Thermodynamics and Phase Transitions of Nonlinear Electrodynamics Black Holes in an Extended Phase Space

We investigate the thermodynamic behavior of nonlinear electrodynamics (NLED) black holes in an extended phase space, which includes the cosmological constant $\Lambda=-3/l^{2}$ and dimensionful couplings $a$ in NLED as thermodynamic variables. For a generic NLED black hole with the charge $Q$, we find that the Smarr relation is satisfied in the extended phase space, and the state of equation can be written as $Tl=\tilde{T}\left( r_{+}/l,Q/l,al^{-c} \right) $, where $\left[ a\right] =L^{c}$, and $T$ and $r_{+}$ are the temperature and horizon radius of the black hole, respectively. For some values of $Q/l$ and $al^{-c}$, the phase structure of the black hole is uniquely determined. Focusing on Born-Infeld and iBorn-Infeld AdS black holes, we obtain the corresponding phase diagrams in the $a/l^{2}$-$Q/l$ plane, which provides a new viewpoint towards the black holes' phase structure and critical behavior. For Born-Infeld black holes, the critical line and the region, where a reentrant phase transition occurs, in the $a/l^{2}$-$Q/l$ plane are both finite and terminate at $\left\{ \tilde{a}_{c}\text{, }\tilde{Q}_{c}\right\} \simeq\left\{ 0.069\text{, }0.37\right\} $. However for iBorn-Infeld black holes, the critical line and the reentrant phase transition region in the $a/l^{2}$-$Q/l$ plane are semi-infinite and extend to $Q/l=\infty$. We also examine thermal and electrical stabilities of Born-Infeld and iBorn-Infeld AdS black holes.