Cubic anisotropy in (Ga,Mn)As layers: Experiment and theory

Historically, comprehensive studies of dilute ferromagnetic semiconductors, e.g., $p$-type (Cd,Mn)Te and (Ga,Mn)As, paved the way for a quantitative theoretical description of effects associated with spin-orbit interactions in solids, such as crystalline magnetic anisotropy. In particular, the theory was successful in explaining uniaxial magnetic anisotropies associated with biaxial strain and nonrandom formation of magnetic dimers in epitaxial (Ga,Mn)As layers. However, the situation appears much less settled in the case of the cubic term: the theory predicts switchings of the easy axis between in-plane $\langle 100\rangle$ and $\langle 110\rangle$ directions as a function of the hole concentration, whereas only the $\langle 100\rangle$ orientation has been found experimentally. Here, we report on the observation of such switchings by magnetization and ferromagnetic resonance studies on a series of high-crystalline quality (Ga,Mn)As films. We describe our findings by the mean-field $p\text{−}d$ Zener model augmented with three new ingredients. The first one is a scattering broadening of the hole density of states, which reduces significantly the amplitude of the alternating carrier-induced contribution. This opens the way for the two other ingredients, namely the so-far disregarded single-ion magnetic anisotropy and disorder-driven nonuniformities of the carrier density, both favoring the $\langle 100\rangle$ direction of the apparent easy axis. However, according to our results, when the disorder gets reduced, a switching to the $\langle 110\rangle$ orientation is possible in a certain temperature and hole concentration range.

Figure 1

Fine lines of lighter shades mark high-resolution x-ray diffraction patterns of the studied (Ga,Mn)As layers: 004 Brag reflection, $2\theta /\omega$ scans. The central narrow features represent reflections from the GaAs substrate and the broader peaks at lower angles are reflections from the layers. Thicker lines of darker shades mark simulations upon which Mn concentrations and the layers thicknesses have been established. Curves are shifted vertically to increase clarity of the presentation.

Figure 2

Uniaxial hard axis magnetic isotherms ${\overline{m}}_{\left[110\right]}\left(H\right)=M_{\left[110\right]}\left(H\right)/M$ at selected temperatures for sample A (solid points) exemplifying different curvatures of mid-field part of the $m\left(H\right)$. The background thick lines of lighter shades indicate the initial slope of each ${\overline{m}}_{\left[110\right]}\left(H\right)$ and serve as references to ease the identification of the curvatures. The thin solid lines of matching colors are calculated from Eq. (2) treating the anisotropy constants as adjustable parameters.

Figure 3

Temperature dependence of uniaxial ($K_U$, squares and pentagons) and cubic ($K_C$, bullets and diamond) anisotropy constants. Solid points are obtained from analysis of uniaxial hard axis magnetization curves ${\overline{m}}_{\left[110\right]}\left(H\right)$ open ones are obtained from analysis of the angular dependence of FRM resonance positions. Dashed lines are guides for the eye only. In the notion adopted here, the positive/negative sign of $K_C$ indicates that the cubic easy axes are aligned along $\langle 100\rangle /\langle 110\rangle$ directions, respectively. The star represents a low-temperature estimation of $K_C$ after removal from the original ${\overline{m}}_{\left[110\right]}\left(H\right)$, a part attributed to nonhomogeneous magnetization originated at the interface regions, as detailed in Sec. 5. Thick solid line represents results of the Zener mean-field model, including a Gaussian broadening of the density of states and a single-ion anisotropy contribution, as described in Sec. 4.

Figure 4

Polar plot of the in-plane angular dependencies of the ferromagnetic resonance (open circles) and the established uniaxial (doted red thick line) and cubic (navy thick line) contribution to the total magnetic energy for sample A at 6 (a) and 50 K (b). (c) Magnetic isotherms $\overline{m}\left(H\right)=M\left(H\right)/M$ measured for the same sample A along three major in-plane crystallographic orientations, $\left[\overline{1}10\right]$ (circles), [110] (squares), and [100] (diamonds), at the same two temperatures [6 K (open) and 50 K (full symbols)]. A black cross at $H=0$ and $\overline{m}=1/\sqrt{2}$ marks the expected magnitude of remnant $\overline{m}$ for the [100] orientation in a single domain approach defined by Eq. (1) expected for $|K_C|<|K_U|$.

Figure 5

(a) Hole density $p$ dependence of cubic anisotropy constant ($K_C$, diamonds) established during incremental low-temperature annealing. The magnitudes of $p$ at each annealing step are obtained from modeling of the corresponding experimental values of Curie temperature $T_{\mathrm{C}}$ and saturation magnetization $M_{\mathrm{S}}$ determined during annealing (b). Dashed line in (a) is a guide for the eye only. Thick solid line represents results of the mean-field Zener model with a Gaussian broadening of the density of states and a single-ion anisotropy contribution included, as detailed in Sec. 4.

Figure 6

The effect of disorder on the magnitude of the carrier-mediated in-plane cubic anisotropy coefficient $K_C$ computed within the mean-field $p\text{−}d$ Zener model extended by including a Gaussian broadening of the hole energy states for various values of the standard deviation $\sigma$.

Figure 7

Single-ion cubic anisotropy coefficient for noninteracting $S=5/2$ spins as a function of magnetization $〈S〉$, in units of $-a$. The curve designated “classical” represents evaluation treating $S$ as a classical vector, while the dashed curve “quantum” is rescaled by the factor of $24/125$ to match the low-temperature limit of the quantum calculation. “Full treatment” corresponds to the mean-field approach to the ferromagnetic case [Eq. (7)]. The thin red curves represent consequent orders of expansion in powers of magnetization, while the blue thick curve has been obtained by the Aitken extrapolation.

Figure 8

(a) Thick black line: expected vertical hole density profile $p\left(z\right)$ for the $d=5$ nm thin (Ga,Mn)As layer. The dashed line indicates the first 7-nm $p\left(z\right)$ in the original (Ga,Mn)As layer before thinning. An exempt from Fig. 3(a) in Ref. [28]. The dark shaded part of the profile marks the central part of the layer where uniform ferromagnetic coupling specific to metallic (Ga,Mn)As prevails. On moving away towards the interfaces, the rapidly decreasing $p$ results in the magnetic phase separation, as indicated by a light blue texture. (b) Temperature $T$ and (c) magnetic field $H$ dependent studies of the same samples. The thermoremnant moments (TRm) are acquired for the uniaxial easy orientation (along $\left[1\overline{1}0\right]$) on warming after field cooling the samples at $H=1$ kOe to about 2 K and quenching $H$ to sub-Oe range. Arrows indicate the magnitudes of Curie temperature $T_{\mathrm{C}}$ established upon TRm. $H$-dependent characteristics for $T=50$ K are shown for both uniaxial easy ($\left[1\overline{1}0\right]$: open symbols) and hard axis ([110]: full symbols) orientations.

Figure 9

(a) Comparison of the changes to $T$ = 50 K uniaxial hard axis magnetic isotherm ${\overline{m}}_{\left[110\right]}\left(H\right)=M_{\left[110\right]}\left(H\right)/M$ (grey bullets) due to lowering temperature to 5 K (red squares) and thinning the layer down to 5 nm (navy diamonds). (b) Illustration of the method used to establish low temperature magnetic field $H$ response specific to interfaces in 10 nm thin sample A (the thick navy line). All measurements are performed along $\left[110\right]$, the uniaxial hard direction. The labeling of the data sets corresponds to the notation used in the text. (c) The same sample. A comparison between the original hard axis magnetization curve (solid squares) and the specific to ferromagnetic (uniform) part of the sample (open circles). The background thick lines of lighter shades indicate the initial slope of each $\overline{m}\left(H\right)$ and serve as references to ease the identification of the opposite mid-field curvatures. The solid lines of matching colors represent the modeling of the data by Eq. (2). The magnitudes of the established uniaxial $K_U$ and cubic $K_C$ anisotropy constants are denoted in the panel (in units of ${\mathrm{erg}/\mathrm{cm}}^3$), together with values of a required extra horizontal shift $H_0$ to align modeled curves with the experimental points at $m\left(H\right)\simeq 0$.