting the polarizer out-of-plane appropriately. Also, we observe improved tunability of frequencies of oscillations from ∼15GHz to ∼75GHz and increments in the power spectral density by current and tilt angles. In addition, our findings and insights pave a simple way for implementation of nanoscale level microwave generators.

Simulation of Strain-Assisted Switching in Synthetic Antiferromagnetic Free Layer-Based Magnetic Tunnel Junction

We perform macrospin simulations on a magnetic tunnel junction structure with perpendicular magnetic anisotropy, in which the conventional free layer is substituted with synthetic antiferromagnetic free layers to improve the thermal stability. In order to reduce the switching current Pb(Zr0.2Ti0.8)O3 material with (001) surface orientation is adopted, which is adjacent to the second free layer so that it applies biaxial stress to the interface under external voltage. We solve the Landau–Lifshitz–Gilbert–Slonczewski equation incorporating thermal and strain fields to describe the transient dynamics of magnetizations in both the free layers. The simulation results show a substantial reduction in switching current and significantly enhanced switching probability by modulating current and strain pulses.

Micromagnetic simulation of current-induced magnetization dynamics in a half-Heusler Co1.5Fe1.5Ge alloy spin-valve nanopillar

We investigated the magnetization switching by means of a spin-polarized current via spin-transfer torque (STT) in a half-metallic Heusler Co1.5Fe1.5Ge alloy by numerically solving the Landau–Lifshitz–Gilbert–Slonczewski equation including the STT term in a spin-valve nanopillar Co1.5Fe1.5Ge/Ag/Co1.5Fe1.5Ge using micromagnetic simulation. In this work, we use Co1.5Fe1.5Ge film of 2 nm thickness, and then we inject a spin polarization current perpendicular to plane of the multilayers in order to flip the free layer magnetization and therefore get the magnetization switching between the two layers. Half-metallic Heusler Co1.5Fe1.5Ge is characterized by a small Gilbert damping constant α = 0.0025 and a complete spin polarization at the Fermi level. Using these specific parameters, we obtain the magnetization switching between the free layer and the pinned layer with a critical current density of about 106 A/cm2, as well the magnetization switching process of nanopillar are discussed and analyzed. Also, we deeply examine the behavior of magnetization switching dynamic as a function of the current density within a weak external magnetic field which helps to trig switching. Finally, we studied the impact of free layer thickness on magnetization switching dynamic and STT energy in order to immunize further the switching time and the critical current density.

Spin–transfer torque oscillator in magnetic tunneling junction with short–wavelength magnon excitation

Bloch–Bloembergen–Slonczewski (BBS) equation is established by extending Bloch–Bloembergen equation, and it is used to study magnetization oscillation in the free magnetic layer of a magnetic tunneling junction. Since both short–wavelength magnon excitation and spin–transfer torque are taken into account in the BBS equation, it is distinguished from Landau–Lifshitz–Gilbert–Slonczewski equation. The macro–spin BBS model predicts that the transverse relaxation time in free magnetic layer should be long enough, as compared with the longitudinal relaxation time, to achieve stable magnetization oscillation for spin–transfer torque oscillator application. Moreover, field–like torque favors the tolerance of fast transverse relaxation, which makes magnetic tunneling junction a better choice than spin valve for the spin–transfer torque oscillator application.

Loading characteristics of a spin-transfer nano-oscillator

A family of loading and control characteristics of a spin-transfer nano-oscillator (STNO) at various values of direct current have been constructed by integrating the Landau–Lifshitz–Gilbert–Slonczewski equation. The obtained characteristics are compared to approximated dependences calculated using truncated equations with respect to the first harmonic. An approximate expression is derived for the load resistance ensuring the maximum STNO output power.

Alternating superlattice textures in driven nanomagnets

Nanomagnets driven with uniform electric currents exhibit a wide variety of spatial textures. In the present work, we investigate alternating superlattice states in nanomagnets, which are spatially periodic textures composed by several spatial modes that oscillate in time. The magnetic system is described in the continuum approach by the Landau–Lifshitz–Gilbert–Slonczewski equation, and direct numerical simulations of this model allow us to characterize the alternating patterns. As a result of this temporal oscillation, textures alternate between different shapes. In particular, we focus on two types of textures, namely a superhexagon and a square-like pattern, which are composed by six and two dominant Fourier modes, respectively. Based on an appropriate modal decomposition, we reveal that the mechanism that originates the alternating superhexagon is a homoclinic bifurcation. In addition, we show that the oscillatory square-like texture emerges through a supercritical Andronov–Hopf bifurcation.

Modeling and Design Space Exploration for Bit-Cells Based on Voltage-Assisted Switching of Magnetic Tunnel Junctions

The effect of voltage on the anisotropy of a magnetic tunnel junction (MTJ) is of substantial interest for low-power nonvolatile memory applications. In this paper, we develop a device-to-bit-cell level simulation framework for voltage-assisted switching of the MTJs, which can satisfactorily reproduce the published experimental data. Our simulation framework is based on a coupled Landau–Lifshitz–Gilbert–Slonczewski equation and nonequilibrium greens function formalism. Using this simulation framework, we investigate the effect of scaling the oxide thickness of an MTJ to enhance the electric field effect. Although it seems attractive for an isolated device, yet, in a bit-cell configuration, reducing the oxide thickness leads to an increase in the supply voltage. In addition, we demonstrate that the unipolar characteristic of the voltage-assisted switching leads to write failures depending on the initial state stored in the MTJ. We, therefore, suggest a read before write scheme exhibiting approximately $2\times $ improvement in the write energy consumption at a bit-cell level compared with a standard STT-MRAM for iso-oxide thickness. However, the latency overhead associated with voltage-assisted MTJs results in a $1.55\times $ degradation of the write speed. Moreover, our layout analysis indicates that the bit-cell area reduction is constrained by the metal pitch, in spite of smaller access transistors.

Magnetic Pattern Recognition Using Injection-Locked Spin-Torque Nano-Oscillators

In this paper, we propose an efficient methodology for detecting magnetic field and thereby magnetic patterns using an injection-locked spin-torque nano-oscillator (STNO) array. We demonstrate the methodology with the implementation of a physical STNO model based on the Landau–Lifshitz–Gilbert–Slonczewski equation, which is benchmarked with experimental data. Based on our simulations, we provide an analysis of how the STNO, as a field-controlled oscillator, together with injection locking, can be used to sense magnetic fields and thereby magnetic patterns. The output can be sensed using simple CMOS peripheral circuitry.

High frequency spin torque oscillators with composite free layer spin valve

We report the oscillations of magnetic spin components in a composite free layer spin valve. The associated Landau–Lifshitz–Gilbert–Slonczewski (LLGS) equation is studied by stereographically projecting the spin on to a complex plane and the spin components were found. A fourth order Runge–Kutta numerical integration on LLGS equation also confirms the similar trajectories of the spin components. This study establishes the possibility of a Spin Torque Oscillator in a composite free layer spin valve, where the exchange coupling is ferromagnetic in nature. In-plane and out-of-plane precessional modes of magnetization oscillations were found in zero applied magnetic field and the frequencies of the oscillations were calculated from Fast Fourier Transform of the components of magnetization. Behavior of Power Spectral Density for a range of current density is studied. Finally our analysis shows the occurrence of highest frequency 150GHz, which is in the second harmonics for the specific choice of system parameters.

Zero-Magnetic-Field Oscillation of Spin Transfer Nano-Oscillator with a Second-Order-Perpendicular-Anisotropy Free Layer**Supported by the National Natural Science Foundation of China under Grant Nos 11204203 and 61274089, and the International Technology Collaboration Program of Shanxi Province under Grant No 201481029-2.

The zero-magnetic-field oscillation behavior of spin torque nano-oscillator (STNO) with a perpendicularly magnetized free layer with second-order uniaxial anisotropy is studied theoretically based on the Landau–Lifshitz–Gilbert–Slonczewski equation. It is demonstrated numerically that the second-order uniaxial anisotropy plays a significant role in the occurrence of a zero-magnetic-field steady-state precession, which can be understood in terms of the energy balance between the energy accumulation due to the spin torque and the energy dissipation due to the Gilbert damping. In particular, a relatively large zero-magnetic-field-oscillation current region, in which the corresponding microwave frequency is increased while the threshold current still maintains an almost constant value, can be obtained by modulating the second-order uniaxial anisotropy of the free layer. These results suggest a tunable zero-magnetic-field STNO, and it may be a promising configuration for STNO's applications in future wireless communications.

On Dynamics of Current-Induced Static Wall Profiles in Ferromagnetic Nanowires Governed by the Rashba Field

This article deals with the analytical study of propagation of static wall profiles in ferromagnetic nanowires under the effect of spin–orbit Rashba field. We consider the governing dynamics for the evolution of magnetization inside the ferromagnetic material as an extended version of Landau–Lifshitz–Gilbert–Slonczewski equation of micromagnetism. It comprises the nonlinear dissipation factors like dry-friction and viscous. We establish the threshold and Walker-type breakdown estimates of the external sources in the steady-regime and also illustrate the obtained results numerically.

Effects of the channel material parameters on the spin-torque critical current of lateral spin valves

The lateral spin valve structures operated by the principle of spin- torque effect have shown great potentials for constructing the all- spin – logic. Like vertical spin valves, the problem of excessive spin-torque critical current also exists in lateral spin valves. Appropriate channel materials may reduce the spin-torque critical current, so finding the right channel material is a direction of many authors’ efforts over years. In this paper, the four-component distributed spin-circuit model is used to calculate the spin-polarized current which is transported from the polarizer (fixed layer) to the detector (free layer). The effects of channel material parameters on the spin-torque critical current in the free layer of the lateral spin valve are studied numerically by using the Landau–Lifshitz–Gilbert–Slonczewski equation. Based on the numerical results, the problem of whether graphene is a suitable channel material is discussed.

Investigations on the Magnetization Switching Dynamics in Spin Valve Multilayers Under Periodic Applied Magnetic Field

Magnetization switching dynamics in a spin valve nanopillar, induced by spin transfer torque in the presence of a periodic applied field is investigated by solving the Landau–Lifshitz–Gilbert–Slonczewski equation. Under steady state conditions, the switching of magnetization occurs in the system, above a threshold current density value Jc. A general expression for the critical current density is derived and it is shown that this further reduces when there is magnetic interface anisotropy present in the free layer of the spin valve. We also investigated the chaotic behavior of the free layer magnetization vector in a periodically varying applied magnetic field, in the presence of a constant DC magnetic field and spin current. Further, it is found that in the presence of a nonzero interfacial anisotropy, chaotic behavior is observed even at much smaller values of the spin current and DC applied field.