Gilbert damping in conducting ferromagnets. II. Model tests of the torque-correlation formula

We report on a study of Gilbert damping due to particle-hole pair excitations in conducting ferromagnets. We focus on a toy two-band model and on a four-band spherical model which provide an approximate description of ferromagnetic (Ga,Mn)As. These models are sufficiently simple that disorder-ladder-sum vertex corrections to the long-wavelength spin-spin response function can be summed to all orders. An important objective of this study is to assess the reliability of practical approximate expressions which can be combined with electronic structure calculations to estimate Gilbert damping in more complex systems.

Figure 1

Dyson equation for the renormalized vertex of the transverse spin-spin response function. The dotted line denotes impurity scattering.

Figure 2

(Color online) M2DEG: Gilbert damping in the absence of spin-orbit coupling. When the intrinsic spin-orbit interaction is small, the first vertex correction is sufficient for the evaluation of Gilbert damping provided that the ferromagnet exchange splitting is large compared to the lifetime broadening of the quasiparticle energies. For more disordered ferromagnets ($E_F{\tau }_0<5$ in this figure) higher-order vertex corrections begin to matter. In either case vertex corrections are significant. In this figure $1/{\tau }_0$ stands for the scattering rate off spin-independent impurities, defined as a two-band average at the Fermi energy, and the spin-dependent and spin-independent impurity strengths are chosen to satisfy $u^0=3u^z$.

Figure 3

(Color online) M2DEG: Gilbert damping for strong SO interactions $\left(\lambda k_F=1.2E_F\simeq 4{\Delta }_0\right)$. In this case higher-order vertex corrections matter (up to 20%) even at low disorder. This suggests that higher-order vertex corrections will be important in real ferromagnetic semiconductors because their intrinsic SO interactions are generally stronger than their exchange splittings.

Figure 4

(Color online) M2DEG: Gilbert damping for moderate SO interactions $\left(\lambda k_F=0.2{\Delta }_0\right)$. In this case there is a crossover between the intraband dominated and the interband dominated regimes, which gives rise to a nonmonotonic dependence of Gilbert damping on disorder strength. The stronger the intrinsic SO relative to the exchange field is, the higher is the value of disorder at which the crossover occurs. This is why the damping is monotonically increasing with disorder in Fig. 2 and monotonically decreasing in Fig. 3.

Figure 5

(Color online) GaMnAs: higher-order vertex corrections make a significant contribution to Gilbert damping due to the prominent spin-orbit interaction in the band structure of GaAs. $x$ is the Mn fraction, and $p$ is the hole concentration that determines the Fermi energy $E_F$. In this figure, the spin-independent impurity strength $u^0$ was taken to be three times larger than the magnetic impurity strength $u^z$. $1/{\tau }_0$ corresponds to the scattering rate off Coulomb impurities and is evaluated as a four-band average at the Fermi energy.

Figure 6

(Color online) GaMnAs: when the spin-orbit splitting is reduced (in this case by reducing the hole density to $0.2{\text{nm}}^{-3}$ and artificially taking ${\gamma }_3=0.5$), the crossover between interband and intraband dominated regimes produces a nonmonotonic shape of the Gilbert damping, much like in Fig. 4. When either ${\gamma }_2$ or $p$ is made larger or $x$ is reduced, we recover the monotonic decay of Fig. 5.

Figure 7

(Color online) M2DEG: comparison of Gilbert damping predicted using spin-flip and torque matrix element formulas, as well as the exact vertex-corrected result. In this figure the intrinsic spin-orbit interaction is relatively weak $\left(\lambda k_F=0.05E_F\simeq 0.06{\Delta }_0\right)$ and we have taken $u^z=0$. The torque-correlation formula does not distinguish between spin-dependent and spin-independent disorders.

Figure 8

(Color online) M2DEG: comparison of Gilbert damping predicted using spin-flip and torque matrix element formulas, as well as the exact vertex-corrected result. In this figure the intrinsic spin-orbit interaction is relatively strong $\left(\lambda k_F=0.5E_F=5{\Delta }_0\right)$ and we have taken $u^z=0$.

Figure 9

(Color online) GaMnAs: comparison of Gilbert damping predicted using spin-flip and torque matrix element formulas, as well as the exact vertex-corrected result. $p$ is the hole concentration that determines the Fermi energy $E_F$ and $x$ is the Mn fraction. Due to the strong intrinsic SO, this figure shows similar features as those of Fig. 8.

Figure 10

(Color online) GaMnAs: comparison of Gilbert damping predicted using spin-flip and torque matrix element formulas, as well as the exact vertex-corrected result. In relation to Fig. 9 the effective spin-orbit interaction is stronger due to a larger $p$ and a smaller $x$.