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

The experimental work [J. Crossno et al., Science 351, 1058 (2016)] that 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.

Figure 1

Longitudinal and transverse Lorenz ratio for MLG as a function of $T$ for various choices of $\mathrm{\Delta }, n$, and $B_z$. The scattering parameters in the cases of $\mathrm{\Delta }=0\mathrm{K}$ (first column) and $\mathrm{\Delta }=60\mathrm{K}$ (second column) are $A/C=2.78\times {10}^{-5}{\mathrm{K}}^{-2}, B/C=4.63\times {10}^{-6}{\mathrm{K}}^{-3}$, and in the case of $\mathrm{\Delta }=600\mathrm{K}$ (third column) are $A/C=2.78\times {10}^{-9}{\mathrm{K}}^{-2}, B/C=4.63\times {10}^{-12}{\mathrm{K}}^{-3}$. The unit of $B_z$ is $0.2C{e}^{-1}{v}_F^{-2}$.

Figure 2

Longitudinal and transverse Lorenz ratio for MLG as a function of $n$ for various choices of $\mathrm{\Delta }, T$, and $B_z$. The scattering parameters and the unit of $B_z$ are the same as in Fig. 1.

Figure 3

Longitudinal and transverse Lorenz ratio for BLG as a function of $T$ for various choices of $\mathrm{\Delta }, n$, and $B_z$. The scattering parameters in the cases of $\mathrm{\Delta }=0\mathrm{K}$ (first column) and $\mathrm{\Delta }=60\mathrm{K}$ (second column) are $A/C=0.0167{\mathrm{K}}^{-1}, B/C=2.78\times {10}^{-3}{\mathrm{K}}^{-2}$, and in the case of $\mathrm{\Delta }=600\mathrm{K}$ (third column) are $A/C=1.67\times {10}^{-6}{\mathrm{K}}^{-1}, B/C=2.78\times {10}^{-9}{\mathrm{K}}^{-2}$. The unit of $B_z$ in the cases of $\mathrm{\Delta }=0,60\mathrm{K}$ is $80C{e}^{-1}{v}_F^{-2}$ and in the case of $\mathrm{\Delta }=600\mathrm{K}$ is $2C{e}^{-1}{v}_F^{-2}$.

Figure 4

Longitudinal and transverse Lorenz ratio for MLG as a function of $n$ for various choices of $\mathrm{\Delta }, T$, and $B_z$. The scattering parameters and the unit of $B_z$ are the same as in Fig. 3.

Figure 5

Longitudinal and transverse Lorenz ratio for MLG (left column) and BLG (right column) as a function of $B_z$ at $\mathrm{\Delta }=60\mathrm{K}, T=40\mathrm{K}, n={10}^9{\mathrm{cm}}^{-2}$. The unit of $B_z$ is $C{e}^{-1}{v}_F^{-2}$ for MLG and $80C{e}^{-1}{v}_F^{-2}$ for BLG. The scattering parameters are the same as in the corresponding cases in Figs. 1 and 3.