Origin of $\pi$-shifted three-dimensional charge density waves in kagome metal AV$_3$Sb$_5$

Understanding the nature of charge density wave (CDW) and superconductivity in kagome metal AV$_3$Sb$_5$ (A=Cs,Rb,K) is a recent subject of intensive study. Due to the presence of van Hove singularities, electron-electron interaction has been suggested to play an important role in the formation of such broken symmetry states. Recent experiments show that the CDW order is three-dimensional and it is staggered across different kagome layers. However, the experimental interpretation for the precise structure of CDW varies in terms of whether it is the star of David (SD), inverse star of David (ISD) or the alternation of the two among neighboring layers. In this work, we show that the origin of these distinct CDW orders can be understood in a unified picture by considering intra- and inter-layer electron-electron interactions as well as the coupling between electrons and lattice distortions. Utilizing an effective 9-band model with V $d$ orbitals and out-of-plane Sb $p$ orbitals, it is demonstrated that the repulsive electron-electron interaction favors charge bond order which induces either SD or ISD upon including lattice distortions. As the inter-layer interaction is introduced, $\pi$-shifted CDW develops with the staggered ordering along the $c$-axis. We also find that the phase with alternating SD and ISD can be stabilized as the ground state under strong inter-layer interaction.