3.2.3 Uniform mode approximation

To this end, we model the thin-film as a flat ellipsoid, characterized by the demagnetizing factors $N_x$, $N_y$, $N_z$. such that $N_x\ll N_z,N_y\ll N_z$. The demagnetizing factors can be computed as function of the aspect ratios $c/a$ and $b/a$ by means of the following expressions [71]:

$$N_x=\frac{\cos\phi \cos\theta}{\sin^3\theta \sin^2\psi}\left[F(k,\theta)-E(k,\theta)\right]\quad,$$ (3.24)

$$N_y=\frac{\cos\phi \cos\theta}{\sin^3\theta \sin^2\psi\cos^2\psi... ...si F(k,\theta) - \frac{\sin^2\psi \sin\theta\cos\theta}{\cos\phi} \right] ,$$ (3.25)

$$N_z=\frac{\cos\phi \cos\theta}{\sin^2\theta \cos^2\psi}\left[\frac{\sin\theta \cos\phi}{\cos\theta}-E(k,\theta)\right]\quad,$$ (3.26)

where $\cos\theta=c/a$, $\cos\phi=b/a$, and the angle $\psi$ is defined by

$$\sin\psi=\left[\frac{1-(b/a)^2}{1-(c/a)^2}\right]^{1/2}= \frac{\sin\phi}{\sin\theta}=k\quad;$$ (3.27)

$F(k,\theta)$ and $E(k,\theta)$ are the incomplete elliptic integrals [72] of the first and second kind respectively. All the angles $\theta, \phi, \psi$ are intended to belong to the interval $[0,\pi/2]$. In our case, the application of the above formulas gives:

$$N_x=0.0062 , N_y=0.0175 , N_z=0.9763\quad,$$ (3.28)

which, give the following values for the $D_x,D_y,D_z$ coefficients (shape and magnetocrystalline anisotropy):

$$D_x=N_x-\frac{2K_1}{\mu_0M_s^2}=1.2\times 10^{-3}, D_y=N_y, D_z=N_z \quad.$$ (3.29)

We assume that the external field $\textbf{h}_$a is applied along the $y$ axis:

$$\textbf{h}_ =h_a \mathbf{e}_y \quad.$$ (3.30)

We compute the critical time instants $t_1,t_2,T_s$ expressed by Eqs. (2.53)-(2.55), slightly generalized [50] for the case $D_x,D_y,D_z\neq 0$:

$$t_1=\int_{u_0}^{u_1} \frac{du}{k(D_z-D_x)\sqrt{1-p^2 \cos^2 u - [a_y-(p/k)\sin u]^2}} \quad,$$ (3.31)

$$t_2=t_1+2 \int_{u_1}^{u_2} \frac{du}{k(D_z-D_x)\sqrt{1-p^2 \cos^2 u - [a_y-(p/k)\sin u]^2}} \quad,$$ (3.32)

$$T_s=\int_{u_0}^{u_2} \frac{du}{k(D_z-D_x)\sqrt{1-p^2 \cos^2 u - [a_y-(p/k)\sin u]^2}} \quad.$$ (3.33)

where the parameters are given by:

$$k^2=(D_z - D_y)/(D_z - D_x)\quad,$$ (3.34)

$$a_y=-h_a/(D_z-D_y) \quad,$$ (3.35)

$$p^2=k^2 a_y^2+\frac{D_z-2 g_0}{D_z-D_x} \quad,$$ (3.36)

$$g_0=\frac{D_x}{2} \quad.$$ (3.37)

Notice that here we are supposing that the initial state is $\textbf{{m}}=\mathbf{e}_x$. Similarly to the derivation presented in section 2.4.3, in equations (3.31)-(3.33) the value of the parameters $u_0$, $u_2$ can be found by using parametric equations

$$m_x=-p\cos u \quad,\quad m_y=a_y+\frac{p}{k} \sin u \quad,$$ (3.38)

of the elliptic curve on which magnetization motion occurs:

$$m_x^2+k^2(m_y-a_y)^2=p^2 \quad,$$ (3.39)

to find the intersections with the unit circle $m_x^2+m_y^2=1$. The value $u_1$ can be found from the intersection between the elliptical trajectory (3.39) with the ellipse $m_x^2+k^2m_y^2=k^2$ delimiting the high energy region. The above technique can be applied to whatever external field applied in the $x,y$ plane. When the field is applied along one axis, as in our case, it is possible to carry out the integration of conservative LLG equation ($\alpha=0$) where $m_x,m_y$ are given in the parametric form (3.38):

$$\frac{du}{\sqrt{1-p^2 \cos^2 u - [a_y-(p/k)\sin u]^2}}=k(D_z-D_x)dt ,$$ (3.40)

which gives the relation between the parametric variable $u$ and time. In fact it is possible [42,50] to bring Eq. (3.40) in the canonical form which can be integrated by means of Jacobi elliptic^3.3 functions [72]. In this way one can obtain the magnetization dependance on time in exact analytical form and exact expression of the critical instants (3.31)-(3.33). It is also shown in [42] that, in the case of $\textbf{h}_$a$=h_a\mathbf{e}_y$, the critical value of the external applied field for precessional switching is still one half of the Stoner-Wohlfarth field:

$$h_c=\frac{D_y-D_x}{2}=\frac{h_\text{SW}}{2} \quad.$$ (3.41)