Magnetization dynamics and damping due to electron-phonon scattering in a ferrimagnetic exchange model

We present a microscopic calculation of magnetization damping for a magnetic “toy model.” The magnetic system consists of itinerant carriers coupled antiferromagnetically to a dispersionless band of localized spins, and the magnetization damping is due to coupling of the itinerant carriers to a phonon bath in the presence of spin-orbit coupling. Using a mean-field approximation for the kinetic exchange model and assuming the spin-orbit coupling to be of the Rashba form, we derive Boltzmann scattering integrals for the distributions and spin conherences in the case of an antiferromagnetic exchange splitting, including a careful analysis of the connection between lifetime broadening and the magnetic gap. For incoherent scattering of itinerant carriers with the phonon bath, i.e., the Elliott-Yafet mechanism, we extract dephasing and magnetization times $T_1$ and $T_2$ from initial conditions corresponding to a tilt of the magnetization vector and draw a comparison to phenomenological equations such as the Landau-Lifshitz (LL) or the Gilbert damping. We also analyze magnetization precession and damping for this system including an anisotropy field and find a carrier mediated dephasing of the localized spin via the mean-field coupling.

Figure 1

Illustration of nonequilibrium spin dynamics in presence of a magnetic field without relaxation (a) and within relaxation (b).

Figure 2

Dynamics of $\delta M_{\perp }$ and $\delta M_{\parallel }$ computed using the Landau-Lifshitz damping $({\omega }_{\mathrm{L}}=1{\mathrm{ps}}^{-1},H={10}^6\frac{\mathrm{A}}{\mathrm{m}}\approx 1.26\times {10}^4\mathrm{Oe}$, and $\lambda ={10}^{-7}\frac{\mathrm{m}}{\mathrm{A}\mathrm{ps}})$. (a) An angle of ${10}^{\circ }$ leads to an exponential decay with well defined $T_1$ and $T_2$ times. (b) For an angle of ${90}^{\circ }$, the decay (solid line) is not exponential as a comparison with the exponential fit (dashed line) clearly shows.

Figure 3

Sketch of the band structure with localized (flat dispersions) and itinerant (parabolic dispersions) electrons. Above the Curie temperature $T_{\mathrm{C}}$, the spin eigenstates are degenerate (a), whereas below $T_{\mathrm{C}}$ a gap between the spin states exists.

Figure 4

(a) Localized spin $\langle \vec{S}\rangle$ and itinerant spin $\langle \vec{s}\rangle$ in thermal equilibrium. (b) Itinerant spin tilted relatively to the localized spin by an angle $\beta$.

Figure 5

Initial spin density-matrix (occupations and coherences) for itinerant electrons corresponding to a relative tilt $\beta ={50}^{\circ }$. The deviation from equilibrium occurs only between the Fermi wave vectors of the “$+$” and “$-$” bands.

Figure 6

Nonequilibrium dynamics of localized (red) and itinerant (blue) spins including electron-phonon scattering without (a) and with (b) spin mixing.

Figure 7

Relaxation dynamics of itinerant spins in the fixed frame.

Figure 8

Relaxation dynamics of itinerant spins in the rotating frame. An exponential fit determines the Bloch decay times to $T_1=0.5\mathrm{ps}$ and $T_2=1.0\mathrm{ps}$.

Figure 9

(Top) Longitudinal (black squares) and transverse (blue circles) Bloch decay times vs electron-phonon coupling strength $D$ together with two fit curves $\propto$$1/D^2$ (solid lines). To a good approximation $T_{1,2}\propto 1/D^2$. (Bottom) Ratio $T_2/T_1$ of both decay times vs coupling strength.

Figure 10

Precession frequency with respect to the damping strength in terms of the decay rate $1/T_2$.

Figure 11

Dynamics of the longitudinal and transverse itinerant spin components in the rotating frame (solid lines) for a tilt angle of $\beta ={140}^{\circ }$, together with exponential fits toward equilibrium (dashed lines). The longitudinal equilibrium polarization is shown as a dotted line. For $s_{\perp }$, the fit cannot be distinguished from dynamics on this scale.

Figure 12

$T_2$ time extracted from an exponential fit to $s_{\perp }$ dynamics in rotating frame for different initial tilting angles $\beta$.

Figure 13

Dynamics of the modulus $|\vec{s}|$ of the itinerant spin for different initial tilt angles $\beta$. Note the slightly different time scale compared to Fig. 11.

Figure 14

Dynamics of the localized spin $\vec{S}$ and itinerant spin $\vec{s}$. At $t=0$, the equilibrium configuration of both spins is tilted $\left(\beta ={10}^{\circ }\right)$ with respect to an anisotropy field ${\vec{H}}_{\mathrm{aniso}}$. The anisotropy field is only experienced by the itinerant subsystem.

Figure 15

Relaxation dynamics of the localized spin toward the anisotropy direction for longitudinal component $S_z$ and transverse component $\sqrt{S_x^2+S_y^2}$. An exponential fit yields Bloch decay times of $T_1^{\mathrm{aniso}}=67.8\mathrm{ps}$ and $T_2^{\mathrm{aniso}}=134.0\mathrm{ps}$.

Figure 16

Larmor frequency ${\omega }_{\mathrm{L}}^{\mathrm{aniso}}$ and Bloch decay time $T_2^{\mathrm{aniso}}$ extracted from the spin dynamics vs anisotropy field $H_{\mathrm{aniso}}$, as well as the corresponding damping parameter ${\alpha }_{\mathrm{aniso}}$.

Figure 17

Damping parameter ${\alpha }_{\mathrm{aniso}}$ vs coupling constant $D$ (black diamonds). The red line is a quadratic fit, indicative of ${\alpha }_{\mathrm{aniso}}\propto D^2$.

Figure 18

Precession frequency of the localized spin around the anisotropy field vs Bloch decay time $1/T_2^{\text{aniso}}$.