Double-Free-Layer Stochastic Magnetic Tunnel Junctions with Synthetic Antiferromagnets

TL;DR

This work tackles realizing fast, energy-efficient stochastic MTJs suitable for probabilistic computing by introducing a double-free-layer MTJ with synthetic antiferromagnet free layers. It combines a spin-circuit model with stochastic LLG dynamics to capture transport, dipolar, exchange, and thermal effects, enabling self-consistent simulations of a four-magnet SAF stack. The results show that low-barrier SAF layers suppress dipolar coupling, yielding near-zero cos θ fluctuations and uncorrelated behavior up to diameters around $D \approx 100\ \text{nm}$ for $t \approx 1$–$2$\,nm, with bias independence and uniform randomness preserved. The authors estimate an energy per random bit of $\approx 3.6$\,fJ and a flip rate of $\approx 3.3$\,GHz per p-bit, and demonstrate integration with CMOS to realize a scalable, energy-efficient probabilistic computing substrate relevant for AI/ML tasks.

Abstract

Stochastic magnetic tunnel junctions (sMTJ) using low-barrier nanomagnets have shown promise as fast, energy-efficient, and scalable building blocks for probabilistic computing. Despite recent experimental and theoretical progress, sMTJs exhibiting the ideal characteristics necessary for probabilistic bits (p-bit) are still lacking. Ideally, the sMTJs should have (a) voltage bias independence preventing read disturbance (b) uniform randomness in the magnetization angle between the free layers, and (c) fast fluctuations without requiring external magnetic fields while being robust to magnetic field perturbations. Here, we propose a new design satisfying all of these requirements, using double-free-layer sMTJs with synthetic antiferromagnets (SAF). We evaluate the proposed sMTJ design with experimentally benchmarked spin-circuit models accounting for transport physics, coupled with the stochastic Landau-Lifshitz-Gilbert equation for magnetization dynamics. We find that the use of low-barrier SAF layers reduces dipolar coupling, achieving uncorrelated fluctuations at zero-magnetic field surviving up to diameters exceeding ($D\approx 100$ nm) if the nanomagnets can be made thin enough ($\approx 1$-$2$ nm). The double-free-layer structure retains bias-independence and the circular nature of the nanomagnets provides near-uniform randomness with fast fluctuations. Combining our full sMTJ model with advanced transistor models, we estimate the energy to generate a random bit as $\approx$ 3.6 fJ, with fluctuation rates of $\approx$ 3.3 GHz per p-bit. Our results will guide the experimental development of superior stochastic magnetic tunnel junctions for large-scale and energy-efficient probabilistic computation for problems relevant to machine learning and artificial intelligence.

Figures (7)

• Figure 1: Stochastic MTJ (sMTJ) designs (a) This work: Double-SAF-free sMTJ where the low barrier free layers are replaced by magnetically inert SAF layers. (b) Double-free-layer sMTJ which no fixed layer but two low-barrier free layers. (c) Standard sMTJ with a fixed layer and a low-barrier free layer: commonly used in literature since it minimally modifies existing stable MTJs with a fixed layer to have a low-barrier free layer.
• Figure 2: Modular spin-circuit model for the double-SAF-free layer structure (a) Close-up look on the spin-circuit transport model for a single SAF stack-layer: represented with two ferromagnet-normal metal (F$|$N) interfaces and a normal metal (NM) block. (b) The full model coupling magnetizaton dynamics (sLLG) with spin-transport modules. The transport module produces spin-polarized currents input to sLLGs. For the two interfaces of the same magnet, we add the spin-currents incident to the magnets vectorially (e.g., $\vec{I}_{s2} = \vec{I}_{s21} + \vec{I}_{s22}$) as an input to the sLLGs, since we are in the monodomain approximation for the magnets. LLGs in turn produce instantaneous magnetization vectors back to the interface modules. In addition, the two interfaces of the same magnet receive the same magnetization vector from their corresponding sLLG's as shown. Dipolar, exchange and thermal noise are considered by the sLLG model. Details of the 4$\times$4 conductances and the sLLG module can be found in Ref.'s camsari2015modularcamsari2015modularAtorunbalci2018modularnanohub:spintronics. Numerical parameters are summarized in Table \ref{['tab:simulation_parameters']}.
• Figure 3: Dipolar coupling in the double-SAF-free layer stucture (a) Dipolar coupling coefficients $\mathrm{D_{ij}}$ between magnets $(i,j)$ are shown at varying diameters (1 nm thickness), numerically calculated based on the approach described in Ref. camsari2021double. The effective dipolar ($\text{D}_\text{eff}$) coupling (Eq. \ref{['eq:deff']}) is also shown. $\mathrm{D_{12}}$ and $\mathrm{D_{34}}$ are combined with exchange tensor ($J_{ij}$) for SAF couples in the numerical model but not shown in the plot. (b) Average angle between layers (2,3), $\langle \cos \ \theta \rangle$, are calculated numerically (markers) and analytically (solid and dashed lines), based on (Eq. \ref{['eq:bessel']}), as a function of varying magnet diameters and thicknesses. Numerical calculations for the sLLG model uses a 1 ps time-step and averages are taken over 5 $\mu s$.
• Figure 4: Autocorrelation of magnetizations (a) shows numerical and theoretical autocorrelation for the in-plane component of layers (2,3). (b) shows the numerical and theoretical autocorrelation of $\cos \theta$. Numerical results are obtained for a diameter of 50 nm and a thickness, t = 1 nm. Numerical results are obtained over 5 $\mu$s using a timestep of 1 ps in sLLG simulations. (c) Time-dependent magnetizations of all layers from the full numerical model. Histogram for the cosine of the angle between layers (2,3) $\cos \ \theta$ demonstrating the necessary near uniform randomness between -1 and +1 for p-bit operation.
• Figure 5: Out-of-plane magnetizations of easy-plane SAFs and single nanomagnets (a) The out-of-plane components of magnets $m_z^2$ are evaluated as a function of diameter analytically (Eq. \ref{['eq:ima-mz']} - Eq. \ref{['eq:saf-mz']}) and numerically for single and SAF easy-plane magnets at zero bias (equilibrium), along with their probability distributions obtained from the Boltzmann law ($\rho\propto \exp(-E)$). (b) $\langle {m}_z^2 \rangle$ is shown numerically at different voltage biases from $-$0.5 V to 0.5 V for a range of diameters from 10 nm to 200 nm with free layer thickness of 1 nm. Each point is averaged over 250 ns simulations with 1 ps transient time step using the .trannoise function in HSPICE.