Off-resonant all-optical switching dynamics in a ferromagnetic model system

We present a theoretical study of the effects of optical fields on a ferromagnetic model system of itinerant carriers that includes both magnetism at the level of a time-dependent mean-field and spin-orbit coupling. In the framework of this model, which contains a band gap and is similar to the one introduced by Qaiumzadeh et al. [Phys. Rev. B 88, 064416 (2013)] we investigate the inverse Faraday effect, i.e., how the magnetization of the ferromagnetic bands can be influenced by the helicity of off-resonant optical fields. The magnetization dynamics are calculated self-consistently by including a dynamic ferromagnetic splitting and the corresponding single-particle states. We study the magnetization dynamics for the case of a rapid ramping-up of an optical field that is off-resonant with respect to the gap and find that the magnetization switching occurs due to the interplay of the coherence between the spin-split ferromagnetic bands and incoherent scattering/dephasing processes.

Figure 1

Schematic structure of the four-band system with two magnetically split conduction bands separated from two filled valence bands by a gap. Dipole transitions between the valence and conduction bands are shown for a few $k$ points schematically by vertical lines. The coherence ${\rho }^{+-}$ between the magnetic bands is also shown. Note that we do not show the $k$ dependence of the valence bands and plot them side by side instead of on top of each other. The different dipole coupling matrix elements are indicated by the thickness of the arrows and the angular momentum transferred to the carriers by a left (right) circularly polarized photon is indicated by $-\hslash (+\hslash$). Both $|+\rangle$ and $|-\rangle$ states are connected to each of the $p$-like states by dipole transitions with both helicities.

Figure 2

Schematic structure of the four-band system without Rashba SOC: two magnetically split conduction bands are separated from two filled valence bands by a gap. Dipole transitions between the valence and conduction bands are shown for a few $k$ points schematically by vertical lines. As there are no coherences ${\rho }^{+,-}$, at each $k$ point, we have effectively uncoupled two-level systems.

Figure 3

(a) Time dependent magnetization (spin polarization) of the ferromagnetic model system with parabolic bands (no spin-orbit coupling in magnetic bands, no quantum coherence) excited by an optical field with right-circular (dashed line) and left-circular (solid line) polarization. The intensity of the driving field is shown, in arbitrary units, as thin grey dashed line. (b) Corresponding energy density.

Figure 4

(a) Time-dependent magnetization (spin polarization) of the ferromagnetic model system excited by a optical field with right-circular (dashed line) and left-circular (solid line) polarization. Magnetization switching occurs only for left-circularly-polarized excitation. (b) corresponding energy density. (c) corresponding density.

Figure 5

(a) Time-dependent magnetization (spin polarization) of the ferromagnetic model system excited by a optical field with right-circular (dashed line) and left-circular (solid line) helicity. Magnetization switching occurs only for left-circularly-polarized excitation. (b) without quantum coherence but with scattering, (c) with quantum coherence, no scattering, (d) no quantum coherence, no scattering.

Figure 6

(a) Spin polarization dynamics for a Rabi energy $\hslash \mathrm{\Omega }=5$ (blue solid line), 10 (red dashed line), 15 (light blue dotted line), and $20\mathrm{meV}$ (green dash-dotted line) for a circularly left polarized optical field. (b) Corresponding energy density. (c) Corresponding particle density.

Figure 7

Spin polarization dynamics a detuning $\delta =50$, 100, 200, 400, and $800\mathrm{meV}$ for a circularly left polarized optical field.

Figure 8

Spin polarization dynamics for $\alpha =0$ (black dashed line), 5 (blue solid line), 10 (light blue dashed line), 10 (green dotted line), and $25\mathrm{meV}\mathrm{nm}$ (red dash-dotted line) for a circularly left polarized optical field.

Figure 9

Spin polarization dynamics for a Stoner parameter $U=50.8$, 51.8, 53.5, 56.8, and $64.0\mathrm{meV}$ for a circularly left polarized optical field.