a Evolution of the bound exciton gas in a bulk semiconductor with increasing carrier density. Two scenarios are possible: (i) bound excitons are ionized into an e–h plasma and the Mott transition to a metallic state takes place; (ii), (iii) e–h correlations still persist in the form of Mahan excitons, i.e. bound states in the Fermi sea in a (ii) chemically-doped and (iii) photodoped semiconductor. EF indicates the Fermi energy; EF,c and EF,v represent the quasi-Fermi energies of the conduction band and valence band, respectively. b Schematic representation of the optical absorption spectrum of a bulk semiconductor in the presence of Wannier excitons (black curve), and its modification at high carrier densities. The Mott transition manifests itself with the ionization of the Wannier exciton (blue curve), whereas the Mahan exciton scenario features the persistence of the Wannier peak and the enhancement of the absorption continuum (red curve). c Absorption spectrum of CH3NH3PbBr3 single crystals as obtained from the ellipsometry data (dots), fitted with Elliott theory (solid line) and resulting in a binding energy Eb = 71 meV, linewidth Γ = 34 meV, and single-particle gap energy Eg = 2.42 eV. The blue and red dotted lines represent the distinct contributions of the Wannier exciton and the continuum, respectively. d Exciton ionization ratio as a function of the excitation density, where nfree∕n = 0 corresponds to an exciton gas and nfree∕n = 1 to a fully ionized plasma, as calculated from the theory of ionization equilibrium (TIE, red dots). The vertical line indicates the Mott critical density, found at nM ~ 8 × 1017 cm−3. The solid line represents the ionization ratio calculated with the Saha equation, and it is added for comparison.