Magnetothermoelectric transport properties of multiterminal graphene nanoribbons

The Peltier effect and the Ettingshausen effect are investigated in graphene nanoribbons, where charge current produces heat current along the longitudinal direction in the former case, and longitudinal charge current generates transverse heat current in the latter case. With the aid of the nonequilibrium Green's function and the Landauer-B\"uttiker formalism, the Peltier coefficient ${\mathrm{\ensuremath{\Pi}}}_c$ and the Ettingshausen coefficient ${\mathrm{E}}_c$ are obtained. We found that the Kelvin relation is always valid for the longitudinal thermoelectric transport, i.e., ${\mathrm{\ensuremath{\Pi}}}_c=\mathcal{T}{S}_c$, with $\mathcal{T}$ the temperature and ${S}_c$ the Seebeck coefficient. In contrast, for transverse magnetothermoelectric transport, the Kelvin relation breaks down and ${\mathrm{E}}_c\ensuremath{\ne}\mathcal{T}{N}_c$ usually, with ${N}_c$ the Nernst coefficient. In the region of weak magnetic field, the Ettingshausen effect depends strongly on device parameters. When the Fermi energy ${E}_F$ is close to the Dirac point, the Ettingshausen effect of the semiconducting armchair graphene nanoribbon is much stronger than that of the metallic one. When ${E}_F$ is far away from the Dirac point, the Ettingshausen coefficient ${\mathrm{E}}_c$ oscillates around zero. When under a strong magnetic field, ${\mathrm{E}}_c$ is independent of the device parameters and swells only near the Dirac point. Further, the dependence of ${\mathrm{E}}_c$ on ${E}_F$ can be scaled by ${E}_F/{k}_B\mathcal{T}$, with a peak value of $(2ln2){k}_B\mathcal{T}/e$ for the three-terminal system and $(\frac{4}{3}ln2){k}_B\mathcal{T}/e$ for the four-terminal system. We also study the impact of disorder on the Ettingshausen effect. Regardless of the magnetic field strength, ${\mathrm{E}}_c$ is robust against moderate disorder scattering. In addition, in the strong magnetic field, ${\mathrm{E}}_c$ with additional regular oscillating structure can be caused by disorder.