### Analytical method

We apply the Landau-Lifshitz-Gilbert (LLG) equation of motion of the free layer magnetization $\dot{\mathbf{m}}=-\gamma \mathbf{m}\times {\mathbf{H}}_{\mathrm{eff}}+{\alpha}_{\mathrm{G}}\mathbf{m}\times \dot{\mathbf{m}}+\gamma {\mathbf{\tau}}_{s}$, where **m** = **M**/*M*_{s}, *M*_{s} is the saturation magnetization, *γ* > 0 is the gyromagnetic ratio, **H**_{eff} is the effective field, and *α*_{G} is the Gilbert damping. We use a spin angular momentum transfer torque in the form suggested by Slonczewski [24], **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 $\mathbf{P}=P\widehat{\mathbf{z}}$. 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*):

${G}_{\mathit{\alpha \beta}}{\dot{X}}_{\beta}=-{\partial}_{\alpha}W+{D}_{\mathit{\alpha \beta}}{\dot{X}}_{\beta}+{F}_{\mathit{ST}}^{\alpha},$

(1)

where

*W* is the total magnetic energy,

*α*,

*β* =

*x*,

*y*, and ∂

_{
α
} = ∂/∂

*X*_{
α
}. The components of the gyrotensor

$\widehat{G}$, damping tensor

$\widehat{D}$, and the spin-torque force can be expressed as follows [

16]:

$\begin{array}{l}{G}_{\mathit{\alpha \beta}}\left(\mathbf{X}\right)=\frac{{M}_{s}}{\gamma}{\displaystyle \int {d}^{3}\mathbf{r}}\left({\partial}_{\alpha}\mathbf{m}\times {\partial}_{\beta}\mathbf{m}\right)\cdot \mathbf{m}\\ {D}_{\mathit{\alpha \beta}}\left(\mathbf{X}\right)=-{\alpha}_{G}\frac{{M}_{s}}{\gamma}{\displaystyle \int {d}^{3}\mathbf{r}}\left({\partial}_{\alpha}\mathbf{m}\times {\partial}_{\beta}\mathbf{m}\right)\\ {F}_{\mathit{ST}}^{\alpha}\left(\mathbf{X}\right)={M}_{s}\mathit{\sigma J}\mathbf{P}\cdot {\displaystyle \int {d}^{3}\mathbf{r}}\left(\mathbf{m}\times {\partial}_{\alpha}\mathbf{m}\right).\end{array}$

(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 ${m}_{x}+i{m}_{y}=2w/\left(1+w\overline{w}\right)$ and ${m}_{z}=\left(1-w\overline{w}\right)/\left(1+w\overline{w}\right)$ expressed via a complex function $w\left(\zeta ,\overline{\zeta}\right)$[25]. Inside the vortex core, the vortex configuration is described as a topological soliton, $w\left(\zeta ,\overline{\zeta}\right)=f\left(\zeta \right)$, |*f*(*ζ*)| ≤ 1, where *f*(*ζ*) is an analytic function. Outside the vortex core region, the magnetization distribution is $w\left(\zeta ,\overline{\zeta}\right)=f\left(\zeta \right)/\left|f\left(\zeta \right)\right|$, |*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 $f\left(\zeta \right)=-\mathit{iC}\left(\zeta -s\right)\left(\overline{s}\zeta -1\right)/c\left(1+{\left|s\right|}^{2}\right)$[26], 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.

The total micromagnetic energy

$W={W}_{m}^{v}+{W}_{m}^{s}+{W}_{\mathit{ex}}+{W}_{Z}$ in Equation

1 including volume

${W}_{m}^{v}$ and surface

${W}_{m}^{s}$ magnetostatic energy, exchange

*W*_{
ex
} energy, and Zeeman

*W*_{
Z
} 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

${\mathbf{H}}_{J}=\left(0,{H}_{J}^{\varphi},0\right)$ of the spin-polarized current,

*W*_{
Z
}(

**X**) = -

*M*_{
s
} ∫

*dV* **m**(

**r**,

**X**) ⋅

**H**_{
J
}. We introduce a time-dependent vortex orbit radius and phase by

*s* =

*u* exp(

*i* Φ). The gyroforce in Equation

1 is determined by the gyrovector

$\mathbf{G}=G\widehat{\mathbf{z}}$, where

*G* =

*G*_{
z
} =

*G*_{
xy
}. The functions

*G*(

*s*) and

*W*(

*s*) depend only on

*u* = |

*s*| due to a circular symmetry of the dot.

*G*(0) = 2

*πpM*_{
s
}*L*/

*γ*, where

*p* is the vortex core polarity. The damping force

$\widehat{D}\dot{\mathbf{X}}$ and spin-torque force

**F**_{
ST
} 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

$i\dot{s}+{\omega}_{G}\left(u\right)s=-d\left(u\right)\dot{s}+\mathit{i\chi}\left(u\right)s-i{d}_{n}\left(s\right)\dot{\overline{s}},$

(3)

where *ω*_{
G
}(*u*) = (*R*^{2}*u*|*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**) [14], where *a* is proportional to the CPP current density *J* and *a*(0) = *πRLM*_{
s
}*σJ*.

To solve Equation

3, we need to answer the following questions: (1) can we decompose the functions

*W*(

*s*),

*G*(

*s*),

*D*_{
αβ
}(

*s*), and

**F**_{
ST
}(

*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

*u*^{4},

**F**_{
ST
}(

*s*) up to

*u*^{3}, and

*G*(

*s*),

*D*_{
αβ
}(

*s*) up to

*u*^{2}-terms to get a cubical equation of the vortex motion. The series decomposition of

*G*(

*s*) does not contain

*u*^{2}-term; it contains only small

*c*^{2}*u*^{2}-term,

*G*(

*u*) =

*G*(0)[1 -

*O*(

*c*^{2}*u*^{2})], 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

$w\left(u\right)=W\left(u\right)/{M}_{s}^{2}V$ is

$w\left(u\right)=w\left(0\right)+\frac{1}{2}\kappa {u}^{2}+\frac{1}{4}{\kappa}^{\prime}{u}^{4},\phantom{\rule{0.5em}{0ex}}u=\left|s\right|,$

(4)

and the coefficients are

$\begin{array}{l}\frac{1}{2}\kappa \left(\beta ,R,J\right)=8\pi {\displaystyle \underset{0}{\overset{\infty}{\int}}\mathit{dt}\frac{f\left(\mathit{\beta t}\right)}{t}}{I}^{2}\left(t\right)-{\left(\frac{{L}_{e}}{R}\right)}^{2}+\frac{2\pi}{15c}\frac{\mathit{JCR\varsigma}}{{M}_{s}}\phantom{\rule{0.5em}{0ex}}\mathrm{and}\\ \frac{1}{4}{\kappa}^{\prime}\left(\beta ,R,J\right)=2\pi {\displaystyle \underset{0}{\overset{\infty}{\int}}\mathit{dt}\frac{f\left(\mathit{\beta t}\right)}{t}}\left[{I}_{2}^{2}\left(t\right)-I\left(t\right){I}_{1}\left(t\right)\right]+\frac{1}{2}{\left(\frac{{L}_{e}}{R}\right)}^{2}+\frac{\pi}{15c}\frac{\mathit{JCR}}{{M}_{s}},\end{array}$

where $I\left(t\right)={\displaystyle {\int}_{0}^{1}\mathit{d\rho \rho}{J}_{1}}\left(\mathit{t\rho}\right)$, ${I}_{1}\left(t\right)={\displaystyle {\int}_{0}^{1}\mathit{d\rho}{\rho}^{-1}{\left(1-{\rho}^{2}\right)}^{2}{J}_{1}}\left(\mathit{t\rho}\right)$, ${I}_{2}\left(t\right)={\displaystyle {\int}_{0}^{1}\mathit{d\rho}\left(1+{\rho}^{2}\right){J}_{2}}\left(\mathit{t\rho}\right)$, *β* = *L*/*R*, ${L}_{e}=\sqrt{2A}/{M}_{s}$, and *ς* = 1 + 15(ln 2 - 1/2)*R*_{
c
}/8*R*.

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*[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 - *u*^{2})/(1 + *u*^{2}). 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.

The nonlinear vortex gyrotropic frequency can be written accounting Equation

4 as

${\omega}_{G}\left(u\right)={\omega}_{0}\left[1+N{u}^{2}\right],$

(5)

where the linear gyrotropic frequency is *ω*_{0} = *γM*_{
s
}*κ*(*β*, *R*, *J*)/2, and *N*(*β*, *R*) = *κ*′(*β*, *R*)/*κ*(*β*, *R*).

The frequency ${\omega}_{0}^{\prime}=\gamma {M}_{s}\kappa \left(\beta ,R,0\right)/2$ 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

${d}_{n}\left(s\right)={d}_{n}^{0}+{d}_{n}^{1}{s}_{x}{s}_{y}$, where

${d}_{n}^{0}=0$, i.e., the term containing

*d*_{
n
}(

*s*) ≈

*α*_{
G
}*u*^{2} <<1 can be neglected. Then, substituting

*s* =

*u* exp(

*i* Φ) to Equation

3, we get the system of coupled equations

$\dot{\mathrm{\Phi}}-{\omega}_{G}\left(u\right)=d\left(u\right)\frac{\dot{u}}{u},\phantom{\rule{0.5em}{0ex}}\dot{u}=\left[\chi \left(u\right)-d\left(u\right)\dot{\mathrm{\Phi}}\right]u.$

(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 $\dot{u}/u=\chi \left(u\right)-d\left(u\right){\omega}_{G}\left(u\right)$ and $\dot{\mathrm{\Phi}}={\omega}_{G}\left(u\right)$ only if the conditions *d*^{2} < < 1 and *dχ* < < *ω*_{
G
} are satisfied and we define the positive/negative damping parameters [13] 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 [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 *d*_{
n
} < < 1, *d*^{2} < < 1, and *dχ* < < *ω*_{
G
} 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

$\dot{u}=0$ yields the steady vortex oscillation solution

*u*_{0}(

*J*) > 0 as root of the equation

*χ*(

*u*_{0}) =

*d*(

*u*_{0})

*ω*_{
G
}(

*u*_{0}) for

*χ*(0) >

*d*(0)

*ω*_{0} (

*J* >

*J*_{c 1}) and

*u*_{0} = 0 otherwise. If we use the power decompositions

*ω*_{
G
}(

*u*) =

*ω*_{0} +

*ω*_{1}*u*^{2},

*d*(

*s*) =

*d*_{0} +

*d*_{1}*u*^{2}, and

*χ*(

*u*) =

*χ*_{0} +

*χ*_{1}*u*^{2} 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

${\omega}_{0}\left(J\right)={\omega}_{0}^{\prime}+{\omega}_{e}J$, where

*ω*_{
e
} = (8

*π*/15)(

*γR*/

*c*)

*ς*[

12,

16], then we get the vortex core steady orbit radius at

*J* >

*J*_{c 1}${u}_{0}\left(J\right)=\lambda \left(J\right)\sqrt{J/{J}_{c1}-1},\phantom{\rule{0.5em}{0ex}}{\lambda}^{2}\left(J\right)=\frac{{d}_{0}{\omega}_{0}^{\prime}}{\left[{d}_{1}{\omega}_{0}\left(J\right)+{d}_{0}{\omega}_{1}\left(J\right)-{\chi}_{1}\left(J\right)\right]}.$

(7)

The model parameters are ${J}_{c1}={d}_{0}{\omega}_{0}^{\prime}/\left(\mathit{\gamma \sigma}/2-{d}_{0}{\omega}_{e}\right)$, *d*_{0} = *α*_{
G
}[5 + 4 ln(*R*/*R*_{
c
})]/8, *d*_{1} = 11*α*_{
G
}/6, *χ*_{0} = *γσJ*/2. The ratio *χ*_{1}/*χ*_{0} = *O*(*c*^{2}*u*^{2}) < < 1, therefore, the nonlinear parameter *χ*_{1} 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.