Novel three-dimensional Fermi surface and electron-correlation-induced charge density wave in FeGe

As the first magnetic kagome material to exhibit the charge density wave (CDW) order, FeGe has attracted much attention in recent studies. Similar to AV$_{3}$Sb$_{5}$ (A = K, Cs, Rb), FeGe exhibits the CDW pattern with an in-plane 2$\times$2 structure and the existence of van Hove singularities (vHSs) near the Fermi level. However, sharply different from AV$_{3}$Sb$_{5}$ which has phonon instability at $M$ point, all the theoretically calculated phonon frequencies in FeGe remain positive. Here, we perform a comprehensive study of the band structures, Fermi surfaces and nesting function of FeGe through first-principles calculations. Surprisingly, we find that the maximum of nesting function is at $K$ point instead of $M$ point. Two Fermi pockets with Fe-$d_{xz}$ and Fe-$d_{x^{2}-y^{2}}$/$d_{xy}$ orbital characters have large contribution to the Fermi nesting, which evolve significantly with $k_{z}$, indicating the highly three-dimensional (3D) feature of FeGe in contrast to AV$_{3}$Sb$_{5}$. Meanwhile, the vHSs are close to the Fermi surface only in a small $k_{z}$ range, and does not play a leading role in nesting function. Considering the effect of local Coulomb interaction, we reveal that the Fermi level eigenstates nested by vector $K$ are mainly distributed from unequal sublattice occupancy, thus the instability at $K$ point is significantly suppressed. Meanwhile, the wave functions nested by vector $M$ have many ingredients located at the same Fe site, thus the instability at $M$ point is enhanced. This indicates that the electron correlation, rather than electron-phonon interaction, plays a key role in the CDW transition at $M$ point.