${\mathrm{Ga}}_{1-\mathit{x}}{\mathrm{Mn}}_{\mathit{x}}\mathrm{As}$: A Frustrated Ferromagnet

Starting from a microscopic description of the exchange interaction in ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$, we derive an effective Mn-Mn interaction. Because of the strong spin-orbit coupling in the valence band, this effective interaction is highly anisotropic and has a spatial structure somewhat similar to dipolar interactions. The corresponding ground state has a finite magnetization but is intrinsically spin disordered even at zero temperature, explaining many features of the experimental data. Our results are relevant for many magnetic semiconductors.