Derivation of the Ginzburg-Landau equations of a ferromagnetic $p$-wave superconductor*

We derive a Ginzburg-Landau free energy for a $p$-wave ferromagnetic superconductor. The starting point is a microscopic Hamiltonian including a spin generalised BCS term and a Heisenberg exchange term. We find that coexistence of magnetisation and superconductivity depends on the sign of the energy-gradient of the DOS at Fermi level. We also compute the tunnelling contribution to the Ginzburg-Landau free energy, and find expressions for the spin-currents and Josephson currents across a tunnelling junction separating two ferromagnetic $p$-wave superconductors.