1.2 The Thermal Model
1.2.2 Magnetization Dynamics, Approaching T C
The deficiency of the Landau-Lifshitz-Gilbert equation in the regions close to TC has been investigated in [16]. It is shown there, how the LLB is advantageous in describing magnetization dynamics around TC. The approach is similar to that of Garanin in [25, 26]. The starting point is an atomistic simulation based on the Landau-Lifshitz equation, in the same manner as in equation 1.4. The Langevin fields are included in the total field, as well as the exchange interaction and Zeeman
contributions. The inclusion of the Zeeman field ensures a non-vanishing magneti-zation even aboveTC.
Two test cases were simulated. In the first case, all spins pinned by an external field were simultaneously rotated by 30◦ out of their equilibrium direction, for a temperature range from below up to above TC. In that case the behavior of trans-verse relaxation τ⊥ was investigated. The result is an exponentially decaying sine oscillation with the transverse decay time τ⊥. By fitting the decaying oscillation function, the result is that for increasing temperatures belowTC the relaxation also increases, i.e. the relaxation time τ⊥ decreases. Above TC on the other hand, the relaxation begins to decrease with increasing temperature, which means that the relaxation timeτ⊥ increases. In the second case, the spins were rotated by an angle of 135◦ out of equilibrium, which ensured that the projection of the external field onto the magnetization changed sign to negative, and was opposed to the exchange field. This opposite direction of both fields is responsible for the decrease of the absolute magnetization value m= |m| in the course of relaxation. In the data the decrease of m shows up in a dip in the course of relaxation. The magnitude of the dip is ∆m∼=m(H, T)−m(0, T), whereH is the absolute value of the external mag-netic field. That means that the absolute value of m decreases by the difference of the equilibrium magnetization atH = 0 and the magnetization at the actual given field. The second feature studied within this simulation set was the longitudinal re-laxation timeτk, which increases slowly with T. Around T ≤TC,τk increases faster, until for temperaturesT ≥TC it starts to decrease again. This quick increase of the longitudinal relaxation time τk in the vicinity of TC was recognized as the critical slowing down and ascribed to a second order transition.
In a next step, the Landau-Lifshitz-Bloch equation was solved, in order to test, whether it exhibits the same physical features, as the atomistic model simulations.
The parameters applied in the calculation, the Curie temperatureTC, and the zero-field equilibrium magnetization me were taken from the mean-field-approximation (MFA). Wherem =B∞(β(mJ0−µ0H)) is taken from the Curie-Weiss law withB∞ being the Langevin function in the approximation B∞(x) = 1/3x−1/45x3..., and β = 1/T. The other parameters are as described in section 1.2.1. For magnetization dynamics calculations of real magnetic materials, these parameters can be taken as input data obtained from experiments. The results are shown in figure 1.3. For the calculations, the LLB in the form (see equation 1.13)
m˙ =−γ[m×H] +γαk
[m·Heff]m
m2 −γα⊥
[m×[m×Heff]]
m2 (1.14)
was applied. Both dimensionless longitudinal αk and transverse α⊥ damping pa-rameters are similar to the relaxation coefficients in equation 1.13 for T < TC:
αk =λ 2T
For T > TC the transverse damping parameter α⊥ → αk. The coupling parameter λ which connects both, the longitudinal and the transverse damping to the bath
1 Ultrafast Spin Dynamics
Figure 1.3: The longitudinal τk and transverse τ⊥ relaxation times versus the temper-ature. Calculated from the atomistic model simulations and by solving the Landau-Lifshitz-Bloch equation for different Temperatures far below, up to aboveTC. Taken from [16].
has the same magnitude, as was used in the atomistic model simulation. In fact, λ itself is also temperature dependent, but it is difficult to include this dependence into a semi-phenomenological approach, and besides that, leaving λ temperature independent, makes the comparison with the atomistic model possible. The much weaker than the exchange interaction effective fieldHeff contains the magnetization m, which differs for the temperature range, as shown in the derivation outlined in section 1.2.1, the anisotropy field HA = −(mxex +myey)/χ˜⊥, and the applied
The second case, for temperatures around and above TC in the third term of Heff can be rewritten, for the longitudinal susceptibility, in −χ˜1
k
1− 5(T3T−TC
C)
m.
A closer look for comparison of the LLB to the atomistic model shows the equiv-alence of both models, and hence the validity for the LLB even for elevated tem-peratures. Firstly, from equation 1.15 one can see that the longitudinal damping parameter monotonically increases with temperature, taking λ as a constant, even linearly, while the transverse damping parameter decreases for temperatures below TC monotonically and increases again for temperatures above TC. However, the change is very small over the temperature range up to TC. Therefore the relaxation rates need to be evaluated, in order to compare the relaxation times of the LLB to the ones obtained form the atomistic simulations. Those are defined as follows:
Γk = γαk
˜
χk(H, T), Γ⊥ = γα⊥
˜
χ⊥(H, T).
In order to gain an insight into the change of the relaxation rates and the inversely proportional relaxation times, a closer look at the susceptibilities and their behav-ior over the temperature range is necessary. From the definition of ˜χk in equation 1.12 it can be seen that Γk ∼ J0. Since the exchange interaction is very strong in a ferromagnet, the relaxation rate is high and therefore the relaxation time τk is very low, as is also the case within the atomistic model. Around TC however, the longitudinal susceptibility increases, which decreases the relaxation rate Γk, and increases τk. This critical slowing down is rather a feature of the temperature de-pendence of χk, than the variation of αk. In the isotropic model, the transverse susceptibility is defined by ˜χ⊥ = m(H, T)/H. Therefore, the transverse relaxation rate Γ⊥ ∼ H, is much weaker than the longitudinal relaxation rate. Additionally, Γ⊥∼1/m(H, T), and sincemdecreases with increasing temperature, also the relax-ation rate increases. For the relaxrelax-ation times τk,⊥ that means, at low temperatures, τk τ⊥, but τk increases, and τ⊥ decreases with increasing temperature. At high temperatures aroundTC both relaxation rates approach each other very closely and merge for temperatures above TC, as is illustrated in figure 1.3.
The vanishing of χk at low temperatures is also the reason for the convergence of m towards me, the equilibrium magnetization. Then the first relaxation term in equation 1.13 and equation 1.14 vanishes, and both equations merge into the Landau-Lifshitz-Gilbert equation.
1 Ultrafast Spin Dynamics