  To understand the essential properties of Dirac crystals, such as their
thermal conductivity, we require models that consider the interaction between
Dirac electrons and dispersive acoustic phonons. The exceptionally high thermal
conductivity in 2D Dirac crystals is attributed to near-ideal phonon quantum
gases, while undesired limitations arise from electron-phonon (e-ph)
interactions which have been shown to limit the thermal conductivity up to
several microns away. The e-ph thermal conductivity is directly linked to the
phonon scattering rate. Conventional calculations overlook phonons with
short-dispersive wavelengths, rendering them inadequate for analyzing 2D Dirac
crystals. The phonon scattering rate is typically calculated up to the
first-order magnitude, considering 3-particle interactions involving the decay
of an electron and phonon (EP-E*) to create a new electron. However, processes
involving the decay of an electron and the creation of a new electron and
phonon (E-E*P*) are neglected. In this study, we present an accurate expression
for the phonon scattering rate and e-ph thermal conductivity in 2D Dirac
crystals, accounting for short-dispersive wavelength phonons. We demonstrate
the significance of the E-E*P* process even at room temperature in calculating
the phonon scattering rate and e-ph thermal conductivity, particularly for
first-order e-ph interactions. Furthermore, we emphasize the importance of
incorporating second-order e-ph interactions, specifically the EP-E*P*
interaction involving the decay of an electron and phonon and the creation of a
new electron-phonon pair, to accurately determine the phonon scattering rate
and e-ph thermal conductivity at high temperatures and low Fermi energies. This
4-particle interaction process plays a crucial role in characterizing these
properties effectively.
