Unusual domain-wall motion in ferromagnetic semiconductor films with tetragonal anisotropy

Magnetic field-driven domain-wall propagation in the flow regime is investigated in (Ga,Mn)As ferromagnetic semiconductor layers. Square-shape magnetic domains with an unexpected orientation of their edges, at $\pi /8$ with respect to the anisotropy axes, are found. This is shown to arise from the effect of tetragonal magnetic anisotropy on domain-wall dynamics. Using a one-dimensional model for domain-wall motion and modeling domain growth by contour dynamics the shape and orientation of domains and their field range for existence are well reproduced. These results point to the key role of the vectorial nature of the order parameter in the dynamics of ferromagnetic domains.

Figure 1

(Color online) Magneto-optical images showing the square domains at $\text{T}=100\text{K}$ for three field values shown by arrows in (e). Images (a)–(c) are differential images (see text). Image (d): square domain edge with up (white) or down (black) magnetization inside the domain showing the $+\pi /8$ or $-\pi /8$ orientation of the diagonal of the square domain with respect to a $⟨110⟩$ axis (dotted black lines). (e) DW velocity curve at $T=100\text{K}$. In the field range where square domains are observed the red triangles and the black circles represent the velocity measured along the diagonal and perpendicularly to the edges of the domains, respectively, as shown in the inset. The error bars represents the width of the velocity distribution measured for a set of domains over a ${\left(536\mu \text{m}\right)}^2$ area. The dashed line represents the asymptotic linear fit of the high-field precessional regime.

Figure 2

(Color online) Experimental (squares) and theoretical (full curves) DW velocity curves at $T=100\text{K}$. In the field range where square domains are observed the red full squares and the blue empty squares represent the velocity measured along the diagonal and perpendicularly to the edges of the domains, respectively. The red (blue) full line is the normal velocity calculated for ${\phi }_p=\pi /8$ $\left(-\pi /8\right)$ orientation of the DW. The full (empty) circles show the maximum (minimum) velocity obtained from contour dynamics for two field values $h=1.65$ and $h=3.8$. The dashed black line is the velocity obtained when neglecting the in-plane magnetic anisotropy. The Walker velocities without anisotropy $\left(v_W\right)$ and with biaxial anisotropy ($v_{W1}$ and $v_{W2}$) are indicated. The parameters are $H_W=4\text{mT}$, $\Delta =3\text{nm}$, $\alpha =0.28$, and ${\chi }_4=0.6$. The insets show the angular dependence of the normal velocity (in units of $v_W$).

Figure 3

Magnetic domain growth from contour dynamics at $h=1.65$ and at $h=3.8$. The coordinates of the DW are scaled with respect to the perimeter of the initial circular shape $L\left(0\right)$. Time is scaled with respect to $L\left(0\right)/v_W$. Domain shapes are shown from $t=0$ to $t=0.4$ by step 0.05 in (a) and from $t=0$ to $t=1.1$ by step 0.1 in (b). The directions of minimal and maximal displacements for the time interval, as calculated from Eqs. (3, 4, 5, 6) are shown by dashed lines.