Efficient Synchronization of Dipolarly Coupled Vortex-Based Spin Transfer Nano-Oscillators

Due to their nonlinear properties, spin transfer nano-oscillators can easily adapt their frequency to external stimuli. This makes them interesting model systems to study the effects of synchronization and brings some opportunities to improve their microwave characteristics in view of their applications in information and communication technologies and/or to design innovative computing architectures. So far, mutual synchronization of spin transfer nano-oscillators through propagating spinwaves and exchange coupling in a common magnetic layer has been demonstrated. Here we show that the dipolar interaction is also an efficient mechanism to synchronize neighbouring oscillators. We experimentally study a pair of vortex-based spin transfer nano-oscillators, in which mutual synchronization can be achieved despite a significant frequency mismatch between oscillators. Importantly, the coupling efficiency is controlled by the magnetic configuration of the vortices, as confirmed by an analytical model and micromagnetic simulations highlighting the physics at play in the synchronization process.


Supplementary materials The vortex magnetization and the Thiele equation
In the absence of crystalline anisotropy and for adequate dimensions, the remanent magnetic state of a cylinder shaped nanomagnet is a magnetic vortex. 1 The magnetization lies in the plane, winding around the centre of the magnet where it pops out of the plane in a small region called the vortex core. The vortex state is characterized by two parameters, namely the polarity P = ±1 giving the direction of the out-of-plane magnetization in the core and the chirality C = ±1 associated to the curling direction of the in-plane magnetization.
The lowest energy mode of a magnetic vortex is the gyrotropic mode, corresponding to a circular displacement of the vortex core around the centre of the ferromagnet. 1 The associated magnetization dynamics can be described through a single collective variable equation, called the Thiele equation: [2][3][4][5][6] where X is the displacement of the vortex core relative to the center of the dot. The gyrotropic term (LHS) is equal to the sum of the effective dissipative and conservative forces acting on the vortex core (RHS).
The LHS terms features the gyrovector G = −PGẑ = −2πPL M S γẑ . P is the vortex polarity, L is the layer thickness, M S is the ferromagnet saturation magnetization, γ is the gyromagnetic ratio.
The RHS terms respectively are : • The viscous damping force (proportional and opposed to the core speed): F = −αηG˙ X, accounting for the dissipative phenomena opposing the motion. α is the Gilbert damping parameter of the ferromagnet (α = 0.01 for NiFe), and η = 1 2 ln R 2l ex − 1 8 is a phenomenological damping coefficient for the vortex . 4,5,7 R is the pillar radius, l ex is the material exchange length (l ex 6nm for NiFe).
• The spring-like confinement force (opposed to the core displacement): F = −k(X) X. W ( X) = 1 2 k(X)X 2 is the shifted vortex energy, with k(X) = ω 0 G 1 + a X R 2 including both the magneto-static and the Oersted field contributions: 4,8 ω 0 = 5 9π γ µ 0 M S L R +C 0.85 2π γ µ 0 RJ (C is the vortex chirality, J is the current density flowing through the pillar).
• A supplementary force added to account for spin-transfer effects: F STT ( X). In the general case of a non-uniform polarizer, the spin transfer force direction depends on the core polarity sign, and can be described by the general expression: where λ (X, J) expresses the spin transfer torque efficiency. 3-6, 9, 10 Using complex coordinates, the Thiele equation writes: where X X X = X i e iϕ is the complex coordinate of the core position.
While equation 4a determines the equilibrium orbit radius for the core gyration, equation 4b determines the frequency of the motion. An important conclusion drawn from this second equation is that the sense of gyration of the vortex core is directly related to its polarity sign P. 1

Derivation of the synchronization equation
Starting with the two coupled Thiele equations: The interaction force is expressed as : , and the spin transfer force is described by the general expression: By using complex coordinates, these equations can then be simplified as : where X X X i = X i e iϕ i (i ∈ {1, 2}) are the complex coordinates of the core positions and X X X 1,2 * = X i e −iϕ i are their complex conjugates.
In the absence of coupling, the auto-oscillator gyration radius is set by the relationship: λ (X 0 i , J i ) = αη i k i (X 0 i ), while the frequency is given by: . In the following, we will consider deviations of the gyration radii around their equilibrium values through the oscillations power: p i = X i R i 2 = p i 0 + δ p i , and the subsequent deviations of the instantaneous frequencies: P iφi = ω i + N i δ p i , and dissipative forces: The terms associated with dipolar coupling are then considered as perturbations of the equilibrium. Inside these terms, the approximation X 1 /X 2 X 0 1 /X 0 2 1 can be used. So that we obtain: From these equations, it appears that the dipolar coupling action comes with two different time rates: (ω 1 + ω 2 ) and |ω 1 − ω 2 |. Phases and radii variations at the frequency (ω 1 + ω 2 ) will be averaged out in the synchronization process, so that only the low frequency contributions at frequency |ω 1 − ω 2 | should be considered. At this step, we must note that the relative polarities signs will decide which terms should be cancelled or kept.
For the sake of simplicity, we will continue the demonstration for the case P 1 = P 2 = +1. In this case, terms associated with µ (+) are cancelled out: For the slow variations of the gyration radii, we can consider that an equilibrium is permanently reached (δ p i = 0):