First-principles calculations and theoretical analysis of π plasmon in graphene and graphite: From 2D to 3D

Plasmon excitations in graphite, graphene, and their intermediate states have been investigated using first-principles linear response time-dependent density functional theory (LR-TDDFT) within the random phase approximation (RPA) framework, as well as through theoretical analysis based on the tight-binding model. Our findings reveal that in three-dimensional graphite, the dispersion relation of $\pi$ plasmons follows the pattern ${\omega }_{\pi }\left(q\right)={\omega }_{p,0}+\alpha q^2$ as q approaches zero, where ${\omega }_{p,0}$ represents the exact excitation energy at q=0 and $\alpha$ is a fitting parameter. In contrast, in two-dimensional graphene, the dispersion relation exhibits a distinct characteristic given by ${\omega }_{\pi }\left(q\right)=\sqrt{E_{g,M}^2+\beta q}$, where $E_{g,M}$ is the band gap at the M point in the Brillouin zone and $\beta$ is another fitting parameter. This significant difference can be attributed to the presence or absence of interlayer Coulomb interactions. Furthermore, we conducted an in-depth analysis of how the excitation energies of $\pi$ plasmons vary with interlayer spacing. The results indicate that the excitation energy follows the form ${\omega }_{p,0}=\sqrt{E_{g,M}^2+\gamma /d}$, where d is the distance between two layers and $\gamma$ is a fitting constant. Specifically, as the interlayer spacing in graphite gradually increases, the excitation energy of plasmons progressively decreases, eventually converging towards the value observed in graphene.