Solving the stochastic Landau-Lifshitz-Gilbert-Slonczewski equation for monodomain nanomagnets : A survey and analysis of numerical techniques

Abstract

The stochastic Landau-Lifshitz-Gilbert-Slonczewski (s-LLGS) equation is widely used by researchers to study the temporal evolution of the macrospin subject to spin torque and thermal noise. The numerical simulation of the s-LLGS equation requires an appropriate choice of stochastic calculus and numerical integration scheme. In this paper, we first comprehensively evaluate the accuracy and complexity of various numerical techniques to solve the s-LLGS equation. We focus on implicit midpoint, Heun, and Euler-Heun methods that converge to the Stratonovich solution of the s-LLGS equation. By performing several numerical tests, for both strong (path-wise) and weak (statistical) convergence, we quantify the accuracy of various numerical schemes used to solve the s-LLGS equation. We demonstrate a new method intended to solve stochastic differential equations (SDEs) with small noise (the RK4-Heun method), and test its capability to handle the s-LLGS equation. We also discuss the circuit implementation of nanomagnets for large-scale SPICE-based simulations in a standard hierarchical circuit simulator. We evaluate the efficacy of SPICE in handling the stochastic dynamics of the multiplicative noise in the s-LLGS equation. Numerical schemes such as Euler and Gear, which are typically used by SPICE-based circuit simulators do not yield the expected outcome when solving the Stratonovich s-LLGS equation. While the trapezoidal method in SPICE does solve for the Stratonovich solution, its accuracy is limited by the minimum time step of integration in SPICE. We implement the s-LLGS equation in both its cartesian and spherical coordinates form in SPICE and compare the stability and accuracy of the two implementations. The results in this paper will serve as guidelines for researchers to understand the tradeoffs between accuracy and complexity of various numerical methods and the choice of appropriate calculus to solve the s-LLGS equation.

Ising Machine Based on Electrically Coupled Spin Hall Nano-Oscillators

“…We introduce an ISF-based macromodel of the SHNO in Verilog-A that emulates the oscillator's electrical behavior, nonlinear phase dynamics, and thermal phase noise characteristic [33][34][35][36][37]. Previous approaches to model the spin torque oscillator at the circuit level have involved numerically simulating the LLGS equation using an equivalent circuit representation [38][39][40], which is computationally expensive, or solving analytical equations that accurately model the electrical behavior, including output power and linewidth, but do not include nonlinear injection locking [10,[41][42][43][44]. On the other hand, our analytical ISF-based approach comprehensively models the nonlinear behavior of the oscillator without requiring heavy computation.…”

SMART: A Secure Magnetoelectric AntifeRromagnet-Based Tamper-Proof Non-Volatile Memory

“…Fig. 4 shows the magnetic eld-induced bit ip in a representative FM, obtained by solving the Landau-Lifshitz-Gilbert equation for FM dynamics [38].…”

Thermal melting of discrete time crystals: a dynamical phase transition induced by thermal fluctuations

“…Numerically, we adopt Stratonovich's formula of the stochastic differential Eq. ( 7), and solve it using the standard Heun method [42] with the time step of ∆t = 10 −3 , the convergence of which has been checked numerically (see the Supplmentary Material(SM) [43]). The system size in our simulation ranges from L = 8 to L = 28, which enable us to systematically analyze the finite-size effect.…”