9.3.6 Phase Diagram

A summary of the results of the equilibrium magnetization distribution of nanodots with different aspect ratios is given in the phase diagram in Fig. 9.25. The transition from the in-plane magnetization to the vortex state is sharp, because this requires that the symmetry of the single domain state with (almost) homogeneous in-plane magnetization breaks in order to form the vortex state with cylindrical symmetry. The line separating the in-plane and out-of-plane remanent states has a slope of 1.8, which is in agreement with the simulations by Ross et al. [105] and analytical calculations [121]. Magneto-optical measurements of hysteresis curves [110] also show a distinct change between these two regimes. The single domain particles retain high remanence and switch at very low fields, whereas a sudden loss in magnetization reducing the external field (cf. Fig. 9.9) is typical of a flux closure configuration (vortex state, cf. Fig. 9.26).

However, the transition from the vortex to the perpendicular magnetization (parallel to the cylinder axis) is not well defined. The numerical experiments show a smooth transition from one state to the other. For decreasing the dot aspect ratio, the magnetization starts to twist and exhibits very inhomogeneous magnetization distributions (Fig. 9.27). So we have defined a magnetization distribution with $M_z > 0.75$ as being a perpendicular ground state. The two-dimensional analytical model cannot describe this transition properly, because it would require, that the dependence of the magnetization on the $z$-coordinate is taken into account.

Nevertheless, the numerical results and the experimental data are in excellent agreement with analytical calculations of this phase diagram. The solid lines in Fig. 9.25 have been taken from the phase diagram presented in [122].

Figure 9.25: Phase diagram of magnetic ground states (axis scaling in units of the exchange length). The data points indicated by the open symbols have been calculated with the FE model. The circles ($\circ$) represent dots with lowest energy in the in-plane magnetization state, squares ($\Box$) those with perpendicular magnetization, and diamonds ($\Diamond$) dots in vortex/multidomain state. The experimental data have been taken from Ross et al. [105] The crosses ($\times$) indicate ``Ni Type A'' samples with out-of-plane (perpendicular) magnetization at remanence, the plus symbols ($+$) indicate ``Ni Type B'' samples with in-plane, and the asterisks ($\ast$)``Ni Type C'' samples with vortex or multidomain states, respectively. The experimental data nicely fit in the phase diagram with one exception, which is indicated by ``(1)''. There, a remanent state with in-plane magnetization has been found, where a vortex state might be expected. The solid lines give the analytical equilibrium single-domain radius calculated by Metlov et al. [122]

Figure 9.26: Equilibrium magnetization distributions in zero field inside the nanodots (cut along the cylinder axis) for dots with aspect ratios $L/R=1$.

[$L/R=1$, $R=10 \mathrm{nm}$]

[$L/R=1$, $R=25 \mathrm{nm}$]

[$L/R=1$, $R=28 \mathrm{nm}$]

Figure 9.27: Magnetization distributions for dots with aspect ratios $L/R=2$.

[$L/R=2$, $R=10 \mathrm{nm}$]

[$L/R=2$, $R=25 \mathrm{nm}$]

[$L/R=2$, $R=40 \mathrm{nm}$]

Experimental data have been obtained from arrays of soft magnetic cylindrical particles by Ross et al. [105] The data of their Ni samples are also shown in Fig. 9.25. The agreement with the numerically calculated phase diagram is very good. Only one data point does not fit in. A remanent state with in-plane magnetization is found, where a vortex state might be expected. However, also the smooth transition from the perpendicular magnetization to the vortex (multidomain) state has been found. Note, that ignoring the existence of the vortex core in nanodots [123] leads to an over-estimation of the total energy and, as a result, to wrong coordinates of the lines separating different magnetic states in soft magnetic cylindrical nanodots.