Derivative relations between electrical and thermoelectric quantum transport coefficients in graphene

We find that the empirical relation between the longitudinal and Hall resistivities (i.e. $R_{xx}$ and $R_{xy})$ and its counterpart between the Seebeck and Nernst coefficients (i.e. $S_{xx}$ and $S_{xy})$, both originally discovered in conventional two-dimensional electron gases [1,2], hold surprisingly well for graphene in the quantum transport regime except near the Dirac point. These empirical relations can be described by the following equations: \[ R_{xx} =\alpha _r \cdot \frac{B}{n}\frac{dR_{xy} }{dB}, \quad S_{yx} =\alpha _s \cdot \frac{B}{n}\frac{dS_{xx} }{dB} \] Here R and S are electrical resistivity and thermoelectric conductivity tensor respectively. The validity of the relations is cross-examined by independently varying the magnetic field and the carrier density in graphene. We demonstrate that the pre-factor, \textit{$\alpha $}$_{s}$, does not depend on carrier density in graphene. By tuning the carrier mobility therefore the degree of disorders, we find that the pre-factor stays unchanged. Our experimental results validate both derivative relations for massless Dirac fermions except near the Dirac point. \\[4pt] [1] A. M. Chang and D. C. Tsui, Solid State Commun. \textbf{56}, 153 (1985).\\[0pt] [2] B. Tieke et al, Phys. Rev. Lett. \textbf{78}, 4621 (1997).