Monte-Carlo Simulations of Magnetic Tunnel Junctions: from Physics to Application A.F. V INCENT∗ , W.S. Z HAO∗ , J.-O. K LEIN∗ , S. G ALDIN -R ETAILLEAU∗ and D. Q UERLIOZ∗ ∗ Institut

d’Électronique Fondamentale, Univ. Paris-Sud, CNRS, Orsay, France e-mail: [email protected]

I NTRODUCTION Magnetic Tunnel Junctions (MTJs) – the basic structures of the Spin-Transfer Torque Magnetic RAMs (STT-MRAM) currently reaching the market - present a complex and probabilistic switching behavior. Although some analytical models describing this behavior exist, they can not describe all the switching regimes of the MTJs. They can model low (“subcritical”) and high (“supercritical”) currents, but not the intermediate currents, which are essential for applications. In this work, we present Monte-Carlo simulations of MTJs that have been used to build an analytical model linking the two different current regimes. This model allowed us to perform system-level simulations of an original neuro-inspired chip that uses MTJs as binary stochastic “synapses”.

linked by a new mathematical expression of our own. Tests on a set of devices with different geometries and Ms provide good agreement with the results of simulation. Furthermore, we studied the statistical distribution of the switching delay: the insets of figure 2 show some examples for different Js values. We empirically developed a model of the delay distribution that uses a gamma law as probability distribution function (PDF). As we can see on the figure 2, its predicts correctly the general shape of the PDF extracted from the simulations results and tends to an exponential law in the weak current regime as the MTJ behavior becomes a Poisson’s process (theoretical derivation in [2]). Figure 3 shows that our analytical model can also be fitted on experimental data, measured in our lab and mentioned in [4], [5].

A BOUT OUR SIMULATOR OF MTJ S Figure 1 presents a typical MTJ. We studied current-driven devices: a tunnel current goes through the MTJ and favors one of the two stable magnetic states thanks to spin-transfer torque (STT). Due to the small dimensions of the free layer, we considered its total magnetic moment as a unique macrospin mf . We simulated its dynamic thanks to the usual Landau-LifschitzGilbert equation (in SI units), augmented of two field-like terms, a Slonczewski’s one - HSTT - for the STT part [1] and a stochastic term hsto to include the thermal agitation of mf : (1 + α2 ) dmf = −µ0 mf × (Heff + hsto ) |γ| dt αµ0 − mf × (mf × (Heff + hsto + HSTT )) Ms V with α the Gilbert’s damping ratio, γ the gyromagnetic ratio, µ0 the magnetic permeability of the vacuum, Ms the saturated magnetization of the free layer and V its volume. Heff is a field-like term including the different anisotropy terms and a possibly applied exterior field. At each time step, we draw each component of hsto according to a Gaussian law determined by thermodynamical considerations [2]. F IRST RESULTS AND DISCUSSION From multiple simulations at different values of the injected current density Js , we obtained < ∆t > the average reversal delay of mf presented on figure 2. With these results, we developed an analytical model that covers a wide range of Js , based on existing analytical models in the extreme regimes [3] c 2014 IEEE 978-1-4799-5433-9/14/$31.00

A N EXAMPLE OF HIGH - LEVEL USE Thanks to a simulator using our analytical model, we performed system-level simulations of an original car counting application. The chip architecture is neuro-inspired: the MTJs are used as binary stochastic “synapses” connecting input and output spiking neurons (figure 4). They learn in an unsupervised way, thanks to a “spike-timing-dependent plasticity” rule. These first results will be published at ISCAS 2014 [6]. C ONCLUSION From a physical equation, we have built up an analytical model that describes the stochastic switching delay of a current-driven MTJ. Thanks to this model, we have explored through system-level simulations the relevance of MTJs as binary stochastic synapses in neuro-inspired chips. ACKNOWLEDGMENT This work was supported by the ANR COGNISPIN project (ANR-13-JS03-0004-01) and the FP7 ICT BAMBI European project (FP7-ICT-2013-C). R EFERENCES [1]

J.C. S LONCZEWSKI Current-driven excitation of magnetic multilayers, J. Magn. Magn. Mater. , 159, (1996). [2] J.L. G ARCÍA -PALACIOS and F.J. L ÁZARO Langevin-dynamics study of the dynamical properties of small magnetic particles, Phys. Rev. B, 58, (1998) [3] Z. D IAO et al. Spin-transfer torque switching in magnetic tunnel junctions and spin-transfer torque random access memory, J. Phys.-Condens. Mat., 19, (2007)

[4]

T. D EVOLDER et al. Single-shot time-resolved measurements of nsscale spin-transfer induced switching: stochastic vs deterministic aspects, Phys. Rev. Lett., 100, (2008) [5] M. M ARINS DE C ASTRO et al., Precessional spin-transfer switching in a MTJ with a synthetic antiferromagnetic perpendicular polarizer, J. Appl. Phys., 111, (2012). [6] A.F. V INCENT, D. Q UERLIOZ et al., Spin-transfer torque magnetic memory as a stochastic memristive synapse, International Symposium on Circuits and Systems, (2014).

Fig. 1. A classical MTJ: a thin (< 1 nm) insulator between two ferromagnetic layers, often with an elliptical shape. The fixed one has its magnetic moment (black arrow) pinned down along one direction whereas that of the free layer can take two stables directions, parallel (P) or antiparallel (AP) with the previous one. It leads to two different resistance states, RP and RAP . A positive current I can switch the device AP → P whereas a negative I can cause a P → AP event.

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Fig. 4. Architecture of the simulated neuro-inspired chip: a crossbar of nanosynapses (purple ) connects input spiking neurons (blue I) to output spiking neurons (red H). Each nanosynapse is made of a single MTJ. Initially, half of the synapses are in an RP state. Thanks to a bio-inspired learning rule, the synapses’ state evolves during the learning process. On the right side is given a post-learning conductance map that makes the associated output neuron fire only when a vehicle passes on a specific lane in the input “video” of cars on a 6-lane freeway.

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Fig. 2. Main figure: comparison between the results provided by our macrospin simulator and our model for < ∆t >. Insets: probability distribution functions of ∆t, for different values of Js ; simulations results (blue bars) vs our model (dashed green lines).