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Controlling the phase locking of unstable magnetic bits for ultra-low power computation

When fabricating magnetic memories, one of the main challenges is to maintain the bit stability while downscaling. Indeed, for magnetic volumes of a few thousand nm3, the energy barrier between magnetic configurations becomes comparable to the thermal energy at room temperature. Then, switches of the magnetization spontaneously occur. These volatile, superparamagnetic nanomagnets are generally considered useless. But what if we could use them as low power computational building blocks? Remarkably, they can oscillate without the need of any external dc drive, and despite their stochastic nature, they can beat in unison with an external periodic signal. Here we show that the phase locking of superparamagnetic tunnel junctions can be induced and suppressed by electrical noise injection. We develop a comprehensive model giving the conditions for synchronization, and predict that it can be achieved with a total energy cost lower than 10-^13 J. Our results open the path to ultra-low power computation based on the controlled synchronization of oscillators.


Introduction
Superparamagnetic tunnel junctions present a number of advantages for computation. First, they can be downscaled to atomic dimensions 1 . In addition, because the energy barrier separating the two magnetic configurations is small, low current densities can lead to significant action of spin torques 2 . But how can they be harnessed for applications? A first option is to use superparamagnetic tunnel junctions as sensors. Indeed, thanks to their high sensitivity to electrical currents they are able to detect weak oscillating signals 3-5 through the effect of stochastic resonance 6 . A second option is to use them as building blocks of computing systems leveraging the synchronization of oscillators for processing 7,8 . It has been recently recognized that coupled nano-oscillators are promising brain-inspired systems for performing cognitive tasks such as pattern recognition [9][10][11][12][13][14][15][16] . Like neurons in some parts of the brain, they compute by synchronizing and desynchronizing depending on sensory inputs 17 . However, such systems require a high number of oscillators, each powered by substantial dc current. Using superparamagnetic tunnel junctions would allow orders of magnitude gain in power consumption. In addition, by shrinking their dimensions they can be fabricated from the same magnetic stack as stable junctions, allowing for densely interweaving oscillations and memory.
Nevertheless there are a number of prerequisites to be able to use superparamagnetic tunnel junctions for computational purposes. In particular, it is necessary to identify handles providing control over their synchronization and to model accurately the associated physics for simulating large scale systems of interacting oscillators. Here we show experimentally that we can induce the phase-locking of a superparamagnetic tunnel junction to a weak periodic signal through the addition of a small electrical noise, and that we can suppress the phase-locking by adding more noise. While the stochastic behavior of most systems becomes unpredictable when shrunk to nanometer scale, the dedicated model we develop here encompasses all our experimental results. The quantitative agreement between model and experiments allows predicting the power consumption of computing systems harnessing phase-locking of superparamagnetic tunnel junctions.

Experimental Results
We study experimentally superparamagnetic tunnel junctions with an MgO barrier and a CoFeB free layer of dimensions 601201.7 nm 3 (details in Methods). As depicted in the inset of Fig. 1a, we evaluate their ability to phase lock to a weak square periodic drive voltage in the presence of electrical white noise, at room temperature. We set the drive frequency at = 50 Hz and the drive amplitude at = 63 mV, which corresponds to approximately 25% of the voltage threshold for deterministic magnetization switching at 0 K. Fig. 1a shows how the mean frequency of the stochastic oscillator evolves when the amplitude of the electrical noise is increased. We observe three different regimes, illustrated in Fig. 1b. As can be seen in the first panel, the jumps in the junction resistance, corresponding to reversals of the magnetization, remain stochastic for small values of injected electrical noise. In addition, the junction mean frequency is lower than the drive frequency. Usually, adding noise to a system tends to destroy its coherence and is detrimental to the occurrence of a synchronized regime. On the contrary, in our case, by increasing the electrical noise amplitude, we can increase the junction's mean frequency towards the drive frequency. Eventually, for an optimal range of electrical noise (between 20 and 30 mV), we observe both frequency locking (as evidenced from the plateau in Fig. 1a), and phase locking to the driving signal (as shown in panel 2). In this second regime, electrical noise optimally assists the periodic drive to overcome the voltage threshold for magnetization switching at every oscillation of the drive voltage [18][19][20] . In the third regime (panel 3), higher amplitude electrical noise induces unwanted switches of the magnetization and prevents synchronization.

Analytical model and simulations of phase-locking
Noise-controlled phase-locking has been experimentally demonstrated in a few non-linear systems such as Schmitt triggers 19,21 or lasers 22 but never in a nanoscale system as achieved here. In order to assess the potential of superparamagnetic tunnel nanojunctions for applications, we now propose a model that accurately describes their noise-mediated synchronization to weak periodic signals. The thermally activated escape rate of a single domain magnetization, modulated by spin transfer torque 23,24 , has a simple expression [25][26][27] : and thermal noises (T=300K). Using Eq.
(1), we can numerically compute the junction's mean frequency as a function of the electrical noise amplitude 28  The analytical models that have been developed in the past to describe noise-induced phase locking 18,20,29 focused on cases for which noise can be taken into account as a time-independent variable in the escape rates, such as temperature in Eq.
(1). However in our case, the escape rates from the parallel and antiparallel states are time varying, random variables because they depend on the electrical noise .
In order to go further, we develop an original and generic method to analytically determine the conditions for synchronization. Starting from Eq. (1), we calculate the probabilities ( ) for the magnetization to switch from out-of-phase with the drive voltage to in-phase (from in-phase to out-ofphase) during half a period. The details of the derivation and the expressions for the phase-locking and phase-unlocking probabilities and are given in Methods. In the vicinity of the plateau, the mean frequency of the junction is described by . Considering a 99% frequency locking requirement, the boundaries of the synchronization region are given by and , as shown by the red arrows in Fig.2a for the case of a drive amplitude of 63 mV. Fig. 2b and c show that our analytical model (dotted lines) quantitatively predicts the boundaries of the experimental synchronization zone (symbols) over the whole range of investigated parameters. As can be seen in Fig. 2b, the range of electrical noise for which phase-locking occurs increases with the drive amplitude. When the drive amplitude is too low (here below 37 mV), synchronization cannot be achieved. On the other hand, at large drive amplitudes (here above 85 mV), phase locking can be achieved through room temperature thermal noise alone, without the need to add up electrical noise. In addition, as shown in Fig. 2c, phase-locking is achievable for frequencies orders of magnitude higher than the natural mean frequency (0.1 Hz here).    When only thermal noise is used, ( ) where V 1 is the minimum drive voltage required to phase-lock the junction in one drive period (see inset in Fig. 3 and Methods). When electrical noise is added, the minimum energy becomes ( ) , where V 0 is the minimum drive voltage required for phase-locking and σ 0 the corresponding noise level (see inset in Fig. 3 and Methods).
Interestingly, for each drive frequency, there is an optimal diameter for which the energy needed to achieve phase-locking is minimal. Indeed the junction diameter determines its natural frequency: large diameters correspond to low frequencies because large magnetic volumes are more difficult to switch. Above , the drive frequency is larger than the junction's mean frequency. To phase-lock, the junction has to be accelerated. In the absence of electrical noise, this can be done through an increase of the drive amplitude, which enhances the ability of the junction to synchronize (stars in Fig.3). Adding electrical noise lowers the drive amplitude required to synchronize and thus decreases the total amount of energy to provide (circles in Fig. 3). Below , the junction has to be slowed down in order to phase lock. As electrical noise always speeds up the oscillator by increasing the number of switches, this can only be achieved by increasing the drive amplitude. Our results indicate that carefully engineering the junctions' dimensions can drastically decrease the energy required to achieve phaselocking, about 8×10 -14 J for a drive frequency of 10 MHz (Figure 3). By comparison, synchronizing a harmonic dc-driven spin-torque oscillator with a 10 GHz frequency 31 to a drive current would require 100 times more energy (see Methods). CMOS implementations of oscillators for bio-inspired computing applications are also more costly in terms of energy, with a consumption above 7x10 -12 J for integrate and fire neurons 32 . In addition they occupy a large area on chip, typically several hundreds of μm 2 .
Because of these issues of size and energy consumptions, bio-inspired computing systems leveraging the synchronization of coupled oscillators for computing have never been implemented in CMOS. Thanks to their small area and low energy consumption, arrays of phase-locked superparamagnetic tunnel junctions are a promising alternative for pattern recognition. We take the example of image classification, which generally requires one oscillator per pixel 15,16 In this study is the effective attempt frequency 27 , is the energy barrier between the two stable states, is the Boltzmann constant, is the temperature, is the applied voltage and is the threshold voltage for deterministic switching 26,27 .
For numerical simulations we compute at each time step the probability for the magnetization to switch during the time interval knowing the initial state: where is the probability to switch from P to AP during when the excitation voltage is .
Therefore, is the probability to switch from out of phase to in-phase during while is the probability to switch from in-phase to out of phase during . With and is a Gaussian distribution over N.
When the level of noise is sub-optimal, synchronization is limited by the junction's ability to phase-lock fast enough when the excitation voltage reverses. Therefore the mean frequency of the junction is . On the other hand, when the noise level is supra-optimal, synchronization is limited by the junction's tendency to jump out of phase with the excitation voltage. Therefore . On the whole, near the plateau, the mean frequency of the junction is . In consequence, and means that the junction is frequency-locked with less than 1% error.
V 0 is computed as the minimum voltage drive V ac for which there is an electrical noise level σ 0 that satisfies P + >99.5% and P -<0.5%. V 1 is the minimum voltage drive V ac for which P + >99.5% and P -<0.5% is satisfied at zero electrical noise.

Energy consumption predictions:
We consider a perpendicularly magnetized tunnel junction and make the assumption that the free layer is a single magnetization element ( ). The thickness of the free layer in the junction is fixed, so that the energy barrier between the two magnetization states scales with the square of the diameter D: . We also consider that the resistance × area product of the junction is constant, so that the resistance of the junction can be written as: .

Additional information
Competing interest: The authors declare no competing financial interests.