We consider the polaron dynamics driven by Froḧlich type, long wavelength dominated electron-phonon interaction at zero temperature, for three different semi-metals: single and bilayer graphene, and semi-Dirac, all grown on polar substrates such as, SiC. Single layer graphene (henceforth called SL graphene), bilayer graphene (henceforth called BL graphene), and semi-Dirac have two dimensional band-structures with point Fermi surfaces in their natural undoped conditions. When these materials are grown on polar substrates, their electrons can interact with the optical phonons (LO) at the surface of the substrates. That gives rise to the possibility of polaron formation in the context of these semi-metals, although they themselves are non-polar. Starting from the Froḧlich type electron-phonon interaction Hamiltonian, perturbation theory is employed to calculate the self energy of the electron due to polaron formation for the three aforementioned systems. The electron self energy, or the polaron energy, calculated analytically for BL graphene, is shown to vary linearly with the electron momentum for small electron momenta. Whereas for ordinary polar crystals (both two and three dimensional), for small electron momentum, the polaron energy is quadratic leading to the mass correction of the electron, for BL graphene the polaron energy disperses linearly, rendering the massive BL graphene electrons effectively massless. Energies for Froḧlich polarons in SL graphene and semi-Dirac on polar substrates, are numerically evaluated. Also, the electron relaxation rate, related to the imaginary part of the analytically continued electron self energy expression, is calculated for the three systems.

我们考虑由Froḧlich类型驱动的极化子动力学，零波长下长波长主导的电子-声子相互作用，对于三种不同的半金属：单层和双层石墨烯以及半狄拉克，它们都生长在极性衬底（例如SiC）上。单层石墨烯（以下称为SL石墨烯），双层石墨烯（以下称为BL石墨烯）和半狄拉克晶体具有二维带状结构，在其自然未掺杂条件下其点为费米面。当这些材料在极性基板上生长时，它们的电子可以与基板表面的光子（LO）相互作用。尽管它们本身是非极性的，但在这些半金属的情况下，这可能会形成极化子。从Froḧlich型电子-声子相互作用哈密顿量开始，采用扰动理论来计算上述三个系统由于极化子形成而产生的电子自能。通过分析BL石墨烯计算出的电子自能量或极化子能量，对于小电子动量，图示随电子动量线性变化。而对于普通的极性晶体（二维和三维），对于较小的电子动量，极化子能量是二次方的，从而导致电子的质量校正；对于BL石墨烯，极化子能量会线性分散，从而使块状BL石墨烯电子有效地无质量。数值评估了SL石墨烯和Semi-Dirac中Froonslich极化子的能量。同样，针对这三个系统，计算出与弛豫连续的电子自能表达式的虚部相关的电子弛豫率。对于BL石墨烯，极化子能量线性分散，从而使大量BL石墨烯电子有效地无质量。数值评估了SL石墨烯和Semi-Dirac中Froonslich极化子的能量。同样，对于这三个系统，计算出与弛豫连续的电子自能表达式的虚部相关的电子弛豫率。对于BL石墨烯，极化子能量线性分散，从而使大量BL石墨烯电子有效地无质量。数值评估了SL石墨烯和Semi-Dirac中Froonslich极化子的能量。同样，针对这三个系统，计算出与弛豫连续的电子自能表达式的虚部相关的电子弛豫率。