## 1 Introduction: Recombination Processes in Perovskites
Non-radiative recombination will create drainage for the recombination of excess free charge carriers and reduces the steady-state charge density, which reduces the splitting of quasi-Fermi levels and thus decreases \(V_{OC}\). The relationship between the V  
Figure 1: Representation of the typical components of lead perovskites and the typical perovskite crystal structure, viewed in three different ways. ABX\({}_{3}\) with A components being methylammonium (MA), formamidinium (FA) or Cs(I) or Rb(I), the B = Pb(II). The X can be iodide (I\({}^{-}\)), bromide or chloride. Note that the octahedral coordination around Pb(II) can be displayed centrally (**a**) or the MA can be shown centrally (**b**). In (**c**) the square unit cell is shown with MA in the center. Reproduced (and adapted) with permission from _Energy Environ. Sci._**2017**, _10(3)_, 710–727, Reference [2]. and recombination can be described by the external photoluminescence quantum efficiency (_PLQE_), a measure for the fraction of radiative versus non-radiative decay (value between 0 and 1):  
\[V_{OC}^{max}=V_{OC}^{rad}-\Delta V_{OC}^{non-rad}=V_{OC}^{rad}+\frac{K_{B}T}{q} \ln(PLQE)\]  
where \(V_{OC}^{rad}\) is the radiative limit of the open circuit voltage when non-radiative (_non-rad_) recombination is fully suppressed, \(K_{B}\) is the Boltzmann constant, \(T\) is the temperature, and \(q\) is the elementary charge. \(\Delta V_{OC}^{non-rad}\) is the non-radiative open circuit voltage loss, the loss of voltage due to non-radiative processes. \(V_{OC}^{max}\) is the maximum attainable voltage. Moreover, recombination at surface and at grain boundaries also influences the charge extraction efficiency and could have an effect on the current collection, which reduces the short-circuit current (\(J_{SC}\)). The voltage depends on the splitting of the electron and hole quasi-Fermi levels and therefore on the excess electron and hole carrier densities \(\Delta n\) and \(\Delta p\), respectively (Figure 2). These densities depend on free charge-carrier lifetime (\(\tau\)), the time that electrons (holes) in the CB (VB) remain free, which is reduced by recombination. We now describe the three most important recombination mechanisms [3]. In radiative recombination (or direct recombination), an electron from the conduction band combines with a hole from the valence band. This mechanism has a higher probability for indirect band gap materials where there is no change in momentum required upon recombination or excitation from valence to conduction band. For direct recombination, we can assume that the rate (\(R_{md}\)) depends on the concentration of electrons (\(n\)) in the conduction band and holes (\(p\)) in the valence band:  
\(R_{rad}=k_{r}\times n\times p\). Recombination can also be facilitated by impurities or lattice defects in Shockley-Read-Hall (SRH) recombination. These defects in the semiconductor structure introduce so-called trap states that can have energies that lie within the band gap. An electron (or hole) can be trapped at the recombination center and thereupon recombine with a hole (or electron) that is attracted. This is a non-radiative process and heat created during the recombination is dissipated into the lattice in the form of vibrations. The rate of SRH recombination is monomolecular: \(R_{SRH}=k_{S}\times n\). When there are three particles involved in the recombination process, this is called Auger recombination [3]. Here, the recombining electron and hole transfer the energy and momentum to another electron or hole. If the energy is transferred to an electron, it is excited into higher energy levels in the conduction band. When it relaxes, the energy of the electron is transferred to vibrational energy of the lattice (phonon modes). A hole will be excited deeper into the valence band before it relaxes back. The rate equation can be expressed as follows: \(R_{auger}=k_{A}\times n^{2}\times p\). Figure 2: Electron and hole energetics and recombination.