Figure 1

Feynman diagram that encodes the transverse spin density induced by a current (ferromagnetic ISGE effect) which results in a change in the steady-state magnetization direction. $a$ and $b$ are band labels for the quasiparticle and the quasihole.Reuse & Permissions

Figure 2

Cartoon of a magnetic thin film (shaded area). The exchange field $\mathbf{\Delta }$ is an effective magnetic field which is parallel to the magnetization only when it points along easy or hard crystalline directions. The orientation of $\mathbf{\Delta }$ can be specified by the polar and azimuthal angles $\theta$ and $\varphi$. The relationship between the direction of $\mathbf{\Delta }$ and the direction of magnetization is altered by an electric current in gyrotropic ferromagnets.Reuse & Permissions

Figure 3

Spin response to a transverse magnetic field ${\mathbf{B}}_{\perp }$ in the presence of a current: perturbation theory to all orders in ${\mathbf{B}}_{\perp }$. The quasiparticles (quasiholes) in these diagrams diagonalize a Hamiltonian whose exchange field is pointing along an easy direction and ${\mathbf{B}}_{\perp }$ is by definition perpendicular to this easy direction. Provided that in Eq. (10) we take the exact eigenstates of the mean-field Hamiltonian (within which the exchange field need not be pointing along an easy direction), all the diagrams of this figure are implicit in the diagram of Fig. 1. In particular, the ferromagnetic ISGE captures the influence of currents on ferromagnetic resonance.Reuse & Permissions

Figure 4

(Color online) Equilibrium anisotropy field (meV per spin) in (Ga,Mn)As for $\varphi =0$, and $\theta ∊\left(0,\pi \right)$. The parameters used for this calculation were: Mn fraction $x=0.08$, hole concentration $p\simeq 0.15{\text{nm}}^{-3}$, ${ϵ}_F\tau =3$, and axial strain ${ϵ}_{\text{ax}}=-0.5\%$. These anisotropy field results were evaluated using the model explained in the text.Reuse & Permissions

Figure 5

(Color online) Equilibrium anisotropy field (meV per spin) in (Ga,Mn)As for $\theta =\pi /2$ and $\varphi ∊\left(0,\pi \right)$. The parameters are: Mn fraction $x=0.08$, hole concentration $p\simeq 0.15{\text{nm}}^{-3}$, ${ϵ}_F\tau =3$, and axial strain ${ϵ}_{\text{ax}}=-0.5\%$. These results were evaluated using the model explained in the text. Due to strain, the in-plane anisotropy is notably weaker than the out-of-plane anisotropy represented in the previous figure.Reuse & Permissions

Figure 6

(Color online) Change in the magnetic anisotropy field of (Ga,Mn)As (in meV per spin) due to the inverse spin-galvanic effect, for an electric field of 1 mV/nm along [010]. The parameters are: Mn fraction $x=0.08$, hole concentration $\simeq 0.25{\text{nm}}^{-3}$, ${ϵ}_F\tau =2$, and axial strain ${ϵ}_{\text{ax}}=-1\%$. We compare between interband and intraband contributions: in contrast to the case of good metals, the interband contributions are not negligible in (Ga,Mn)As. For the present case, had we neglected the interband contribution the minimum electric field needed to reorient the magnetization by $90°$ would be off by approximately 20%. The sum of interband and intraband pieces gives rise to a smooth curve that reflects the Dresselhaus symmetry of the axial strain. Reversing the sign of the axial strain (i.e., making it tensile) leads to a sign reversal of $\delta H_{\varphi }$.Reuse & Permissions

Figure 7

(Color online) Reorientation of the magnetization due to an electric current. An initial magnetization along [100] can be rotated (assisted by damping) into [010] by applying a sufficiently strong electric field with a nonzero projection along the [010] direction (a current along [100] would not destabilize the [100] easy axis). For the parameters of this figure ($x=0.08$, $p\simeq 0.15{\text{nm}}^{-3}$, ${ϵ}_F\tau =3$, ${ϵ}_{\text{ax}}=-0.5\%$) the critical electric field is $\simeq 5\text{mV}/\text{nm}$, which corresponds roughly to a critical current density of $5\times {10}^7\text{A}/{\text{cm}}^2$.Reuse & Permissions

Figure 8

(Color online) Dependence of the critical electric field (at which the magnetization gets reoriented by $90°$) on (compressive) axial strain. The critical current is (roughly) inversely proportional to ${ϵ}_{\text{ax}}$. The reason behind this relationship is that the equilibrium, azimuthal anisotropy is largely indifferent to ${ϵ}_{\text{ax}}$. For $x=0.04$ and ${ϵ}_{\text{ax}}=-2\%$ we find $E_c\simeq 0.25\text{mV}/\text{nm}$, which corresponds to a critical current on the order of ${10}^6\text{A}/{\text{cm}}^2$. These results are for a (Ga,Mn)As model with carrier density $p\simeq 0.15{\text{nm}}^{-3}$ and ${ϵ}_F\tau =3$.Reuse & Permissions