Nature of Perpendicular-to-Parallel Spin Reorientation in a Mn-doped GaAs Quantum Well: Canting or Phase Separation?

It is well known that the magnetic anisotropy in a compressively strained Mn-doped GaAs film changes from perpendicular to parallel with increasing hole concentration $p$. We study this reorientation transition at $T=0$ in a quantum well with delta-doped Mn impurities. With increasing $p$, the angle $\theta$ that minimizes the energy $E$ increases continuously from 0 (perpendicular anisotropy) to $\pi /2$ (parallel anisotropy) within some range of $p$. The shape of $E^{\min }(p)$ suggests that the quantum well becomes phase separated with regions containing low hole concentrations and perpendicular moments interspersed with other regions containing high hole concentrations and parallel moments. However, because of the Coulomb energy cost associated with phase separation, the true magnetic state in the transition region is canted with $0<\theta <\pi /2$.

Figure 1

The energies of the bottom of the carrier bands with ${\mathbf{k}}_{\perp }=0$ versus $J_c$ when only ${\psi }_1(z)$ is considered. The chemical potential increases with the hole filling.

Figure 2

The minimum energy $E^{\min }/N$ and the angle $\theta$ versus $p$ for $c=0.4$ and $\Lambda =10$. Inset is a plot of $p$ versus $\mu$, revealing an “$S$” shaped curve. A Maxwell construction yields phase separation between the two solid circles.

Figure 3

The total energy gain of a phase-separated mixture (dot-dashed curve), the Coulomb cost associated with the domain wall (dashed curve), and the total energy when both contributions are added (solid curve) versus the size of the region for $c=0.5$ and $\Lambda =10$. The higher and lower values of $2{\lambda }_F$ are associated with perpendicular and parallel Mn moments, respectively. The results below $2{\lambda }_F$ have no physical meaning since $D_1$ and $D_2$ would have to be negative.

Figure 4

The phase diagram of the magnetic orientation in a quantum well with $\Lambda =10$, for Mn concentration $c$ versus holes per cation $p$, showing regions of perpendicular and parallel anisotropy, as well as a possible phase-separated region.