## 2 Defect States in Perovskite Materials
Density-functional theory (DFT) is a high-end computational method, a quantum mechanical modelling method that derives properties of molecules and materials based on a determination of the electron density of the molecule, based on functionals of the spatially dependent electron density. Several computational studies on perovskite crystals employing DFT have been carried out in the past few years, which successfully characterized various defects, called point defects [14; 17; 18; 19; 20; 21; 22; 23]. A crystal structure can accommodate several of these point defects which are defects at a single lattice point, not extended in space in any dimension. Here, we distinguish between the following type of point defects that are frequently reported in perovskite material: Vacancies--when one atom leaves its lattice site; Schottky defects--when two oppositely charged ions leave their lattice sites; interstitial defects--when an atom occupies a site in the crystal structure at which there is usually not an atom; substitutional defects--when an impurity atom is incorporated at a regular atomic site in the crystal structure; anti-site defects--when atoms of different type exchange positions. Using DFT, the stability of the defect can be evaluated by calculating the defect formation energy (DFE). The stable charged states of a defect can be obtained by analyzing the DFE as function of the Fermi level. From this the thermodynamic ionization levels between different charged states of a defect (\(q\)/\(q^{\prime}\)) can be calculated, which is defined as the Fermi level where DFEs of two charge states are equal (\(DFE(D^{q})=DFE(D^{q})\)). Negative defects can capture holes (acceptors, \(q^{0}\)/\(q^{-}\)) and positive defects can capture electrons (donors, \(q^{0}\)/\(q^{+}\)). Neutral defects can be ionized by the capture of electrons or holes. The thermodynamic ionization level corresponds to the oxidation and reduction potential of the defect and its position with respect to the VBM and CBM determines whether the defect will act as a (shallow or deep) trap for photogenerated carriers. Figure 7 depicts transition energy levels and the DFEs for point defects in MAPbI\({}_{3}\) calculated by [PERSON] et al. using first-principle calculations [17]. They identified the following defects with low formation energies that can be found near the valence band (acceptors): Iodine interstitials (I\({}_{\rm i}\)), MA substitutional defects (MA\({}_{\rm Pb}\)) and vacancies (V\({}_{\rm MA}\) and V\({}_{\rm Pb}\)). In the conduction band they found that MA\({}_{\rm i}\), Pb\({}_{\rm MA}\), and, V\({}_{\rm I}\) with low formation energies. These defects all have transition energy levels within or close above (under) the valence (conduction) band and are, therefore, considered to be shallow defects. Moreover, the defects with deep transition levels have relatively high formation energies. Figure 6: Proposed mechanism for I\({}_{2}\) formation in MAPbI\({}_{3}\). (**a**) (110) plane of perovskite showing four I\({}^{-}\) ions. (**b**) Iodine at I2 position moves to interstitial I3 position. (**c**) I3 is now surrounded by three iodine ions at approximately same distances. (**d**) This leads I2 to move to the I2A position, creating a bond and producing a I\({}_{2}\) molecule (bond between I3 and I2A). Reproduced with permission from _Nat. Commun._**2017**, \(8\), 15152. Reference [14]. These results suggest that deep level traps are less predominant than defects that create shallow traps. Some defects have transitions where, for example, 2 electrons are involved (Pb\({}_{i}\), (1+/3+)). These defects have low capture cross sections which makes trapping more unlikely. Similar to [PERSON] et al. other theoretical work suggests that Pb\({}^{2+}\), I\({}^{-}\) and MA\({}^{+}\) vacancies are predominant and only create shallow traps [18]. On the other hand, it can be argued that simple trapping and un-trapping of photogenerated charge might not give a complete picture.