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The experimental work [J. Crossno et al., Science 351, 1058 (2016)], whichreported the violation of the Wiedemann-Franz law in monolayer graphenecharacterized by a sharp peak of the Lorenz ratio at a finite temperature, hasnot been fully explained. Our previous work [Y.-T. Tu and S. Das Sarma, Phys.Rev. B 107, 085401 (2023)] provided a possible explanation through aBoltzmann-transport model with bipolar diffusion and an energy gap possiblyinduced by the substrate. In this paper, we extend our calculation to include aweak magnetic field perpendicular to the graphene layer, which isexperimentally relevant, and may shed light on the possible violation or not ofthe Wiedemann-Franz law. We find that the magnetic field enhances the size ofthe peak of the Lorenz ratio but has little effect on its position, and thatthe transverse component of the Lorenz ratio can be either positive or negativedepending on the parameter regime. In addition, we do the same calculation forbilayer graphene in the presence of a magnetic field and show the qualitativesimilarity with monolayer graphene. Our work should motivatemagnetic-field-dependent experiments elucidating the nature of the chargecarriers in graphene layers.

Wiedemann-Franz law in graphene in the presence of a weak magnetic field