Nonlinear magnetic vortex dynamics in a circular nanodot excited by spin-polarized current
© Guslienko et al.; licensee Springer. 2014
Received: 15 June 2014
Accepted: 1 August 2014
Published: 8 August 2014
We investigate analytically and numerically nonlinear vortex spin torque oscillator dynamics in a circular magnetic nanodot induced by a spin-polarized current perpendicular to the dot plane. We use a generalized nonlinear Thiele equation including spin-torque term by Slonczewski for describing the nanosize vortex core transient and steady orbit motions and analyze nonlinear contributions to all forces in this equation. Blue shift of the nano-oscillator frequency increasing the current is explained by a combination of the exchange, magnetostatic, and Zeeman energy contributions to the frequency nonlinear coefficient. Applicability and limitations of the standard nonlinear nano-oscillator model are discussed.
Spin torque microwave nano-oscillators (STNO) are intensively studied nowadays. STNO is a nanosize device consisting of several layers of ferromagnetic materials separated by nonmagnetic layers. A dc current passes through one ferromagnetic layer (reference layer) and thus being polarized. Then, it enters to an active magnetic layer (so-called free layer) and interacts with the magnetization causing its high-frequency oscillations due to the spin angular momentum transfer. These oscillation frequencies can be tuned by changing the applied dc current and external magnetic field [1–3] that makes STNO being promising candidates for spin transfer magnetic random access memory and frequency-tunable nanoscale microwave generators with extremely narrow linewidth . The magnetization in the free layer can form a vortex configuration that possesses a periodical circular motion driven by spin transfer torque [1, 5–11]. For practical applications of such nanoscale devices, some challenges have to be overcome, e.g., enhancing the STNO output power. So, from a fundamental point of view as well as for practical applications, the physics of STNO magnetization dynamics has to be well understood.
In this paper, we show that a generalized Thiele approach  is adequate to describe the magnetic vortex motion in the nonlinear regime and calculate the nanosize vortex core transient and steady orbit dynamics in circular nanodots excited by spin-polarized current via spin angular momentum transfer effect.
We apply the Landau-Lifshitz-Gilbert (LLG) equation of motion of the free layer magnetization , where m = M/Ms, Ms is the saturation magnetization, γ > 0 is the gyromagnetic ratio, Heff is the effective field, and αG is the Gilbert damping. We use a spin angular momentum transfer torque in the form suggested by Slonczewski , τ s = σJ m × (m × P), where σ = ℏη/(2|e|LM s ), η is the current spin polarization (η ≅ 0.2 for FeNi), e is the electron charge, P is direction of the reference layer magnetization, and J is the dc current density. The current is flowing perpendicularly to the layers of nanopillar and we assume . The free layer (dot) radius is R and thickness is L.
We assume that the dot is thin enough and m does not depend on z-coordinate. The magnetization m(x,y) has the components and expressed via a complex function . Inside the vortex core, the vortex configuration is described as a topological soliton, , |f(ζ)| ≤ 1, where f(ζ) is an analytic function. Outside the vortex core region, the magnetization distribution is , |f(ζ)| > 1. For describing the vortex dynamics, we use two-vortex ansatz (TVA, no side surface charges induced in the course of motion) with function f(ζ) being written as , where C is the vortex chirality, ζ = (x + iy)/R, s = s x + is y , s = X/R, c = R c /R, and R c is the vortex core radius.
where ω G (u) = (R2u|G(u)|)- 1∂W(u)/∂u is the nonlinear gyrotropic frequency, d(u) = - D(u)/|G(u)| is the nonlinear diagonal damping, D = D xx = D yy , d n (s) = - D xy (s)/|G(u)| is the nonlinear nondiagonal damping, and χ(u) = a(u)/|G(u)|. It is assumed here that F ST (s) = a(u)(z × s) , where a is proportional to the CPP current density J and a(0) = πRLM s σJ.
where , , , β = L/R, , and ς = 1 + 15(ln 2 - 1/2)R c /8R.
There is an additional contribution to κ/2, 2(L e /R)2, due to the face magnetic charges essential for the nanodots with small R. The contribution is positive and can be accounted by calculating dependence of the equilibrium vortex core radius (c) on the vortex displacement. This dependence with high accuracy at cu < < 1 can be described by the function c(u) = c(0)(1 - u2)/(1 + u2). Here, c(0) is the equilibrium vortex core radius at s = 0, for instance c(0) = 0.12 (R c = 12 nm) for the nanodot thickness L = 7 nm.
where the linear gyrotropic frequency is ω0 = γM s κ(β, R, J)/2, and N(β, R) = κ′(β, R)/κ(β, R).
The frequency was calculated in  and was experimentally and numerically confirmed in many papers. The nonlinear coefficient N(β,R) depends strongly on the parameters β and R, decreasing with β and R increasing. The typical values of N(β,R) at J = 0 are equal to 0.3 to 1.
Equation 3 and the system (6) are different from the system of equations of the nonlinear oscillator approach . Equations 6 are reduced to the autonomous oscillator equations and only if the conditions d2 < < 1 and dχ < < ω G are satisfied and we define the positive/negative damping parameters  as Γ+(u) = d(u)ω G (u) and Γ-(u) = χ(u). We note that reducing the Thiele equation (1) to a nonlinear oscillator equation  is possible only for axially symmetric nanodot, when the functions W(s), G(s), d(s) and χ(s) depend only on u = |s| and the additional conditions d n < < 1, d2 < < 1, and dχ < < ω G are satisfied. The nonlinear oscillator model  cannot be applied for other nanodot (free layer) shapes, i.e., elliptical, square, etc., whereas the generalized Thiele equation (1) has no such restrictions.
The model parameters are , d0 = α G [5 + 4 ln(R/R c )]/8, d1 = 11α G /6, χ0 = γσJ/2. The ratio χ1/χ0 = O(c2u2) < < 1, therefore, the nonlinear parameter χ1 can be neglected. The statement about linearity of the ST-force agrees also with our simulations and the micromagnetic simulations performed in [12, 19]. The coefficient λ(J) describes nonlinearity of the system and decreases smoothly with the current J increasing.
We have simulated the vortex motion in a single permalloy (Fe20Ni80 alloy, Py) circular nanodot under the influence of a spin-polarized dc current flowing through it. Micromagnetic simulations of the spin-torque-induced magnetization dynamics in this system were carried out with the micromagnetic simulation package MicroMagus (General Numerics Research Lab, Jena, Germany) . This package solves numerically the LLG equation of the magnetization motion using the optimized version of the adaptive (i.e., with the time step control) Runge-Kutta method. Thermal fluctuations have been neglected in our modeling, so that the simulated dynamics corresponds to T = 0. Material parameters for Py are as follows: exchange stiffness constant A = 10-6 erg/cm, saturation magnetization Ms = 800 G, and the damping constant used in the LLG equation α G = 0.01. Permalloy dot with the radius R = 100 nm and thickness L = 5, 7, and 10 nm was discretized in-plane into 100 × 100 cells. No additional discretization was performed in the direction perpendicular to the dot plane, so that the discretization cell size was 2 × 2 × L nm3. In order to obtain the vortex core with a desired polarity (spin polarization direction of dc current and vortex core polarity should have opposite directions in order to ensure the steady-state vortex precession) and to displace the vortex core from its equilibrium position in the nanodot center, we have initially applied a short magnetic field pulse with the out-of-plane projection of 200 Oe, the in-plane projection H x = 10 Oe, and the duration Δt = 3 ns. Simulations were carried out for the physical time t = 200 to 3,000 ns depending on the applied dc current because for currents close to the threshold current Jc1, the time for establishing the vortex steady-state precession regime was much larger than for higher currents (see Equation 8 below).
Results and discussion
Our comparison of the calculated dependences u0(J) and ω G (J) with simulations is principally different from the comparison conducted in a paper , where the authors compared Equations 5 and 7 with their simulations fitting the model-dependent nonlinear coefficients N and λ from the same simulations. One can compare Figures 1 and 2 with the results by Grimaldi et al. , where the authors had no success in explaining their experimental dependences u0(J) and ω G (J) by a reasonable model. The realistic theoretical nonlinear frequency parameter N for Py dots with L = 5 nm and R = 250 nm should be larger than 0.11 that the authors of  used. N = 0.25 can be calculated from pure magnetostatic energy in the limit β → 0 (inset of Figure 2). Accounting all the energy contributions in Equation 4 yields N = 0.36, which is closer to the fitted experimental value N = 0.50.
where u(0) is the initial vortex core displacement and is the inverse relaxation time for J > Jc 1 (order of 100 ns). at t → ∞ and J = Jc 1. If J < Jc 1, the orbit radius u(t, J) decreases exponentially to 0 with the relaxation time . The divergence of the relaxation times τ± at J = Jc 1 allows considering a breaking symmetry second-order phase transition from the equilibrium value u0 = 0 to finite defined by Equation 7. Equations 7 and 8 represent mean-field approximation to the problem and are valid not too very close to the value of J = Jc 1, where thermal fluctuations are important [13, 21].
Typical experiments on the vortex excitations in nanopillars are conducted at room temperature T = 300 K without initial field pulse, i.e., a thermal level u(0) should be sufficient to start vortex core motion to a steady orbit. To find the thermal amplitude of u(0), we use the well-known relation between static susceptibility of the system and magnetization fluctuations . The in-plane components are , and M = ξM s s, where ξ = 2/3 within TVA . This leads to the simple relation . It is reasonable to use for interpretation of the experiments. u T (0) ≈ 0.05 (5 nm in absolute units) for the dot made of permalloy with L = 7 nm and R = 100 nm.
The nonlinear frequency coefficient N(β, R, J) = κ′(β, R, J)/κ(β, R, J) is positive (because of κ, κ′ >0 for typical dot parameters), and it is a strong function of the dot geometrical sizes L and R and a weak function of J. For the dot radii R > > L e , N(β, R, 0) ≈ 0.21 - 0.25 (the magnetostatic limit, see inset of Figure 2). If R > > L e and β → 0, then N(β, R, 0) ≈ 0.25 . For the realistic sizes of free layer in a nanopillar (R is about 100 nm and L = 3 to 10 nm), this coefficient is essentially larger due to finite β and exchange contribution, and it can be of order of 1. The exchange nonlinear contribution κ′ex is important for R < 300 nm. However, the authors of [19–21] did not consider it at all. Note that N(0.089, 300 nm, 0) ≈ 0.5 recently measured  is two times larger than 0.25. The authors of  suggested to use an additional term ~u6 in the magnetic energy fitting the nonlinear frequency due to accounting a u4-contribution (N = 0.26) that is too small based on , while the nonlinear coefficient N(β, R) calculated by Equation 5 for the parameters of Py dots (L = 4.8 nm, R = 275 nm)  is equal to 0.38. Moreover, the authors of  did not account that, for a high value of the vortex amplitude u = 0.6 to 0.7, the contribution of nonlinear gyrovector G(u) ∝ c2u2 to the vortex frequency is more important than the u6-magnetic energy term. The gyrovector G(u) decreases essentially for such a large u resulting in the nonlinear frequency increase. The TVA calculations based on Equation 5 lead to the small nonlinear Oe energy contribution κ′Oe, whereas Dussaux et al.  stated that κ′Oe is more important than the magnetostatic nonlinear contribution.
We demonstrated that the generalized Thiele equation of motion (1) with the nonlinear coefficients (2) considered beyond the rigid vortex approximation can be successfully used for quantitative description of the nonlinear vortex STNO dynamics excited by spin-polarized current in a circular nanodot. We calculated the nonlinear parameters governing the vortex core large-amplitude oscillations and showed that the analytical two-vortex model can predict the parameters, which are in good agreement with the ones simulated numerically. The Thiele approach and the energy dissipation approach [12, 19] are equivalent because they are grounded on the same LLG equation of magnetization motion. The limits of applicability of the nonlinear oscillator approach developed for saturated nanodots  to vortex STNO dynamics are established. The calculated and simulated dependences of the vortex core orbit radius u(t) and phase Φ(t) can be used as a starting point to consider the transient dynamics of synchronization of two coupled vortex ST nano-oscillators in laterally located circular nanopillars  or square nanodots with circular nanocontacts  calculated recently.
This work was supported in part by the Spanish MINECO grant FIS2010-20979-C02-01. KYG acknowledges support by IKERBASQUE (the Basque Foundation for Science).
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