Stochastic magnetic tunnel junctions (sMTJ) using low-barrier nanomagnetshave shown promise as fast, energy-efficient, and scalable building blocks forprobabilistic 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 biasindependence preventing read disturbance (b) uniform randomness in themagnetization angle between the free layers, and (c) fast fluctuations withoutrequiring external magnetic fields while being robust to magnetic fieldperturbations. Here, we propose a new design satisfying all of theserequirements, using double-free-layer sMTJs with synthetic antiferromagnets(SAF). We evaluate the proposed sMTJ design with experimentally benchmarkedspin-circuit models accounting for transport physics, coupled with thestochastic Landau-Lifshitz-Gilbert equation for magnetization dynamics. We findthat the use of low-barrier SAF layers reduces dipolar coupling, achievinguncorrelated fluctuations at zero-magnetic field surviving up to diametersexceeding ($D\approx 100$ nm) if the nanomagnets can be made thin enough($\approx 1$-$2$ nm). The double-free-layer structure retains bias-independenceand the circular nature of the nanomagnets provides near-uniform randomnesswith fast fluctuations. Combining our full sMTJ model with advanced transistormodels, 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 guidethe experimental development of superior stochastic magnetic tunnel junctionsfor large-scale and energy-efficient probabilistic computation for problemsrelevant to machine learning and artificial intelligence.
We show that we can harness two recent experimental developments to build acompact hardware emulator for the classical Heisenberg model in statisticalphysics. The first is the demonstration of spin-diffusion lengths in excess ofmicrons in graphene even at room temperature. The second is the demonstrationof low barrier magnets (LBMs) whose magnetization can fluctuate rapidly even atsub-nanosecond rates. Using experimentally benchmarked circuit models, we showthat an array of LBMs driven by an external current source has a steady-statedistribution corresponding to a classical system with an energy function of theform $E = -1/2\sum_{i,j} J_{ij} (\hat{m}_i \cdot \hat{m}_j$). This may seemsurprising for a non-equilibrium system but we show that it can be justified bya Lyapunov function corresponding to a system of coupledLandau-Lifshitz-Gilbert (LLG) equations. The Lyapunov function we constructdescribes LBMs interacting through the spin currents they inject into the spinneutral substrate. We suggest ways to tune the coupling coefficients $J_{ij}$so that it can be used as a hardware solver for optimization problems involvingcontinuous variables represented by vector magnetizations, similar to the roleof the Ising model in solving optimization problems with binary variables.Finally, we train a Heisenberg XOR gate based on a network of four coupledstochastic LLG equations, illustrating the concept of probabilistic computingwith a programmable Heisenberg model.