Anisotropic magnetic molecular dynamics of cobalt nanowires

An investigation of thermally induced spin and lattice dynamics of a cobalt nanowire on a (111)Pt substrate is presented via magnetic molecular dynamics. This dynamical simulation model treats each atom as a particle supporting a classical spin. A coordinate dependent on both exchange and anisotropic functions ensures a minimal coupling between the spin and the lattice degrees of freedom to translate the magnetostrictive behavior of most magnetic materials. A spin-pair model of anisotropy is proposed to connect to the lattice thermodynamics. In order to solve linked spin-coordinate equations of motion, the efficiencies of algorithms based on Suzuki-Trotter decompositions are compared. The temperature dependence of the magnetic behavior of Co nanowires is investigated through thermal stochastic connections with mechanical and spin Langevin noises. From a magnetic Hamiltonian parametrized on ab initio calculations, the size dependence of the energy barriers and characteristic time scales of the magnetization relaxation are computed. In the superparamagnetic limit, it is shown that all spins in a nanowire evolve in a coherent rotation. When the size of the single nanowire increases, nucleations of domain walls let the activation energy be independent of the length of the wire.

Figure 1

The lattice, electron, and spin thermodynamic reservoirs with their corresponding temperatures ($T_L$, $T_E$, and $T_S$). Arrows show the exchange of energy between each reservoir and their typical relaxation times.

Figure 2

Exchange energy as a function of interatomic distance for cobalt. The solid (black) line shows the $J\left(r\right)$ results from a fit on values of exchange integrals [filled (red) circles] obtained from ab initio calculations on hcp[5] and fcc Co[40] structures. The dashed (blue) line shows the $J\left(r\right)$ results obtained from a rescaling of the previous function such that the rescaled function fits the value of the first-nearest-neighbor exchange interaction $J_{1\mathrm{NN}}=50$ meV estimated for Co clusters deposited on various nonmagnetic substrates.[39] Considered parameters are for a bulk system, with $\epsilon =44.6928$ meV, $\delta =3.496\times {10}^{-3}$, and $\sigma =1.4885$ Å, and for nano-objects (in particular, a one-dimensional chain), with $\epsilon =106$ meV and $\delta$ and $\sigma$ unchanged. Inset: The influence of the smooth cutoff function, showing the region between $R_c=3.75$ Å and 4 Å.

Figure 3

Total energy of a system of 1342 atoms of cobalt in the fcc configuration. (a) A pure ferromagnet with a mechanical perturbation is applied at $t=0$, corresponding to $T=1$ K. (b) Without the initial kinetic velocity, one spin is randomly moved at $t=0$. Inset (a): A zoom-in on the variations of energy. Inset (b): Density of states between all algorithms, revealing the spreading of the PRSRP method.

Figure 4

MMD simulation of an hcp Co illustrating the coupling between the lattice and the magnetic subsystems.

Figure 5

Variation of the MCAE of an infinite cobalt wire as a function of the interatomic distance (circles) obtained from ab initio calculations. Positive values of the MCAE correspond to an easy axis parallel to the wire. An anisotropy function $K\left(r\right)$ of analytical form from Eq. (2) is fitted to the ab initio data and is represented in by the solid (red) line. The parameters of this anisotropy function are $\epsilon =-0.00330282$ eV, $\delta =0.864159$, and $\sigma =2.13731$ Å. Inset: One MCAE angular variation for $d=2.5$ Å and its ${\sin }^2\left(\theta \right)$ fit.

Figure 6

Energy decomposition of a 20-atom linear chain separated by $d=2.40$ Å and periodic boundary conditions. Spin directions are random for $t=0$ and $\alpha =0$. Initial impulses are taken from a distribution with a temperature of 200 K. (a) Kinetic and magnetic energies; (b) total and mechanical energies

Figure 7

Magnetization behavior of a 20-atom cobalt chain: (a) $x$ component and (b) $z$ component of the average magnetization for four lattice spacings. Simulations were performed with exchange and anisotropy interactions with the same parameters as presented in Figs. 2 and 5, including $\alpha =0.1$ and $T=0$.

Figure 8

Snapshot of a monatomic chain of eight atoms of Co deposited on a (111)Pt substrate.

Figure 9

(a) Time fluctuations of both $z$ components (with respect to the norm) of the average magnetization for a chain of $L=8$ atoms at 75 K [orange, with respect to green]; the dashed (black) line is from the so-called two-state random telegraph noise analysis.[73] (b) Logarithm of the frequency of reversal versus the temperature, plotted for varying lengths. Solid lines correspond to the linear fit. (c) Probability of nonswitching versus waiting time, plotted for five temperatures and for a chain of eight Co atoms. Lines correspond to the exponential decay fit.

Figure 10

Activation energy versus number of atoms from MMD calculations (circles) and from the Stoner-Wolhfrath model, where the chain magnetization is approximated by a single macrospin of rigid magnitude [dashed (black) line]. The horizontal (orange) line is the minimal energy for the formation of a domain wall given by Ref. 7 ($E=2\sqrt{2JK}$).

Figure 11

Transversal and longitudinal components of the magnetization for a chain of 54 Co atoms after a nucleation time corresponding to the zero component of the average magnetization along the easy axis.

Figure 12

Histograms of the switching time of a 54-atom chain during a simulation of 3.2 ns at two temperatures.