Quantum Landau-Lifshitz-Bloch equation and its comparison with the classical case

The detailed derivation of the quantum Landau-Lifshitz-Bloch (qLLB) equation for simple spin-flip scattering mechanisms based on spin-phonon and spin-electron interactions is presented and the approximations are discussed. The qLLB equation is written in a form suitable for comparison with its classical counterpart. The temperature dependence of the macroscopic relaxation rates is discussed for both mechanisms. It is demonstrated that the magnetization dynamics is slower in the quantum case than in the classical one.

Figure 1

Schematic illustrating the relation between the reduced magnetization $\mathbf{m}$ and the vectors ${\mathbf{H}}_E$, $\mathbf{y}$, $\mathbf{h}$, ${\mathbf{H}}^{\mathrm{MFA}}$, and ${\mathbf{y}}_0$ at instant $t$ in a nonequilibrium state.

Figure 2

Longitudinal relaxation time [Eq. (43)] versus temperature using constant $\lambda =0.02$, $T_c=650$ K, and ${\mu }_{\mathrm{at}}=0.5{\mu }_B$ in the three spin cases with $S=1/2$, $S=9/2$, and $S=\infty$, respectively. The case $S=\infty$ was done by taking the limit $S\to \infty$ in Eq. (43), which is equivalent to the classical LLB equation.

Figure 3

In-field longitudinal relaxation time calculated via direct integration of the qLLB equation with a small deviation from the equilibrium. The following parameters were used: $T_c=650$ K, ${\mu }_{\mathrm{at}}=0.5{\mu }_B$, $\lambda =0.02$, and zero anisotropy constant.

Figure 4

LLG damping [Eq. (45)] versus temperature using $K_1=K_2$, constant $\lambda =0.02$, $T_c=650$ K, and ${\mu }_{\mathrm{at}}=0.5{\mu }_B$ for the three spin cases $S=1/2$, $S=9/2$, and $S=\infty$, respectively. The case $S=\infty$ was done by taking the limit $S\to \infty$ in Eq. (45), which is equivalent to the classical case.

Figure 5

Dynamics of the $m_z$ component for the longitudinal plus transverse dynamics at 630, 645, and 654 K for $S=1/2,\infty$, where the initial magnetization is $\mathbf{m}=(0.05,0,0.2)$ and the applied field is $H_z=-1$ T.

Figure 6

Dynamics of the $m_z$ component at $T=649$ K without and with an external field $H_z=-1\mathrm{T}$ for $S=1/2,\infty$, where the initial magnetization is $\mathbf{m}=(0.05,0,0.2)$.

Figure 7

Reversal time versus temperature for $S=1/2,\infty$ and $S=\infty$. In the pure longitudinal dynamics the initial magnetization is set to $\mathbf{m}=(0,0,0.2)$, and in the longitudinal plus transverse dynamics the initial magnetization is set to $\mathbf{m}=(0.05,0,0.2)$.