Nernst and Seebeck effect in a graphene nanoribbon

The thermoelectric power, including the Nernst and Seebeck effects, in graphene nanoribbon is studied. By using the non-equilibrium Green function combining with the tight-binding Hamiltonian, the Nernst and Seebeck coefficients are obtained. Due to the electron-hole symmetry, the Nernst coefficient is an even function of the Fermi energy while the Seebeck coefficient is an odd function regardless of the magnetic field. In the presence of a strong magnetic field, the Nernst and Seebeck coefficients are almost independent of the chirality and width of the nanoribbon, and they show peaks when the Fermi energy crosses the Landau levels. The height of $n$-th (excluding $n=0$) peak is $[\ln2/|n|]$ for the Nernst effect and is $\ln2/n$ for the Seebeck effect. For the zeroth peak, it is abnormal with height $[2\ln2]$ for the Nernst effect and the peak disappears for the Seebeck effect. When the magnetic field is turned off, however, the Nernst effect is absent and only Seebeck effect exists. In this case, the Seebeck coefficient strongly depends on the chirality of the nanoribbon. The peaks are equidistant for the nanoribbons with zigzag edge but are irregularly distributed for the armchair edge. In particular, for the insulating armchair ribbon, the Seebeck coefficient can be very large near the Dirac point. When the magnetic field varies from zero to large values, the differences among the Seebeck coefficients for different chiral ribbons gradually vanish and the nonzero value of Nernst coefficient appears first near the Dirac point then gradually extents to the whole energy region.