Dynamical stability by spin transfer in nearly isotropic magnets

Abstract

Spin transfer torques (STTs) control magnetisation by electric currents, enabling a range of nano-scale spintronic applications. They can destabilise the equilibrium magnetisation state by counteracting magnetic relaxation. Here, we maximise the STT effect through a dedicated growth-annealing protocol for CoFeB thin films, such that magnetic anisotropies originating from the interface and shape almost cancel each other. The nearly isotropic magnets enable low-current dynamical stabilisation of the magnetisation in the direction opposite to an applied magnetic field, thereby realising a spintronic analogue of the Kapitza pendulum. In an intermediate current regime, the STT drives large magnetisation vector fluctuations that cover the entire Bloch sphere. The continuous variable associated with the stochastic magnetisation direction may serve as a resource for probabilistic computing and neuromorphic hardware. Our results establish isotropic magnets as a platform to study as-yet-uncharted, far-from-equilibrium spin dynamics including anti-magnonics, with promising implications for unconventional computing paradigms.

Figures (19)

• Figure 1: Dynamical stability of the spintronic Kapitza pendulum.a, Schematics of the mechanical Kapitza pendulum with and without an external drive. $mg$ represents the gravitational force on the bob. b-d, Schematics of different magnetisation switching and stability processes illustrated within the angular dependence of magnetic energy ($U$) (b field-induced switching, c STT switching, d dynamical stability). $\theta$ is defined by the relative angle between the magnetic field ($\bm{H}_{\rm ext}$) and moment ($\bm{M}$). See the main text for the mechanism of each process. e, Analogy between the Kapitza pendulum and dynamical stabilisation by spin transfer in an isotropic magnet. The magnetic field provides a minimum and maximum of potential energy as the gravitational field does, and the STT plays the role of dynamical driving that controls their stability. Note that the arrows piercing through the conduction electrons (solid spheres) represent their spin pointing opposite to their magnetic moment. f, A typical solution of the stochastic LLG equation for a macrospin in an isotropic magnet. The initial condition is set at the south pole (red dot) and due to the anti-damping STT exceeding the Gilbert damping torque, the state precesses away from the field direction and settles around the inverted state (yellow dot). Coloured dots on the sphere have the corresponding potential energy values indicated in d.
• Figure 2: Current-induced magnetic damping probed by ferromagnetic resonance.a-b, Field-swept FMR voltages measured for (a) $\phi=225^{\circ}$ and (b) $\phi=45^{\circ}$. The inset is a sketch of our device and the coordinate system. c, Linewidth extracted from FMR at 4 GHz with $\phi=45^{\circ}$ (blue) and $\phi=225^{\circ}$(red), for different $I_\text{dc}$. Solid lines show the results of a linear fitting excluding points below -4 mA for 45$^{\circ}$ and above 4 mA for 225$^{\circ}$. d-f, Field-swept FMR voltages measured for various frequencies and three values of $I_\text{dc}$ (d 0 mA, e -4 mA and f 4 mA). Magnetic fields were applied along $\phi=225^{\circ}$. g, Linewidth as a function of frequency for different $I_\text{dc}$, i.e. +5, 0 and -5 mA, represented by red, grey and blue dots, respectively. The measurements were carried out at $\phi=225^{\circ}$. The grey line is a linear fitting of the 0 mA data. Both red and blue curves are calculated using Eq. \ref{['eq:linewidth']} (see the main text).
• Figure 3: Electric resistance probes for magnetic fluctuations and the inverted states.a-b, Observed angular MR in the coordinate system of Fig. \ref{['fig:stfmr']}a for negative (a) and positive (b) currents, normalised by the resistance at $\phi =0^{\circ }$ for each current. c, $1- \langle M_y^2 \rangle /M_\text{s}^2$ calculated by the stochastic LLG equation in the macrospin approximation. d-f, Time-dependent trajectories of the macrospin moment calculated by the stochastic LLG equation for currents 0.1 mA (d), 4.1 mA (e) and 7 mA (f), and $\phi=270^{\circ}$. The field strength is 125 mT for panels a-f. g, Resistance vs. dc current measurements for applied magnetic fields 25-200 mT along $\phi=270^{\circ}$. h, Calculated $1- \langle M_y^2\rangle /M_{\rm s}^2$ as a function of current and applied magnetic field. i, $I_\text{c}$ extracted in g for different field strengths and angles (dots) and linear fits (lines). The inset shows the slope (dots) and a fit by $\sin\phi$ (curve).
• Figure 4: Stability diagram, bistability and probabilistic distribution for nearly isotropic magnets.a, Bifurcation diagram constructed from the stationary solutions of the LLG equation. b, ST-FMR for different $I_\text{dc}$ close to the damping compensation condition. When a larger $I_\text{dc}$ increases the anti-damping torques beyond the critical value, the magnetisation precesses around the dynamically stable state at the north pole, generating an opposite dc voltage at resonance. c, The ST-FMR results highlight the current-induced magnetisation reversal. The inset shows the FMR frequencies as functions of field for the two magnetisation directions. d-f, Time-dependent traces of the normalised $M_y$ component for small(d)/medium(e)/large(f) anti-damping torques. Here $M_y=-1$ is along the magnetic field direction.
• Figure 5: Two-dimensional image generation by a continuous restricted Boltzmann machine with electrically controlled zero-damping states in isotropic magnets. a, Samples of the Fashion-MNIST training data (t-shirt class). b & c, Generated data using binary and continuous variables for the visible nodes respectively.