Harmonic transition-state theory of thermal spin transitions

A rate theory for thermally activated transitions in spin systems is presented. It is based on a transition-state approximation derived from Landau-Lifshitz equations of motion and quadratic expansion of the energy surface at minima and first order saddle points. While the flux out of the initial state vanishes at first order saddle points, the integrated flux over the hyperplanar transition state is nonzero and gives a rate estimate in good agreement with direct dynamical simulations of test systems over a range in damping constant. The preexponential factor obtained for transitions in model systems representing nanoclusters with 3 to 139 transition metal adatoms is on the order of ${10}^{11}$ to ${10}^{13}{\mathrm{s}}^{-1}$, similar to that of atomic rearrangements.

Figure 1

Comparison of the rate of transitions in a spin trimer obtained directly from dynamics given by the Landau-Lifschitz-Gilbert equation of motion as a function of the damping constant $\alpha$ at $T=23$ K (solid line) and a harmonic TST estimate (dotted line). Inset: the energy surface near a first order saddle point, representation of a hyperplanar transition state dividing surface (thick line) and the spin velocity (arrows).

Figure 2

Minimum energy path (solid line) for a magnetic transition in a rectangular shaped 139 Fe atom island on W(110) surface. A relaxation starting from a straight line interpolation (dotted line) representing a uniform rotation of the spins between spin up and spin down states revealed an intermediate metastable state, as shown by the insets. The discretization points used in the NEB calculation are shown with filled circles but the minimum for the metastable state and a saddle point obtained by subsequent optimization are marked with X. Insets: noncollinear spin configurations at various points along the path.

Figure 3

Minimum energy path for a transition between a parallel (P) and antiparallel (AP) state of a Fe trimer on a metal substrate described with the Alexander-Anderson model. Spin configurations corresponding to locations marked with X on the path are shown with arrows denoting magnitude and direction of the magnetic moments. The direction of spin 1 is taken to be fixed but the relative angles, ${\theta }_2$ and ${\theta }_3$, are variable. The energy is given in units of the $d$-level width, $\Gamma$, due to $s$-$d$ hybridization. Inset: energy surface showing minima corresponding to P and AP states, and the calculated minimum energy path for the transition. Saddle point (SP) is also shown.