The ferromagnetic semiconductor (Ga,Mn)As has been utilized to demonstrate proof-of-concept devices since its first synthesis in 1996^[1]. (Ga,Mn)As exhibits hole mediated ferromagnetism, in which the localized Mn spin couples anti-ferromagnetically with valence band holes^{[2, 3]}. In this review, we describe the recent research on (Ga,Mn)As, mainly focusing on the magnetization dynamics and its related phenomena. Comprehensive reviews of the synthesis, physics, and applications of ferromagnetic semiconductors have been updated continuously by many researchers, and are available elsewhere^[4–6]. We present also recent results on the dynamics in a single crystalline ferromagnetic metal/semiconductor hybrid structures.

Because the thermal-equilibrium solubility of Mn in GaAs is small, non-equilibrium growth method, i.e., low-temperature molecular-beam epitaxy at substrate temperature ~ 250 °C, is developed to synthesize single crystalline (Ga,Mn)As films with on the order of percent of nominal Mn composition x. Its p-type conductivity indicates that most of the Mn atoms (Mn_Ga) act as acceptors by replacing Ga atoms in divalent states (d⁵ configuration with localized spin of 5/2)^{[7, 8]}. A part of the Mn atoms, Mn_I, sit also at the interstitial sites due to the self-compensation effect^[9]. The Mn_I acts as a double donor, and its spin couples antiferromagnetically with that of Mn_Ga^[10]. The presence of Mn_I thus affects the electrical and magnetic properties^[11]. The number of Mn_I can be reduced by post-growth annealing at ~ 200 °C through its diffusion towards surface^[12].

The itinerant holes, which reside in the valence band of GaAs, mediate Ruderman-Kittel-Kasuya-Yosida (RKKY)-type exchange interaction among Mn spins^[13], and bring about the ferromagnetism in (Ga,Mn)As^{[2, 3]}. Therefore, there is a correlation between the electrical and magnetic properties, as explained by the p-d Zener model. The model is an adaption to the RKKY model specific to semiconducting materials, and can describe many properties of (Ga,Mn)As including the spin-orbit coupling related phenomena, such as the magneto-crystalline anisotropy, through band calculations based on effective mass approximations.

In (Ga,Mn)As, the exchange interaction among As 4p and Mn 3d electrons (p–d exchange interaction) results in the ferromagnetic state. According to the p–d Zener model, the Curie temperature T_C is proportional to the effective Mn composition participating in the ferromagnetic order, and the density of state at the Fermi level. Therefore, T_C increases as the hole concentration p increases by decreasing the number of Mn_I through annealing. The enhancement of the annealing effect is expected for a (Ga,Mn)As nanowire because of its larger ratio of the surface region than films^[14]. This was confirmed by magneto-transport measurements on a 310-nm wide nanowire of (Ga,Mn)As with x = 0.13, which exhibits a T_C of 200 K after annealing (as shown in Fig. 1)^[15].

(Ga,Mn)As shows sizable magnetic anisotropy depending on the spontaneous magnetization M, p, and lattice strain. For example, a metallic (Ga,Mn)As film with compressive strain on a GaAs (001) substrate exhibits an in-plane easy axis for the magnetization, while that with tensile strain on an (In,Ga)As buffer layer shows perpendicular easy axis^[16]. The in-plane magnetic anisotropy shows four-fold anisotropy along as well as two two-fold anisotropies along and . Because (Ga,Mn)As exhibits a clear anomalous Hall effect and anisotropic magnetoresistance (AMR)^{[1, 17–19]}, the direction and magnitude of the anisotropies can be determined by magneto-transport measurements^[20]. Ferromagnetic resonance (FMR) can be also used to determine the magnetic anisotropies^[21]. The origin of the anisotropies is explained by the spin-orbit interaction in the valence band of (Ga,Mn)As with (virtual) lattice strain^{[2, 3, 22–26]}.

Magneto-transport measurements show that (Ga,Mn)As is in the vicinity of the metal-insulator transition (MIT)^[27], and the interplay between magnetic properties and carrier localization has been probed experimentally. In doped semiconductors near MIT, there are two regions occupied by itinerant carriers and localized carriers^[28]. In (Ga,Mn)As, the former region exhibits ferromagnetic behavior, and the latter superparamagnetic-like behavior^[29]. The ratio between the two regions depends on the degree of the localization^[30].

By making a capacitor structure with one of the two electrodes as a thin (Ga,Mn)As film, one can apply an electric-field onto (Ga,Mn)As to change p and the degree of MIT, which in turn alters the magnetic properties of (Ga,Mn)As due to carrier induced ferromagnetism. So far, the electric-field modulation of T_C^[31], magnetic anisotropies^[32], net magnetic moment^[28], anomalous Hall coefficient^[33], and Gilbert damping constant α has been demonstrated^[30]. The electric-field effect is now being investigated also in ferromagnetic metals^[34].

There is an excellent review on the magnetization precession in (Ga,Mn)As induced by optical means^[6], and also a comprehensive review on FMR and spin-wave resonance in (Ga,Mn)As^[35]. Here, we introduce recent topics of FMR and its related phenomena in (Ga,Mn)As-based structures.

FMR spectrum is usually measured as the derivative of the microwave absorption, which is well fitted by the derivative of the Lorentz function, − (16I/π){ΔH(H− H_R)/[4(H− H_R)²+ ΔH²]²}, similar to other conventional ferromagnets. From the fitting, one can determine the absorption coefficient I, resonance field H_R, and the linewidth (the full width at half maximum) ΔH. The magnetization dynamics is known to be described by the Landau-Lifshitz-Gilbert (LLG) equation^[36],

$ \frac{{{\rm{d}}{{M}}(t)}}{{{\rm{d}}t}} = - \gamma {{M}}(t) \times {\mu _0}{{{H}}_{{\rm{eff}}}} + \frac{\alpha }{{|{{M}}|}}{{M}}(t) \times \frac{{{\rm d}{{M}}(t)}}{{{\rm{d}}t}}, $  (1)

and this is also the case for (Ga,Mn)As. Here, M is the spontaneous magnetization vector, t the time, μ₀ the magnetic constant, H_eff the effective magnetic field, α the damping constant, γ the gyromagnetic ratio (γ = gμ_B/ħ), g the Landé g factor, μ_B the Bohr magneton, and ħ the Dirac constant. The first term on the right side of Eq. (1) corresponds to the magnetization precession about H_eff, while the second term to the relaxation of M towards H_eff. H_eff is determined from the magnetostatic energy density F^{[35, 37]},

$ \begin{aligned} F = & \frac{M}{2}\bigg\{ { - 2H\left[ {\cos {\theta _{ M}}\cos {\theta _{ H}} + \sin {\theta _{ M}}\sin {\theta _{ H}}\cos \left( {{\varphi _{ M}} - {\varphi _{ H}}} \right)} \right]} \bigg.\\ & + {H_{ K}}{{\cos }^2}{\theta _{ M}} - \displaystyle\frac{{{H_{\rm{B}}}}}{2}\frac{{3 + \cos 4{\varphi _{ M}}}}{4}{{\sin }^4}{\theta _{{{ M}}}}\\ & - {H_{\rm{U}}}{{\sin }^2}{\theta _{ M}}{{\sin }^2}\left( {{\varphi _{ M}} - \displaystyle\frac{\pi }{4}} \right) - {H_{{\rm{U}}2}}{{\sin }^2}{\theta _{ M}}{{\sin }^2}\left. {\left( {{\varphi _{ M}} - \displaystyle\frac{\pi }{2}} \right)} \right\}. \end{aligned} $  (2)

as H_eff = –∂F/∂M. Here M = |M|, H the external magnetic field, H_K the effective perpendicular uniaxial magnetic anisotropy field including the demagnetizing field along [001], H_B the in-plane biaxial magnetic field along 100 , H_U the in-plane uniaxial anisotropy along , and H_U2 the in-plane uniaxial anisotropy along [100]. The magnetic field angles θ_H and φ_H as well as the magnetization angles θ_M and φ_M are polar angles θ measured from the [001] and azimuthal angles φ measured from the [100] orientation. θ_M and φ_M are determined from the energy minimum conditions, ∂ F/∂ θ_M = 0, ∂²F/∂ θ_M² > 0, ∂ F/∂ φ_M = 0, and ∂²F/∂ φ_M² > 0. In a linear-response regime, the resonance condition is given by ^[38],

$ {\left( {\frac{\omega }{\gamma }} \right)^2} = \frac{1}{{{\mu _0}{{ M}^2{{\sin^2}}}{\theta _{ M}}}}\left[ {\frac{{{\partial ^2}F}}{{\partial \theta _{ M}^2}} - \left( {\frac{{{\partial ^2}F}}{{\partial {\theta _{ M}}\partial {\varphi _{ M}}}}} \right)} \right], $  (3)

with ω = 2πf. From the fitting of Eq. (3) to the magnetic-field angle dependence of H_R, one can determine the values of g and magnetic anisotropy fields.

The linewidth induced by intrinsic damping is expressed as^[38],

$ \Delta {H_{\rm{G}}} = \frac{\alpha }{ M}\left( {\frac{{{\partial ^2}F}}{{\partial \theta _{ M}^2}} - \frac{1}{{{{\sin }^2}{\theta _{ M}}}}\frac{{{\partial ^2}F}}{{\partial \varphi _{ M}^2}}} \right){\left| {\frac{{{\rm d}(\omega /\gamma )}}{{{\rm d}{H_{\rm{R}}}}}} \right|^{ - 1}}. $  (4)

Extrinsic contributions such as the dispersion of the magnetic anisotropies and roughness as well as two-magnon scattering can also be included phenomenologically in the analysis of the magnetic-field angle θ_H (or φ_H) dependence of ΔH to determine the magnitude of α^[39–42].

The FMR measurements can be used to determine the magnitude of magnetic anisotropy fields, α, and g. By measuring the FMR spectra under electric-fields, one can determine the electric-field dependence of these parameters. The electric-field effects on the FMR spectra were investigated for a 4-nm thick (Ga,Mn)As film with x = 0.13 annealed at 200 °C for 10 min in a capacitor structure^[30]. The maximum applied voltage V_G is 24 V, which corresponds to an electric field of ~4 MV/cm. The resistance changes by about ±10% by the application of ±24 V.

Figs. 2(a) and 2(b) show the V_G dependence of anisotropic fields H_ani and damping constant. The modulation of the magnetic anisotropy fields and g, whose values are expected to be determined by the spin-orbit interaction, is relatively small (~1%). On the other hand, α becomes larger by decreasing p through an electric field, and its modulation reaches ~12%, suggesting that the modulation is not determined only by the spin-orbit interaction. We measure the electrical resistance and magnetization under electric fields and those of some other samples with different conductivities in addition to α. As shown in Fig. 2(c), there is a clear correlation between the resistance, the portion of a superparamagnetic-like component, and α, i.e., a larger portion of the superparamagnetic-like component and larger α for samples with higher resistance. This observation suggests strongly that α in (Ga,Mn)As is determined mainly by the magnetic disorder induced by carrier localization.

The electric-field effect on α was also observed in a thin CoFeB/MgO structure^[42], and the origin of the modulation is still to be elucidated.

In conductive ferromagnets, the relationship between electric field E and current density j is phenomenologically expressed as^[43],

$ { E} = \rho {{j}} + \left( {{\rho _{\parallel }} - {\rho _ {\bot }}} \right){{n}}({{j}} \cdot {{n}}) + {\rho _{\rm{H}}}{{n}} \times {{j}}, $  (5)

where ρ is the resistivity, n the unit magnetization vector (n = M/M), and the resistivity perpendicular and parallel to j, and ρ_H the anomalous Hall resistivity. The first term corresponds to Ohm’s law, the second term to the AMR effect, and the third term to the anomalous Hall effect. The electric part of the microwave excites an oscillating current j, and the magnetic part of the microwave causes the magnetization n to precess at the same frequency as j. Thus, according to Eq. (5), a DC voltage is generated in the material under FMR.

We prepared a 20-nm thick (Ga,Mn)As film with x = 0.065 on a semi-insulating GaAs (001) substrate, and annealed it at 250 °C for 30 min. T_C of the (Ga,Mn)As was 118 K. We measured FMR spectra and DC voltages V along the [110] orientation simultaneously by sweeping H orthogonal to the [110] orientation at f = 9 GHz and at 45 K^[37]. The electric-field component of the microwave was along the orientation and the magnetic-field component along the [110] orientation in our configuration. As shown in Fig. 3(b), the measured V is well fitted by the sum of the symmetric and anti-symmetric Lorentz functions, L_sym = ΔH²/[4(H− H_R)²+ ΔH²] and L_a-sym = −4ΔH(H− H_R)/[4(H− H_R)²+ ΔH²]. The fitting gives identical H_R and ΔH with those determined from the FMR spectra shown in Fig. 3(a). According to Eq. (5), the transverse AMR effect (planar Hall effect) and the anomalous Hall effect contribute to V. The planar Hall effect reflects the real part of in-plane component of the dynamic magnetization Re(m_x), and thus exhibits a symmetric lineshape, V_PHE ~ Re(m_x). The anomalous Hall effect reflects the out-of-plane component Re(m_y), and thus exhibits an anti-symmetric lineshape, V_AHE ~ Re(m_y). As shown in Figs. 3(c) and 3(d), the magnetic-field angle dependence of V_sym and V_a-sym can be well fitted by planar Hall effect and anomalous Hall effect of (Ga,Mn)As, respectively.

It was also shown that electrical detection of FMR is possible in (Ga,Mn)As through electric-field excitation^[44].

Similar to metallic systems^[45–47], the spin-pumping creates pure spin current from (Ga,Mn)As into adjacent p-GaAs. Due to the spin-orbit interaction in p-GaAs, the pure spin current is converted into a DC voltage through the inverse spin Hall effect, which is superimposed on the DC voltage induced by magnetogalvanic effects.

The measurements of the inverse spin Hall effect was done on virtually identical (Ga,Mn)As films as described in the previous section on a 20-nm thick p-GaAs layer with p = 9.5 × 10¹⁸ cm^–3 grown on a semi-insulating GaAs substrate^[37]. The measurement configuration and condition are the same as those in the previous section. We rotated the sample about the [110] orientation to measure the magnetic-field angle θ_H dependence of FMR and dc voltage. The presence of spin pumping is confirmed by the enhancement of ΔH of the (Ga,Mn)As/p-GaAs comparing to that of (Ga,Mn)As, which is shown in Fig. 4. The linewidth analysis gives the effective mixing conductance at the (Ga,Mn)As/p-GaAs to be 1.5 × 10¹⁹ cm^–2.

The DC voltage induced by inverse spin Hall effect V_ISHE is proportional to the damping term in Eq. (1), which can be derived as V_ISHE ~ Re(m_x)Im(m_y) − Im(m_x)Re(m_y). The lineshape of V_ISHE is symmetric, which is the same as the planar Hall effect. Note that, being different from V_PHE, V_ISHE results from the combination of the real and imaginary part of the dynamic magnetization. Thus, the separation of the signals induced by the two effects is possible by utilizing the different magnetic-field angle θ_H dependence of the two signals as shown in Fig. 5. The fitting reproduces well the experimental observation, and indicates contributions of 12% from the inverse spin Hall effect and 88% from the planar Hall effect to symmetric DC voltage^[37]. This separation method has been applied to other material systems^[48–50].

Due to the crystal and structural symmetry breaking in a strained (Ga,Mn)As film, one can induce the effective spin-orbit magnetic field in (Ga,Mn)As by applying electric current, which was utilized to manipulate the magnetization direction and to excite FMR^{[44, 51, 52]}. Thus, one can imagine not making GaAs ferromagnetic, but putting a layer of single crystalline ferromagnetic metal on top of GaAs^{[53, 54]}. Here, we focus on the FMR induced by spin-orbit torques in a Fe film on GaAs (001) at room temperature. A thin single crystalline Fe film can be grown on GaAs by MBE thanks to relatively small lattice mismatch (~ 1.4%) between the two materials^{[53, 54]}. Fe/GaAs heterostructure has long been used for spin-injection experiments^[55–57], and interest in the system has recently been revived in view of spin-orbitronics due to the presence of Bychkov-Rashba- and Dresselhaus-like spin-orbit interactions at the interface^[58]. The effective in-plane spin-orbit fields, h_eff, in momentum space can be written as^[59],

$ {\mu _0}{{{h}}_{\rm eff}} = \frac{1}{{{\mu _{\rm B}}}}\left( { - \beta {k_x} - \alpha {k_y},\alpha {k_x} + \beta {k_y}} \right), $  (6)

where k_x (k_y) is a [100] ([010]) component of the wavevectork, and α (β) characterizes the strength of the Bychkov-Rashba (Dresselhaus) spin-orbit interaction. The presence of the interfacial spin-orbit interaction was evidenced previously by tunneling anisotropic magnetoresistance measurements in a Au/GaAs/Fe trilayer structure^[53].

The strength of the interfacial spin-orbit fields at the Fe/GaAs interface can be quantified using spin-orbit-torque FMR (SO-FMR). The applied RF current to Fe/GaAs produces RF spin-orbit fields, which induce magnetization precession when the resonance condition is fulfilled. The precession results in the periodic change in resistance with precessional frequency through the magnetogalvanic effect in Fe, and thus produces DC voltage V (Eq. (5))^[44].

The sample used for SO-FMR measurements is a 5-nm thick single crystalline Fe grown by MBE on undoped GaAs (001). The Fe layer was patterned into stripes (6.4 × 100 μm²) along different crystal orientations of GaAs, i,e., [100], [010], [110], and orientations. The DC voltage V was measured at RF current with frequency of 12 GHz as functions of current density up to ~1.9 × 10¹¹ A/m² and the direction φ_H of an external in-plane magnetic field by sweeping the external field. The external magnetic-field dependence of V contains both symmetric and anti-symmetric lineshapes, which can be decomposed into the symmetric and anti-symmetric Lorentz functions. The symmetric component results mainly from the out-of-plane components of effective spin-orbit fields and the anti-symmetric component from the in-plane components. By analyzing the magnetization-angle φ_M dependence of the anti-symmetric component of V, one can determine the crystal-orientation dependence of the magnitude of the effective spin-orbit fields as shown in Fig. 6 at j = 10¹¹ A/m^2[58]. The obtained results are explained by the coexistence of Bychkov-Rashba (h_R) and Dresselhaus (h_D) components of spin-orbit fields in agreement with theoretical predictions.

The spin galvanic effect induced by spin pumping at Fe/GaAs spin-orbit interface was also demonstrated by exciting FMR using an out-of-plane component of an Oersted field in a coplanar waveguide (putting Fe wires a gap between the signal and ground lines of a coplanar waveguide)^[58]. Similar observations have been reported for Bi/Ag and LaAlO₃/SrTiO₃ Rashba interface^{[60, 61]}.

The magnitude of β and α at the Fe/GaAs interface depends linearly on the interfacial electric-field. Thus, it is possible to control the effective spin-orbit field by an external electric-field^[62]. The sample investigated in this study was an Fe (4 nm)/n-GaAs (electron concentration, n = 4 × 10¹⁶ cm^–3) Schottky junction. The DC voltage V induced by SOT-FMR was measured under a gate voltage V_G applied through the depleted n-GaAs underneath Fe, and the strength of spin-orbit effective field was determined by the method described above. Fig. 7 shows the polar plot of the effective spin-orbit fields at V_G = –0.88 V and +0.07 V (arrows are field vectors and lines are spin-orbit energy splitting Δε_SO = 2μ_Bμ₀|h_eff|). It clearly exhibits the electric-field modulation of the spin-orbit field vectors. For [100] and [010] orientations, both the direction and strength of the fields is modifiedV_G, while for [110] and orientations, only the strength of the fields is modified. The result shows also that the modulation of the Bychkov-Rashba spin-orbit fields is several-time larger than that of the Dresselhaus spin-orbit field.

The electric-field modulation of the interfacial spin-orbit effects is attracting much attention also from the view point of practical applications. For instance, the electric-field induced precessional magnetization switching through the modulation of the interfacial magnetic anisotropy was demonstrated, and this switching scheme is fast with low-power consumption^[63].

The interplay of Bychkov-Rashba and Dresselhaus spin-orbit interaction can modify the density of states at the Fe/GaAs interface. This has caused a rich variety of interfacial spin-orbit related phenomena. It has been found that the symmetry of the anisotropic magneto-resistance^[64], the polar magneto-optic Kerr effect^[65] and the Gilbert damping^[66] is governed by the two-fold interfacial C_2v symmetry rather than its bulk fold C_4v symmetry when the thickness of Fe is decreased to a few monolayers. Here we show the emergence of anisotropic damping in ultrathin Fe film on GaAs (001).

Fig. 8(a) shows the angular dependence of Gilbert damping for Fe thickness of 1.9 nm determined by analyzing the in-plane magnetic angle φ_H dependence and the microwave frequency f dependence of the linewidth. Isotropic behavior is observed for 1.9 nm Fe on GaAs. However, clear anisotropic Gilbert damping has been found when the Fe thickness is reduced to 1.3 nm as shown in Fig. 8(b). The anisotropic damping shows two-fold symmetry, coinciding with the symmetry observed for tunneling anisotropic magnetoresistance^[53], crystalline anisotropic magnetoresistance^[64] and the polar magneto-optic Kerr effect^[65], can be explained in terms of the anisotropic density of states (open symbols in Fig. 8(b)) induced by the interfacial spin-orbit interaction.

We have described the fundamental properties of (Ga,Mn)As focusing on its ferromagnetic-resonance (FMR) related phenomena such as the spin pumping and the electric-field modulation of the damping constant. We have described also recent topics on FMR-related phenomena in single crystalline Fe/GaAs structures such as the electric-field modulation of the interfacial spin-orbit fields and the emergence of the anisotropic damping in the structures with an ultrathin Fe. In both systems, (Ga,Mn)As and Fe/GaAs, the observation of the magnetization dynamics provides us the opportunities to investigate a variety of physics based on their spin-orbit interaction.

The authors thank T. Dietl, J. Fabian, M. Gmitra, S. Mankovsky, H. Ebert, M. Kronseder, D. Schuh, D. Bougeard for fruitful discussions. L. C. thanks the German Science Foundation (DFG) via SFB 1277 for support. The work at Tohoku University was partially supported by Grant-in-Aids from MEXT and JSPS.