Origins of ferromagnetism and antiferromagnetism in Gd$_{5}$Ge$_{4}$

The origin of ferromagnetism appearing as a result of a magnetic-field-induced first-order phase transition in Gd$_{5}$Ge$_{4}$ is explored by calculating the total energy, local exchange splitting, density of states, and magnetic moments. The calculations were performed using density functional approaches including the on-site Coulomb interaction parameter. The total energy as a function of shear distortion along the $a$ axis for two different orthorhombic structures is in agreement with experiment, indicating a first-order magnetostructural transition in Gd$_{5}$Ge$_{4}$. The rearrangement of Gd 5$d$ and Ge 4$p$ densities of states, the substantial differences in atom-projected band energies, the exchange splitting, and the magnetic moments calculated with ferromagnetic spin arrangements in the orthorhombic Sm$_{5}$Ge$_{4}$-type and Gd$_{5}$Si$_{4}$-type structures of Gd$_{5}$Ge$_{4}$ help to clarify the differences in the magnetic states of these two structures. Our calculations indicate that the Sm$_{5}$Ge$_{4}$-type structure of Gd$_{5}$Ge$_{4}$ is the structural ground state and that it is antiferromagnetic.