Synchronization properties and reservoir computing capability of hexagonal spintronic oscillator arrays

Highlights

• Magnetic coupling in oscillator arrays depends on the geometry choice.

• Replacing rectangular with hexagonal geometry increases synchronization capability.

• As a result, higher output power and lower linewidth can be achieved.

• It is also possible to increase selected reservoir computing capabilities.

Abstract

The influence of array geometry on synchronization properties of a 2-D oscillator array is investigated based on a comparison between a rectangular and a hexagonal grid. The Kuramoto model is solved for a nearest-neighbor case with periodic boundary conditions and for a small-scale, realistic coupling case with 1/${\mathrm{r}}^3$ decay characteristic of spintronic oscillators. In both cases, it is found that the hexagonal grid choice leads to lower synchronization threshold and higher emission power than its rectangular counterpart, which results from increased connectivity, as well as, in the realistic-coupling case, from decreased contributions of the array edges. Additionally, a more general spin-torque oscillator model including both amplitude and phase as degrees of freedom is employed for reservoir computing simulations, showing that by using hexagonal grid one can increase the short-term memory capacity but not the parity-check capacity of the system.

Introduction

Spintronic oscillators have been drawing a lot of attention as devices that could be utilized for radio-frequency signal generation and transmission as well as for bio-inspired neuromorphic computing [1], [2], [3], [4], [5], [6], [7]. Multiple physical implementations, including spin-torque nano-oscillators [8], [9], [10], [2], [6], [7],spin Hall nano-oscillators [11], [12], [2], [13], oscillators based on antiferromagnets [14], [15], [16] or uncompensated ferrimagnets [17] have been proposed, featuring advantages such as the ability to work in room temperature, energy efficiency, small footprint and high frequency ranges being available [1], [2], [3]. The common thread that remains the key to both of the main application areas, namely the DC-to-RF signal conversion and unconventional computing, is making use of oscillator networks and their ability to synchronize by mutually influencing each oscillator’s dynamics [1], [2], [3], [18], [19], [20]. Therefore, a thorough understanding of the coupling behavior is expected to be crucial for further advancing the spintronic oscillator research.

At the same time, synchronization theory in complex networks of oscillators has seen substantial progress within the last several years, creating solid theoretical foundations for understanding most dynamical phenomena occurring in these systems [21], [22], [23]. In this approach, a single oscillator is usually considered to be an abstract object characterized only by its phase, natural frequency and the coupling presence, leading to a simple dynamics equation known as Kuramoto equation [24], [21]. Recently, a number of works have shown that Kuramoto equation can be successfully used to describe arrays of spintronic oscillators as well, both in its basic form and in the modified form where the phase degree of freedom is replaced by two degrees of freedom corresponding to oscillation phase and amplitude or power [25], [26], [20]. Typically, such a description assumes a coupling form that scales with distance as $1/{\mathrm{r}}^3$, which is characteristic to magnetodipolar coupling that naturally occurs in physical devices such as spin-torque oscillators or spin Hall oscillators [26], [20], [27].

There are, however, important barriers that often make it difficult to directly apply the findings of the oscillation networks theory in the applied spintronics research. Most notably, there exists a vast number of different Kuramoto equation models, from which only a relatively small number can be considered immediately relevant to the spintronic oscillator landscape: many of the commonly made assumptions, such as using periodic boundary conditions, arbitrarily rewired coupling paths in order to increase disorder or an all-to-all coupling scheme, do not hold for a finite-size, magnetically coupled network [28], [21], [22], [23], [29], [30]. One direction of research that remains relatively unexplored is the influence of grid geometry on synchronization properties. While for an all-to-all coupling scheme the relative physical distances and positions of oscillators can be mostly ignored, in the case of a 2D system with spatially-dependent coupling they can be expected to influence the network dynamics significantly. Despite this, a simple arrangement of oscillators on a cubic Cartesian grid is often assumed in literature by default [31], [32], [30], [26], [20], with the notable exception of the recent works on nanoconstriction-based oscillators, where it was found that the synchronization can be helped by using a tilted rectangular array that assists in spin wave propagation along certain directions. [33], [34].

This work investigates synchronization properties and selected neuromorphic computation capabilities for a 2-D oscillator array on a hexagonal grid and compares the obtained results with a rectangular (cubic) grid. Two important cases are examined: firstly, large-scale systems with periodic boundary conditions and nearest-neighbor coupling are considered, in order to highlight some of the differences between the hexagonal arrangement and the results known from the existing synchronization theory literature. Secondly, a more realistic case of a small-scale network with no periodic boundary conditions and $1/{\mathrm{r}}^3$ coupling decay is analyzed as a direct model of a spintronic oscillators array with magnetic coupling. The synchronization threshold, emission power and short-memory capabilities are all shown to depend on the grid choice significantly, opening new possibilities for oscillator networks research and applications.