SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Nano Express

Nonlinear magnetic vortex dynamics in a circular nanodot excited by spin-polarized current

Konstantin Y Guslienko12*, Oksana V Sukhostavets1 and Dmitry V Berkov3

Author Affiliations

1 Depto. Física de Materiales, Facultad de Química, Universidad del País Vasco, UPV/EHU, San Sebastián 20018, Spain

2 IKERBASQUE, Basque Foundation for Science, Bilbao 48011, Spain

3 General Numerics Research Laboratory, Jena 07745, Germany

For all author emails, please log on.

Nanoscale Research Letters 2014, 9:386  doi:10.1186/1556-276X-9-386


The electronic version of this article is the complete one and can be found online at: http://www.nanoscalereslett.com/content/9/1/386


Received:15 June 2014
Accepted:1 August 2014
Published:8 August 2014

© 2014 Guslienko et al.; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Abstract

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.

Keywords:
Magnetic nanodot; Nano-oscillator; Vortex; Spin torque transfer

Background

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 [4]. 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 the present paper, we focus on the magnetic vortex dynamics in a thin circular nanodot representing a free layer of nanopillar (see inset of Figure 1). Circular nanodots made of soft magnetic material have a vortex state of magnetization as the ground state for certain dot radii R and thickness L. The vortex state is characterized by in-plane curling magnetization and a nanosize vortex core with out-of-plane magnetization. Since the vortex state of magnetization was discovered as the ground state of patterned magnetic dots, the dynamics of vortices have attracted considerable attention. Being displaced from its equilibrium position in the dot center, the vortex core reveals sub-GHz frequency oscillations with a narrow linewidth [2,7,12]. The oscillations of the vortex core are governed by a competition of the gyroforce, Gilbert damping force, spin transfer torque, and restoring force. The restoring force is determined by the vortex confinement in a nanodot. Vortex core oscillations with small amplitude can be well described in the linear regime, but for increasing of the STNO output power, a large-amplitude motion has to be excited. In the regime of large-amplitude spin transfer-induced vortex gyration, it is important to take into account nonlinear contributions to all the forces acting on the moving vortex. The analytical description and micromagnetic simulations of the magnetic field and spin transfer-induced vortex dynamics in the nonlinear regime have been proposed by several groups [12-22], but the results are still contradictory. It is unclear to what extent a standard nonlinear oscillator model [13] is applicable to the vortex STNO, how to calculate the nonlinear parameters, and how the parameters depend on the nanodot sizes.

thumbnailFigure 1. Magnetic vortex dynamics in a thin circular FeNi nanodot. Vortex core steady-state orbit radius u0(J) in the circular FeNi nanodot of thickness L = 7 nm and radius R = 100 nm vs. current J perpendicular to the dot plane. Solid black lines are calculations by Equation 7; red circles mark the simulated points. Inset: sketch of the cylindrical vortex state dot with the core position X and used system of coordinates.

In this paper, we show that a generalized Thiele approach [23] 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.

Methods

Analytical method

We apply the Landau-Lifshitz-Gilbert (LLG) equation of motion of the free layer magnetization <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M1">View MathML</a>, 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 [24], τs = σJm × (m × P), where σ = ℏη/(2|e|LMs), 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 <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M2">View MathML</a>. The free layer (dot) radius is R and thickness is L.

We apply Thiele's approach [23] for the magnetic vortex motion in circular nanodot (inset of Figure 1). We assume that magnetization distribution can be characterized by a position of its center X(t) that can vary with time and, therefore, the magnetization as a function of the coordinates r and X(t) can be written as m(r,t) = m(r,X(t)). Then, we can rewrite the LLG equation as a generalized Thiele equation for X(t):

<a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M3">View MathML</a>

(1)

where W is the total magnetic energy, α,β = x,y, and ∂α = ∂/∂Xα. The components of the gyrotensor <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M4">View MathML</a>, damping tensor <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M5">View MathML</a>, and the spin-torque force can be expressed as follows [16]:

<a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M6">View MathML</a>

(2)

We assume that the dot is thin enough and m does not depend on z-coordinate. The magnetization m(x,y) has the components <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M7">View MathML</a> and <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M8">View MathML</a> expressed via a complex function <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M9">View MathML</a>[25]. Inside the vortex core, the vortex configuration is described as a topological soliton, <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M10">View MathML</a>, |f(ζ)| ≤ 1, where f(ζ) is an analytic function. Outside the vortex core region, the magnetization distribution is <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M11">View MathML</a>, |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 <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M12">View MathML</a>[26], where C is the vortex chirality, ζ = (x + iy)/R, s = sx + isy, s = X/R, c = Rc/R, and Rc is the vortex core radius.

The total micromagnetic energy <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M13">View MathML</a> in Equation 1 including volume <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M14">View MathML</a> and surface <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M15">View MathML</a> magnetostatic energy, exchange Wex energy, and Zeeman WZ energy of the nanodot with a displaced magnetic vortex is a functional of magnetization distribution W[m(r, t)]. Using m = m(r, X(t)) and integrating over-the-dot volume and surface, the energy W can be expressed as a function of X within TVA [16]. The Zeeman energy is related to Oersted field <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M16">View MathML</a> of the spin-polarized current, WZ(X) = - Ms ∫ dVm(r, X) ⋅ HJ. We introduce a time-dependent vortex orbit radius and phase by s = u exp(iΦ). The gyroforce in Equation 1 is determined by the gyrovector <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M17">View MathML</a>, where G = Gz = Gxy. The functions G(s) and W(s) depend only on u = |s| due to a circular symmetry of the dot. G(0) = 2πpMsL/γ, where p is the vortex core polarity. The damping force <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M18">View MathML</a> and spin-torque force FST are functions not only on u = |s| but also on direction of s. Nonlinear Equation 1 can be written for the circular dot in oscillator-like form

<a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M19">View MathML</a>

(3)

where ωG(u) = (R2u|G(u)|)- 1W(u)/∂u is the nonlinear gyrotropic frequency, d(u) = - D(u)/|G(u)| is the nonlinear diagonal damping, D = Dxx = Dyy, dn(s) = - Dxy(s)/|G(u)| is the nonlinear nondiagonal damping, and χ(u) = a(u)/|G(u)|. It is assumed here that FST(s) = a(u)(z × s) [14], where a is proportional to the CPP current density J and a(0) = πRLMsσJ.

To solve Equation 3, we need to answer the following questions: (1) can we decompose the functions W(s), G(s), Dαβ(s), and FST(s) in the power series of u = |s| and keep only several low-power terms? and (2) what is the accuracy of such truncated series accounting that u = |s| can reach values of 0.5 to 0.6 for a typical vortex STNO? Some of these functions may be nonanalytical functions of u = |s|. If the answer to the first question is yes, then we should decompose W(s) up to u4, FST(s) up to u3, and G(s), Dαβ(s) up to u2-terms to get a cubical equation of the vortex motion. The series decomposition of G(s) does not contain u2-term; it contains only small c2u2-term, G(u) = G(0)[1 - O(c2u2)], although G(u) essentially decreases at large u, when the vortex core is close to be expelled from the dot [16]. The result of power decomposition of the total energy density <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M20">View MathML</a> is

<a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M21">View MathML</a>

(4)

and the coefficients are

<a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M22">View MathML</a>

where <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M23">View MathML</a>, <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M24">View MathML</a>, <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M25">View MathML</a>, β = L/R, <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M26">View MathML</a>, and ς = 1 + 15(ln 2 - 1/2)Rc/8R.

There is an additional contribution to κ/2, 2(Le/R)2, due to the face magnetic charges essential for the nanodots with small R[27]. 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 (Rc = 12 nm) for the nanodot thickness L = 7 nm.

The nonlinear vortex gyrotropic frequency can be written accounting Equation 4 as

<a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M27">View MathML</a>

(5)

where the linear gyrotropic frequency is ω0 = γMsκ(β, R, J)/2, and N(β, R) = κ′(β, R)/κ(β, R).

The frequency <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M28">View MathML</a> was calculated in [26] 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.

The last term in Equation 3 prevents its reducing to a nonlinear oscillator equation similar to the one used for the description of saturated STNO in [13]. Calculation within TVA yields the decomposition <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M29">View MathML</a>, where <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M30">View MathML</a>, i.e., the term containing dn(s) ≈ αGu2 <<1 can be neglected. Then, substituting s = u exp(iΦ) to Equation 3, we get the system of coupled equations

<a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M31">View MathML</a>

(6)

Equation 3 and the system (6) are different from the system of equations of the nonlinear oscillator approach [13]. Equations 6 are reduced to the autonomous oscillator equations <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M32">View MathML</a> and <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M33">View MathML</a> only if the conditions d2 < < 1 and  < < ωG are satisfied and we define the positive/negative damping parameters [13] as Γ+(u) = d(u)ωG(u) and Γ-(u) = χ(u). We note that reducing the Thiele equation (1) to a nonlinear oscillator equation [13] 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 dn < < 1, d2 < < 1, and  < < ωG are satisfied. The nonlinear oscillator model [13] 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 system (6) at <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M34">View MathML</a> yields the steady vortex oscillation solution u0(J) > 0 as root of the equation χ(u0) = d(u0)ωG(u0) for χ(0) > d(0)ω0 (J > Jc1) and u0 = 0 otherwise. If we use the power decompositions ωG(u) = ω0 + ω1u2, d(s) = d0 + d1u2, and χ(u) = χ0 + χ1u2 for the nonlinear vortex frequency, damping, and spin-torque terms, respectively, and account that the linear vortex frequency contains a contribution proportional to the current density <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M35">View MathML</a>, where ωe = (8π/15)(γR/c)ς[12,16], then we get the vortex core steady orbit radius at J > Jc1

<a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M36">View MathML</a>

(7)

The model parameters are <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M37">View MathML</a>, d0 = αG[5 + 4 ln(R/Rc)]/8, d1 = 11αG/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.

Numerical method

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) [28]. 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 αG = 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 Hx = 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

Calculated analytically, the vortex core steady orbit radius in circular dot u0(J) as a function of current J is compared with the simulations (see Figure 1). There is no fitting except only taking the critical current Jc1 value from simulations. Agreement is quite good, confirming that all the nonlinear parameters of the vortex motion were accounted correctly. The steady orbit radius u0(J) allows finding the STNO generation frequency <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M38">View MathML</a>, which increases approximately linearly with J increasing up to the second critical current value Jc2 when the steady oscillation state becomes unstable (see Figure 2). The instability is related with the vortex core polarity reversal reaching a core critical velocity or the vortex core expelling from the dot increasing the current density J[12,16]. We simulated the values of Jc2 = 2.7, 5.0, and 10.2 MA/cm2 for the dot thickness L = 5, 7, and 10 nm, respectively. The calculated STNO frequency is 15 to 20% higher than the simulated one due to overestimation of <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M39">View MathML</a> within TVA for β =0.1. The calculated nonlinear frequency part is very close to the simulated one, except the vicinity of Jc2, where the analytical model fails.

thumbnailFigure 2. The vortex steady-state oscillation frequency vs. current. The nanodot thickness L is 5 nm (1), 7 nm (2), and 10 nm (3), and radius is R = 100 nm. The frequency is shown within the current range of the stable vortex steady-state orbit, Jc1 < J < Jc2. Solid black lines are calculations by Equation 5; red squares mark the simulated points. Inset: the nonlinear vortex frequency coefficient vs. the dot thickness for R = 100 nm and J = 0 accounting all energy contributions (1) and only magnetostatic contribution (2).

Our comparison of the calculated dependences u0(J) and ωG(J) with simulations is principally different from the comparison conducted in a paper [19], 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. [20], where the authors had no success in explaining their experimental dependences u0(J) and ωG(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 [21] 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.

The system (6) can be solved analytically in nonlinear case. Its solution describing transient vortex dynamics is

<a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M40">View MathML</a>

(8)

where u(0) is the initial vortex core displacement and <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M41">View MathML</a> is the inverse relaxation time for J > Jc1 (order of 100 ns). <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M42">View MathML</a> at t → ∞ and J = Jc1. If J < Jc1, the orbit radius u(t, J) decreases exponentially to 0 with the relaxation time <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M43">View MathML</a>. The divergence of the relaxation times τ± at J = Jc1 allows considering a breaking symmetry second-order phase transition from the equilibrium value u0 = 0 to finite <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M44">View MathML</a> 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 = Jc1, where thermal fluctuations are important [13,21].

Equation 8 describes approaching of the vortex orbit radius to a steady value u0(J) under influence of dc spin-polarized current. We note that the corresponding relaxation time is determined by only linear parameters, whereas the orbit radius (7) depends on ratio of the nonlinear and linear model parameters. The solution of Equation 8 is plotted in Figure 3 as a function of time along with micromagnetic simulations for circular Py dot with thickness L = 7 nm and radius R = 100 nm. The vortex was excited by in-plane field pulse during approximately the first 5 ns, and then the vortex core approached the stationary orbit of radius u0(J). We estimated u(0) after the pulse as u(0) = 0.1 and plotted the solid lines without any fitting except using the simulated value of the critical current Jc1. Overall agreement of the calculations by Equation 8 and simulations is quite good, especially for large times t ≥ 3τ+, although the calculated relaxation time τ+ is smaller than the simulated one due to overestimation of <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M45">View MathML</a> within TVA. The typical simulated ratio Jc2/Jc1 ≈ 1.5; therefore, minimal τ+ ≈ 20 to 30 ns. But the transient time of saturation of u(t, J) is about of 100 ns and can reach several microseconds at J/Jc1 < 1.1. The simulated value of λ = 0.83, whereas the analytic theory based on TVA yields the close value of λ(Jc1) = 0.81.

thumbnailFigure 3. Instant vortex core orbit radius vs. time for different currents. The results are within the current range of the stable vortex steady-state orbit, Jc1 < J < Jc2 (5.0 MA/cm2). The nanodot thickness is L = 7 nm and the radius is R = 100 nm. Solid lines are calculations of the vortex transient dynamics by Equation 8, and symbols (black squares, red circles, green triangles, and blue rhombi) mark the simulated points.

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 <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M46">View MathML</a> and magnetization fluctuations <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M47">View MathML</a>. The in-plane components are <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M48">View MathML</a>, and M = ξMss, where ξ = 2/3 within TVA [26]. This leads to the simple relation <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M49">View MathML</a>. It is reasonable to use <a onClick="popup('http://www.nanoscalereslett.com/content/9/1/386/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/9/1/386/mathml/M50">View MathML</a> for interpretation of the experiments. uT(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 > > Le, N(β, R, 0) ≈ 0.21 - 0.25 (the magnetostatic limit, see inset of Figure 2). If R > > Le and β → 0, then N(β, R, 0) ≈ 0.25 [14]. 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 κex 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 [29] is two times larger than 0.25. The authors of [19] 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 [14], while the nonlinear coefficient N(β, R) calculated by Equation 5 for the parameters of Py dots (L = 4.8 nm, R = 275 nm) [19] is equal to 0.38. Moreover, the authors of [19] 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 κOe, whereas Dussaux et al. [19] stated that κOe is more important than the magnetostatic nonlinear contribution.

Conclusions

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 [13] 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 [30] or square nanodots with circular nanocontacts [31] calculated recently.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

KYG formulated the problem and carried out the analytical calculations. OVS and DVB conducted the micromagnetic simulations. KYG supervised the work and finalized the manuscript. All authors have read and approved the final manuscript.

Acknowledgements

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).

References

  1. Rowlands GE, Krivorotov IN: Magnetization dynamics in a dual free-layer spin torque nano-oscillator.

    Phys Rev B 2012, 86(094425):7. OpenURL

  2. Pribiag VS, Krivorotov IN, Fuchs GD, Braganca PM, Ozatay O, Sankey JC, Ralph DC, Buhrman RA: Magnetic vortex oscillator driven by d.c. spin-polarized current.

    Nat Phys 2007, 3:498-503. Publisher Full Text OpenURL

  3. Mistral Q, Van Kampen M, Hrkac G, Kim JV, Devolder T, Crozat P, Chappert C, Lagae L, Schrefl T: Current driven vortex oscillations in metallic nano-contacts.

    Phys Rev Lett 2008, 100(257201):4. OpenURL

  4. Katine JA, Fullerton EE: Device implications of spin-transfer torques.

    J Magn Magn Mater 2008, 320:1217-1226. Publisher Full Text OpenURL

  5. Abreu Araujo F, Darques M, Zvezdin KA, Khvalkovskiy AV, Locatelli N, Bouzehouane K, Cros V, Piraux L: Microwave signal emission in spin-torque vortex oscillators in metallic nanowires.

    Phys Rev B 2012, 86(064424):8. OpenURL

  6. Sluka V, Kákay A, Deac AM, Bürgler DE, Hertel R, Schneider CM: Spin-transfer torque induced vortex dynamics in Fe/Ag/Fe nanopillars.

    J Phys D Appl Phys 2011, 44(384002):10. OpenURL

  7. Locatelli N, Naletov VV, Grollier J, de Loubens G, Cros V, Deranlot C, Ulysse C, Faini G, Klein O, Fert A: Dynamics of two coupled vortices in a spin valve nanopillar excited by spin transfer torque.

    Appl Phys Lett 2011, 98(062501):4. OpenURL

  8. Manfrini M, Devolder T, Kim J-V, Crozat P, Chappert C, Roy WV, Lagae L: Frequency shift keying in vortex-based spin torque oscillators.

    J Appl Phys 2011, 109(083940):6. OpenURL

  9. Martin SY, de Mestier N, Thirion C, Hoarau C, Conraux Y, Baraduc C, Diény B: Parametric oscillator based on nonlinear vortex dynamics in low-resistance magnetic tunnel junctions.

    Phys Rev B 2011, 84(144434):9. OpenURL

  10. Petit-Watelot S, Kim J-V, Rutolo A, Otxoa RM, Bouzehouane K, Grollier J, Vansteenkiste A, Wiele BV, Cros V, Devolder T: Commensurability and chaos in magnetic vortex oscillations.

    Nat Phys 2012, 8:682-687. Publisher Full Text OpenURL

  11. Finocchio G, Pribiag VS, Torres L, Buhrman RA, Azzerboni B: Spin-torque driven magnetic vortex self-oscillations in perpendicular magnetic fields.

    Appl Phys Lett 2010, 96(102508):3. OpenURL

  12. Khvalkovskiy AV, Grollier J, Dussaux A, Zvezdin KA, Cros V: Vortex oscillations induced by spin-polarized current in a magnetic nanopillar.

    Phys Rev B 2009, 80(140401):7. OpenURL

  13. Slavin AN, Tiberkevich V: Nonlinear auto-oscillator theory of microwave generation by spin-polarized current.

    IEEE Trans Magn 2009, 45:1875-1918. OpenURL

  14. Gaididei Y, Kravchuk VP, Sheka DD: Magnetic vortex dynamics induced by an electrical current.

    Intern J Quant Chem 2010, 110:83-97. Publisher Full Text OpenURL

  15. Guslienko KY, Heredero R, Chubykalo-Fesenko O: Non-linear vortex dynamics in soft magnetic circular nanodots.

    Phys Rev B 2010, 82(014402):9. OpenURL

  16. Guslienko KY, Aranda GR, Gonzalez J: Spin torque and critical currents for magnetic vortex nano-oscillator in nanopillars.

    J Phys Conf Ser 2011, 292(012006):5. OpenURL

  17. Guslienko KY: Spin torque induced magnetic vortex dynamics in layered F/N/F nanopillars.

    J Spintron Magn Nanomater 2012, 1:70-74. OpenURL

  18. Drews A, Krüger B, Selke G, Kamionka T, Vogel A, Martens M, Merkt U, Möller D, Meier G: Nonlinear magnetic vortex gyration.

    Phys Rev B 2012, 85(144417):9. OpenURL

  19. Dussaux A, Khvalkovskiy AV, Bortolotti P, Grollier J, Cros V, Fert A: Field dependence of spin-transfer-induced vortex dynamics in the nonlinear regime.

    Phys Rev B 2012, 86(014402):12. OpenURL

  20. Bortolotti P, Grimaldi E, Dussaux A, Grollier J, Cros V, Serpico C, Yakushiji K, Fukushima A, Kubota H, Matsumoto R, Yuasa S: Parametric excitation of magnetic vortex gyrations in spin torque nano-oscillators.

    Phys Rev B 2013, 88(174417):10. OpenURL

  21. Grimaldi E, Dussaux A, Bortolotti P, Grollier J, Pillet G, Fukushima A, Kubota H, Yakushiji K, Yuasa S, Cros V: Response to noise of a vortex based spin transfer nano-oscillator.

    Phys Rev 2014, 89(104404):12. OpenURL

  22. Sanches F, Tyberkevych V, Guslienko KY, Sinha J, Hayashi M, Slavin AN: Current driven gyrotropic mode of a magnetic vortex as a non-isochronous nanoscale auto-oscillator.

    Phys Rev B 2014, 89(140410):5. OpenURL

  23. Thiele AA: Steady-state motion of magnetic domains.

    Phys Rev Lett 1973, 30:230-233. Publisher Full Text OpenURL

  24. Slonczewski JC: Current driven excitations of magnetic multilayers.

    J Magn Magn Mater 1996, 159:L1-L7.

    Excitations of spin waves by an electric current.ibid. 1999, 195:L261-268

    Publisher Full Text OpenURL

  25. Guslienko KY, Metlov KL: Evolution and stability of a magnetic vortex in cylindrical ferromagnetic nanoparticle under applied field.

    Phys Rev B 2001, 63(100403):4. OpenURL

  26. Guslienko KY, Ivanov BA, Novosad V, Shima H, Otani Y, Fukamichi K: Eigenfrequencies of vortex state excitations in magnetic submicron-size disks.

    J Appl Phys 2002, 91:8037-8039. Publisher Full Text OpenURL

  27. Metlov KL: Vortex precession frequency in cylindrical nanomagnets.

    J Appl Phys 2013, 114(223908):6. OpenURL

  28. Berkov DV, Gorn NL:

    MicroMagus.

    http://www.micromagus.de webcite

    OpenURL

  29. Sukhostavtes OV, Pigeau B, Sangiao S, de Loubens G, Naletov VV, Klein O, Mitsuzuka K, Andrieu S, Montaigne F, Guslienko KY: Probing anharmonicity of the potential well for a magnetic vortex core in a nanodot.

    Phys Rev Lett 2013, 111(247601):5. OpenURL

  30. Belanovsky AD, Locatelli N, Skirdkov PN, Abreu Araujo F, Grollier J, Zvezdin KA, Cros V, Zvezdin AK: Phase locking dynamics of dipolarly coupled vortex-based spin transfer oscillators.

    Phys Rev B 2012, 85(100409):5.

    Numerical and analytical investigation of the synchronization of dipolarly coupled vortex spin-torque nano-oscillators. Appl Phys Lett 2013, 103(122405):4

    OpenURL

  31. Erokhin S, Berkov DV: Robust synchronization of an arbitrary number of spin-torque driven nano-oscillators.

    Phys Rev B 2014, 89(144421):12. OpenURL