We investigate numerically the dynamics and statistics of inertial particles transported by stratified turbulence, in the case of particle density intermediate in the average density profile of the fluid. Under these conditions, particles tend to form a thin layer around the corresponding fluid isopycnal. The thickness of the resulting layer is determined by a balance between buoyancy (which attracts the particle to the isopycnal) and inertia (which prevents them from following it exactly). By means of extensive numerical simulations, we explore the parameter space of the system and we find that in a range of parameters particles form fractal clusters within the layer.