Direct/Dual electronic thermal conductivity on graphene: Gate-potential and ripples

A functional relationship allows to connect the thermodynamic properties between the Direct and Dual bands of a graphene sheet. For low temperature, analytical expressions for internal energy, heat capacity and entropy are obtained when a positive gate potential is considered. Through a numerical study, it is shown that thermal conductivity decreases in both branches when the gate potential is applied, whereas ripples increase electronic thermal conductivity.