Temperature and Thickness Dependence of Statistical Fluctuations of the Gilbert Damping in $\mathrm{Co}$-$\mathrm{Fe}$-$\mathrm{B}$/$\mathrm{Mg}\mathrm{O}$ Bilayers

We theoretically investigate the temperature and thickness dependence of the effective Gilbert damping constant $(\alpha )$ in the $\mathrm{Co}$-$\mathrm{Fe}$-$\mathrm{B}$/$\mathrm{Mg}\mathrm{O}$ system using atomistic spin dynamics. We consider a high damping constant at the interface layer and a low damping constant for the bulklike layers due to large interfacial spin-orbit coupling. We find a strong dependence of the effective Gilbert damping with the film thickness, in quantitative agreement with experimental data. The temperature dependence of the effective damping arising from thermal-spin fluctuations up to temperatures of 400 K is weak, with no apparent change over the studied temperature range. Interestingly, we find that the temperature produces a different effect: a statistical fluctuation of the Gilbert damping parameter for a given relaxation induced solely from the finite size of the system. This statistical variation of the Gilbert damping is an intrinsic effect and is important for spintronic devices operating at gigahertz frequencies, where the dynamic response must be carefully controlled.

Figure 1

A schematic of the simulated system: the white and gold spheres represent the interface layer and bulklike $\mathrm{Co}$-$\mathrm{Fe}$-$\mathrm{B}$ layer, respectively.

Figure 2

The magnetization dynamics of a $\mathrm{Co}$-$\mathrm{Fe}$-$\mathrm{B}$ layer with various external fields obtained from the atomistic model (top) and the field dependence of the precession frequency of magnetization in the $\mathrm{Co}$-$\mathrm{Fe}$-$\mathrm{B}$ layer (bottom).

Figure 3

(a) The time evolution of the magnetization component of the $\mathrm{Co}$-$\mathrm{Fe}$-$\mathrm{B}$ system with a thickness of 2 nm at 300 K after a ${30}^{\circ }$ rotation of the equilibrated spin configuration. The points show the analytical magnetization components and the dashed lines represent the simulated magnetization. (b) A comparison of the effective Gilbert damping constant of the $\mathrm{Co}$-$\mathrm{Fe}$-$\mathrm{B}$ system with different thicknesses at 300 K, obtained from simulations and experiments in Ref. [1]: the line is a linear fit on $1/t_{\mathrm{Co}\mathrm{Fe}\mathrm{B}}$, ${\alpha }_{\mathrm{eff}}=0.003+0.0189677/t_{\mathrm{Co}\mathrm{Fe}\mathrm{B}}$. The simulated thickness dependence of the effective damping quantitatively agrees with the experimental data.

Figure 4

(a) The simulated temperature dependence of the mean effective Gilbert damping constant of the $\mathrm{Co}$-$\mathrm{Fe}$-$\mathrm{B}$/$\mathrm{Mg}\mathrm{O}$ structure for different thicknesses extracted from 20 stochastic simulation runs. The data show a weak temperature dependence and strong thickness dependence. (b) The temperature dependence of the standard deviation ($\sigma$) of the distribution of the simulated Gilbert damping constants as a function of the film thickness. The data show a strong finite size dependence on the distribution of the Gilbert damping parameter, with a 30% deviation for 2-nm-thick films at 400 K.

Figure 5

The simulated relaxation dynamics (points) for a 2-nm-thick $\mathrm{Co}$-$\mathrm{Fe}$-$\mathrm{B}$/$\mathrm{Mg}\mathrm{O}$ film at $T=400\mathrm{K}$. For different random thermal fluctuations, simulated through a different pseudorandom sequence, the same system displays (a) low and (b) high values of the effective Gilbert damping constant (fits shown by lines).