Electronic structure and magnetic phase transition in helicoidal Fe$_{1-x}$Co$_{x}$Si ferromagnets

LSDA + U + SO calculations of the electronic structure of helicoidal Fe$_{1-x}$Co$_{x}$Si ferromagnets within the virtual crystal approximation have been supplemented with the consideration of the Dzyaloshinski–Moriya interaction and ferromagnetic fluctuations of the spin density of collective d electrons with the Hubbard interactions at Fe and Co atoms randomly distributed over sites. The magnetic-state equation in the developed model describes helicoidal ferromagnetism and its disappearance accompanied by the occurrence of a maximum of uniform magnetic susceptibility at temperature $T_C$ and chiral fluctuations of the local magnetization at $T >T_C$. The reasons why the magnetic contribution to the specific heat at the magnetic phase transition changes monotonically and the volume coefficient of thermal expansion (VCTE) at low temperatures is negative and has a wide minimum near $T_C$ have been investigated. It is shown that the VCTE changes sign when passing to the paramagnetic state (at temperature $T_S$).