Stability of Neel skyrmions in ultra-thin nanodots considering Dzyaloshinskii-Moriya and dipolar interactions

An analytical expression for the energy of N'eel skyrmions in ultra-thin nanodots considering exchange, uniaxial anisotropy, Dzyaloshinskii-Moriya, and dipolar contributions has been obtained. In particular, we have proposed for the N'eel skyrmion, a general ansatz for the component of the magnetization perpendicular to the dot, given by $m_z(r)=[1-(r\mathrm{/}{R}_s{)}^n]\mathrm{/}[1+(r\mathrm{/}{R}_s{)}^n]$, where $R_s$ is the radius of the skyrmion and $n$ is an integer and even number. As proof of concept, we calculate the energy of a N'eel skyrmion in an ultra-thin Co/Pt dot, and we find that the dipolar contribution cannot be neglected and that both Dzyaloshinskii-Moriya interaction and anisotropy play an important role to stabilize the skyrmion. Additionally, we have obtained a good agreement between our analytical calculations and previously published micromagnetic simulations for $n=10$. For this reliable value of $n$, we have obtained that for a Dzyaloshinski Moriya constant $D=5.5(mJ\mathrm{/}{m}^2)$, it is possible to stabilize a N'eel skyrmion for $K_u$ in the range, $0.4 , (MJ/m^3)< K_u <1.3 , (MJ/m^3)$,whereasfor$K_u = 0.8 , (MJ/m^3)$, the skyrmion stabilizes for $5.0(mJ\mathrm{/}{m}^2)<D<6.0(mJ\mathrm{/}{m}^2)$. Thus, this analytical equation can be widely used to predict stability ranges for the N'eel skyrmion in spintronic devices.