The braneworld stability and large-scale correction in graphene like background

In this work, we consider a graphene-like background in braneworld scenario which is one of the interesting models in cosmology and theoretical physics. Indeed, this paper is an application of holography in condense matter. We use the geometric form of potential which help to obtain field equations and solve it to obtain the energy spectrum. In that case we calculate superpotential and energy density of this model. Thermodynamical study of this model suggest that the entropy should modified due to the large-scale corrections. An important point is stability of this model in braneworld scenario. We study model stability by two separated methods. We calculate heat capacity, Gibbs and Helmholtz free energy to analyze thermodynamics stability, then consider small perturbations in graphene like metric background to investigate gravitational stability. We find that the model is completely stable unless in very low temperatures where the graphene structure may break. Hence, we confirm that our braneworld model is thermodynamically stable and also small perturbations yields to an oscillation around the equilibrium point.