Charge and spin modulation in ferromagnetic semimetals

We show that in ferromagnetic semimetals, in which the Ruderman-Kittel-Kasuya-Yosida indirect exchange predominates, the magnetically ordered state is unstable against inhomogeneous fluctuations. At a temperature slightly lower than the Curie temperature, the system enters into a state in which both the carrier density and the magnetization are modulated. The modulation scale is fixed by the Coulomb screening of the carriers, and can be tuned by changing the ratio of the electron and hole concentrations. At much lower temperatures a reentrant transition into a uniformly ordered state occurs. The modulated state is strongly suppressed by an external magnetic field. The results explain qualitatively the magnetic properties of ${\mathrm{EuB}}_6.$