Angle-dependent spin-wave resonance spectroscopy of (Ga,Mn)As films

A modeling approach for standing spin-wave resonances based on a finite-difference formulation of the Landau-Lifshitz-Gilbert equation is presented. In contrast to a previous study [C. Bihler et al., Phys. Rev. B 79, 045205 (2009)], this formalism accounts for elliptical magnetization precession and magnetic properties arbitrarily varying across the layer thickness, including the magnetic anisotropy parameters, the exchange stiffness, the Gilbert damping, and the saturation magnetization. To demonstrate the usefulness of our modeling approach, we experimentally study a set of (Ga,Mn)As samples grown by low-temperature molecular-beam epitaxy by means of angle-dependent standing spin-wave resonance spectroscopy and electrochemical capacitance-voltage measurements. By applying our modeling approach, the angle dependence of the spin-wave resonance data can be reproduced in a simulation with one set of simulation parameters for all external field orientations. We find that the approximately linear gradient in the out-of-plane magnetic anisotropy is related to a linear gradient in the hole concentrations of the samples.

Figure 1

Relation between the two coordinate systems employed. The $(x,y,z)$ frame of reference is spanned by the cubic crystal axes, while the $(1,2,3)$ coordinate system is determined by the equilibrium orientation of the magnetization (3 direction) and two transverse directions, the 2 direction being parallel to the film plane; the latter system is $z$ and ${\mu }_0\mathbf{H}$ dependent, as described in the text.

Figure 2

Simulation to demonstrate the influence of the uniform resonance field ${\mu }_0{H}_{\mathrm{uni}}^{001}$ on the SWR modes for ${\mathbf{m}}_0\left|\right|\left[001\right]$, assuming circular precession. In (a), ${\mu }_0{H}_{\mathrm{uni}}^{001}$ is set to be constant across the layer, while in (c) it varies linearly [dashed (blue) lines], in analogy with a square potential and a triangular potential, respectively. Dotted (black) lines are the resonance fields, calculated assuming boundary conditions of natural freedom (see text). Solid (red) lines show the eigenmodes of the system, i.e., the precession amplitude $\tilde{m}$ of the magnetization; for each mode the dotted line corresponds to $\tilde{m}=0$. As shown in (a), for a constant uniform resonance field the first mode occurs at the uniform resonance field and exhibits a constant precession amplitude across the layer, i.e., an FMR mode. The second and third modes (higher order modes are not shown) exhibit a nonuniform magnetization profile. In order to couple to the driving field the modes need to have a finite net magnetic moment. As shown in (a), the positive and negative areas of the second and third modes are equal, thus these modes are not visible in the SWR spectrum (b). This is in contrast to the case of the linearly varying uniform resonance field (c), where the mode profile is given by Airy functions, which have a nonzero net magnetic moment also for the second and third modes, resulting in a finite SWR intensity of these modes (d). Spectra in (b) and (d) were calculated by integrating over the eigenmodes $\tilde{m}$ and convoluting the square of the result with Lorentzians.

Figure 3

Atlas illustrating the influence of gradients in the anisotropy parameters on SWR spectra. In (a) all anisotropy parameters are kept constant with the values given in the text, except $B_{001}$ which is varied linearly. Correspondingly, in (b) and (c) $B_{4\perp }$ and $B_{4\left|\right|}$ were varied linearly, respectively. Panels (i) show the first derivative of simulations using Eq. (12) with respect to ${\mu }_{0}H$ and panels (ii)-(iv) show the precession cone $\text{Im}(m_1^{*}{m}_2-m_1{m}_2^{*})$ in a color plot together with the uniform resonance field ${\mu }_0{H}_{\text{uni}}\left(z\right)$ (dashed blue lines) at three different external field orientations; the black dotted lines indicate the resonance field positions of the modes. Panel (a i) additionally shows the influence of a linear gradient in the exchange stiffness parameter on the spin-wave spectra, see text for further details and discussion.

Figure 4

(a) The hole concentration in the different (Ga,Mn)As samples is shown as a function of the depth within the layers as determined by ECV profiling. (b) Uniform resonance fields ${\mu }_0{H}_{\text{uni}}^{001}\left(z\right)$ for the four samples obtained from simulations for the out-of-plane orientation of the external field ($\psi ={90}^{\circ }$) as a function of the depth.

Figure 5

The spin-wave resonance data [dotted (blue) lines] are shown together with simulations [solid (red) lines] using the numerical procedure described in the text and in Appendix 2. Data were obtained as a function of the external magnetic field orientation and magnitude for samples with a V/III flux ratio of (a) 1.1, (b) 1.3, (c) 1.5, and (d) 3.5. The rotation angle $\psi$ is defined in the inset and the parameters used for the simulations are summarized in Table .

Figure 6

Simulated magnetization mode profile and uniform resonance field of sample A. The contour plots show the magnetization precession amplitude $\text{Im}(m_1^{*}{m}_2-m_1{m}_2^{*})$ as a function of the position within the film and the external magnetic field for the external field aligned (a) along [001], (b) at an angle of 50${}^{\circ }$ with respect to [110] (cf. the inset in Fig. 5), and (c) along [110]. Dashed (blue) lines in (a–c) show the uniform resonance field, obtained by numerically solving Eq. (8) for each given field orientation. Dotted (black) lines in (a) indicate the resonance magnetic fields. (d–f) A magnification of the data [dotted (blue) lines] and simulation [solid (red) lines] from Fig. 5a shown using the same scale for all orientations. In (e), a simulation with a different set of parameters is shown for comparison (black, dashed line), see text.