Quasi-Fermi level splitting and sub-bandgap absorptivity from semiconductor photoluminescence

A unified model for the direct gap absorption coefficient (band-edge and sub-bandgap) is developed that encompasses the functional forms of the Urbach, Thomas-Fermi, screened Thomas-Fermi, and Franz-Keldysh models of sub-bandgap absorption as specific cases. We combine this model of absorption with an occupation-corrected non-equilibrium Planck law for the spontaneous emission of photons to yield a model of photoluminescence (PL) with broad applicability to band-band photoluminescence from intrinsic, heavily doped, and strongly compensated semiconductors. The utility of the model is that it is amenable to full-spectrum fitting of absolute intensity PL data and yields: (1) the quasi-Fermi level splitting, (2) the local lattice temperature, (3) the direct bandgap, (4) the functional form of the sub-bandgap absorption, and (5) the energy broadening parameter (Urbach energy, magnitude of potential fluctuations, etc.). The accuracy of the model is demonstrated by fitting the room temperature PL spectrum of GaAs. It is then applied to Cu(In,Ga)(S,Se)2 (CIGSSe) and Cu2ZnSn(S,Se)4 (CZTSSe) to reveal the nature of their tail states. For GaAs, the model fit is excellent, and fitted parameters match literature values for the bandgap (1.42 eV), functional form of the sub-bandgap states (purely Urbach in nature), and energy broadening parameter (Urbach energy of 9.4 meV). For CIGSSe and CZTSSe, the model fits yield quasi-Fermi leveling splittings that match well with the open circuit voltages measured on devices made from the same materials and bandgaps that match well with those extracted from EQE measurements on the devices. The power of the exponential decay of the absorption coefficient into the bandgap is found to be in the range of 1.2 to 1.6, suggesting that tunneling in the presence of local electrostatic potential fluctuations is a dominant factor contributing to the sub-bandgap absorption by either purely electrostatic (screened Thomas-Fermi) or a photon-assisted tunneling mechanism (Franz-Keldysh). A Gaussian distribution of bandgaps (local Eg fluctuation) is found to be inconsistent with the data. The sub-bandgap absorption of the CZTSSe absorber is found to be larger than that for CIGSSe for materials that yield roughly equivalent photovoltaic devices (8% efficient). Further, it is shown that fitting only portions of the PL spectrum (e.g., low energy for energy broadening parameter and high energy for quasi-Fermi level splitting) may lead to significant errors for materials with substantial sub-bandgap absorption and emission.