A universal exponential factor in the dimensional crossover from graphene to graphite

TL;DR

This paper reveals a universal exponential factor, γ_c=π/2, governing the transition from two-dimensional graphene to three-dimensional graphite, supported by experimental data and phonon spectrum analysis, linking it to a Helmholtz-like equation.

Contribution

It identifies a universal exponential factor in the dimensional crossover of FLG, supported by experimental data and theoretical phonon spectrum analysis, connecting it to a fundamental equation.

Findings

Universal exponential factor γ_c=π/2 identified.

The same exponential factor appears in phonon mode crossover.

The exponential factor relates to a Helmholtz-like molal equation.

Abstract

A universal exponential factor, $\gamma_{c}=\pi/2$, is disclosed for the dimensional crossover of few-layer graphene (FLG) from two-dimensional graphene to three-dimensional graphite. $\gamma_{c}$ is found by analyzing available experimental data on different properties of FLG with varying thickness. A theoretical study on the phonon spectrum of the vertical acoustic mode in FLG is carried out to further check this exponential factor $\gamma_{c}$. Interestingly, the same exponential factor appears in the dimensional crossover of the phonon mode. It turns out that the exponential factor $\gamma_{c}$ is related to the homogeneous Helmholtz-like molal equation in the mass transfer with a first order chemical reaction. The finding should provide valuable information for experimentalists and theorists in the future investigation on thickness dependent properties of FLG.