2. Consider electron thermoelectric transport along an isolated atomic sheet (such as a graphene sheet that...

 

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2. Consider electron thermoelectric transport along an isolated atomic sheet (such as a graphene sheet that is made of one atomic layer of carbon atoms) at temperature T. We assume that the atomic sheet is a semiconductor. The conduction band dispersion is given by ħ² (k² + k²) E = E + 2m The atomic sheet is doped to n-type such that the electron density per unit area is n. We also assume that electrons in the sheet have a constant relaxation time t, and that the doping is not heavy such that the chemical potential is below the conduction band edge and the Boltzmann statistics is valid, i.e., the factor one in the denominator of the Fermi-Dirac distribution is much smaller than the exponential factor and hence it is negligible. Answer the following questions: (1) Determine an expression for the chemical potential of electrons in terms of given variables. (2) Derive expressions the Seebeck coefficient and the Peltier coefficient and show that the Onsager relation is valid. (3) Derive expressions for the electrical conductivity and electronic contribution to the thermal conductivity. 2. Consider electron thermoelectric transport along an isolated atomic sheet (such as a graphene sheet that is made of one atomic layer of carbon atoms) at temperature T. We assume that the atomic sheet is a semiconductor. The conduction band dispersion is given by ħ² (k² + k²) E = E + 2m The atomic sheet is doped to n-type such that the electron density per unit area is n. We also assume that electrons in the sheet have a constant relaxation time t, and that the doping is not heavy such that the chemical potential is below the conduction band edge and the Boltzmann statistics is valid, i.e., the factor one in the denominator of the Fermi-Dirac distribution is much smaller than the exponential factor and hence it is negligible. Answer the following questions: (1) Determine an expression for the chemical potential of electrons in terms of given variables. (2) Derive expressions the Seebeck coefficient and the Peltier coefficient and show that the Onsager relation is valid. (3) Derive expressions for the electrical conductivity and electronic contribution to the thermal conductivity.

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1 Chemical Potential of Electrons The chemical potential of electrons in the atomic sheet can be expressed as E F 2 k2 k3 2m Where E F is the Fermi en... View the full answer