Dirac fermions duality in graphene: Ripples and fractional dimensions as function of temperature

Graphene consists of coupled direct/dual fermionic sub-systems and, consequently, the thermal properties of both are intrinsically correlated. The dual is characterized by negative temperatures, and its free energy keeps opposite sign concerning the direct. The growth of ripples in graphene becomes related to temperature rises with fractional spatial dimension ~2.19 at 300 °K. An analytical, and suitable, expression for ripples dimension as a function of temperature is presented. Further, internal energy, entropy, specific heat and free energy are evaluated as a function of temperature and dimension for both sub-systems. Free energy supports a simple, functional expression inversely proportional to ripples dimension.