D3/D7 holographic Gauge theory and Chemical potential

N=2 supersymmetric Yang-Mills theory with flavor hypermultiplets at finite temperature and in the dS${}_4$ are studied for finite quark number density ($n_b$) by a dual supergravity background with non-trivial dilaton and axion. The quarks and its number density $n_b$ are introduced by embedding a probe D7 brane. We find a critical value of the chemical potential at the limit of $n_b=0$, and it coincides with the effective quark mass given in each theory for $n_b=0$. At this point, a transition of the D7 embedding configurations occurs between their two typical ones. The phase diagrams of this transition are shown in the plane of chemical potential versus temperature and cosmological constant for YM theory at finite temperature and in dS${}_4$ respectively. In this phase transition, the order parameter is considered as $n_b$. % and the critical value of the chemical potential This result seems to be reasonable since both theories are in the quark deconfinement phase.