The charge density wave (CDW) is a widely regarded emergent phenomenon in condensed matter physics. To establish a systematic understanding of the CDW, we develop a diagrammatic self-consistent field approach for a cubic Holstein model employing the fluctuation exchange approximation, and explore the emergence and transition of three-dimensional CDWs. A commensurate CDW (c-CDW) locked at $(\pi ,\pi ,\pi )$ is favored near half filling, and the transition temperature is predicted around half of the nearest-neighbor hopping. Large hole doping leads to a suppression of the CDW transition temperature and the emergence of incommensurate CDW (i-CDW), which is evidenced by a drifting of the ordering vector away from $(\pi ,\pi ,\pi )$ towards $(\pi ,\pi ,0)$. The phonon frequency significantly impacts the transition temperature and the phase boundary between c-CDW and i-CDW, and the optimal frequency for enlarging the CDW regime is also predicted near half of the nearest-neighbor hopping. These theoretical results provide a systematic understanding of CDW and a fresh perspective on emergent phenomena dominated by the electron-phonon interaction.

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• Charge density waves
• Electron-phonon coupling
• Electronic structure
• Green's function methods
• Holstein model