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

An analytical expression for the energy of Neel skyrmions in ultra-thin nanodots considering exchange, uniaxial anisotropy, Dzyaloshinskii-Moriya, and dipolar contributions has been obtained. In particular, we have proposed for the Neel skyrmion, a general ansatz for the magnetization component perpendicular to the dot, given by m_z(r)=[1-(r/R_s)ⁿ]/[1+(r/R_s)ⁿ], where R_s is the radius of the skyrmion and n is an even integer number. As proof of concept, we calculate the energy of a Neel skyrmion in an ultra-thin Co/Pt cylinder, 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/m²), it is possible to stabilize a Neel skyrmion for K_u in the range 0.4 (MJ/m³) < K_u < 1.3 (MJ/m³), whereas for K_u=0.8(MJ/m³), the skyrmion stabilizes for 5.0 (mJ/m²) < D < 6.0 (mJ/m²). Thus, this analytical equation can be widely used to predict stability ranges for the Neel skyrmion in spintronic devices.