  The experimental work [J. Crossno et al., Science 351, 1058 (2016)], which
reported the violation of the Wiedemann-Franz law in monolayer graphene
characterized by a sharp peak of the Lorenz ratio at a finite temperature, has
not been fully explained. Our previous work [Y.-T. Tu and S. Das Sarma, Phys.
Rev. B 107, 085401 (2023)] provided a possible explanation through a
Boltzmann-transport model with bipolar diffusion and an energy gap possibly
induced by the substrate. In this paper, we extend our calculation to include a
weak magnetic field perpendicular to the graphene layer, which is
experimentally relevant, and may shed light on the possible violation or not of
the Wiedemann-Franz law. We find that the magnetic field enhances the size of
the peak of the Lorenz ratio but has little effect on its position, and that
the transverse component of the Lorenz ratio can be either positive or negative
depending on the parameter regime. In addition, we do the same calculation for
bilayer graphene in the presence of a magnetic field and show the qualitative
similarity with monolayer graphene. Our work should motivate
magnetic-field-dependent experiments elucidating the nature of the charge
carriers in graphene layers.
