Giant Huang–Rhys Factor for Electron Capture by the Iodine Intersitial in Perovskite Solar Cells

Improvement in the optoelectronic performance of halide perovskite semiconductors requires the identification and suppression of nonradiative carrier trapping processes. The iodine interstitial has been established as a deep level defect and implicated as an active recombination center. We analyze the quantum mechanics of carrier trapping. Fast and irreversible electron capture by the neutral iodine interstitial is found. The effective Huang–Rhys factor exceeds 300, indicative of the strong electron–phonon coupling that is possible in soft semiconductors. The accepting phonon mode has a frequency of 53 cm–1 and has an associated electron capture coefficient of 1 × 10–10 cm3 s–1. The inverse participation ratio is used to quantify the localization of phonon modes associated with the transition. We infer that suppression of octahedral rotations is an important factor to enhance defect tolerance.

• a Jupyter Notebook with additional analysis of the harmonic phonon data Further details of the calculation procedure for defect properties Geometry relaxation procedure: So that no preference was given to a particular combination of octahedral tilts the starting point for atomic relaxation was MAPI in the pseudocubic phase. Upon adding an iodine interstitial the structure was found to have a number of local minima and multiple relaxations between charge states were required to reach a global minimum. To calculate the collective atomic displacement (∆Q) between charge states accurately the starting point for relaxation of the charged states must be the neutral charge state geometry. The atomic relaxation procedure is outlined in Figure S1. Figure S1: Atomic relaxation procedure for point defects in hybrid halide perovskites in the pseudo-cubic phase. IP indicates that the defect is lying in the ab-plane, and OP indicates that the defect is lying along the c-axis. The lowest energy structures are in a dash-line box. The hybrid halide perovskite structure has a number of local minima and multiple relaxations were required to break symmetry and reach a global minimum. For the higher energy defect structures the energy above the global minimum for that charge state is given.
where E d (q) is the total energy of the supercell, E b is the total energy of the pristine lattice, µ i is the chemical potential of species i and n i is the number of atoms that are added or removed. E d (q), E b and µ i were calculated using DFT, as outlined in the previous section.
The defect correction E corr consists of two terms -the image charge correction (calculated using Ref. S3 ) for charged defects and a tilting correction that is specific to pseudo-cubic perovskites.
The static dielectric constant of MAPI is large (¯ 0 = 22.67) S4 and so the lattice can S-3 effectively screen charged defects, in addition the point defects in this study have a maximum charge of one and the supercell is relatively large (193 atoms). These factors lead to a small image charge correction of −0.057 eV.
All of the defect formation energies are referenced to the minimum energy of the pristine lattice. However the geometry of the pristine lattice in the pseudo-cubic phase corresponds to a time average and is not the minimum energy structure ( Figure S2). To calculate the minimum energy of the lattice the pseudo-cubic structure can be distorted along the soft mode at the R-point in q-space. This 'modemapping' procedure S5 is not compatible with the supercell expansion used for the 197-atom supercell, but the correction energy can be inferred from modemapping other supercell expansions ( Table S2).
Note that in our model we do not consider possible finite-temperature fluctuations in the trap energy. In the case of the bromine vacancy in hybrid perovskite this leads to shifts in trap energy up to 1 eV. S6 Table S2: Tilting corrections for supercell expansions of the pseudo-cubic perovskite lattice. The 768-atom supercell is eight times larger than the 96-atom supercell, so we expect the tilting correction to be eight times larger. It is calculated to be 7.5 times larger. The tilting correction for the 192-atom expansion, which cannot be calculated directly, is twice as large as the correction for the 96-atom supercell.
supercell expansion # atoms tilting correction (meV) Charge transition levels: The calculated charge transition levels are given in Figure   S3. As is found in the previous literature S1, S2 there is negative-U behaviour, where the (+/0) transition level is higher in energy than the (0/-) transition. S-4 Figure S2: A schematic of the tilting correction. The green line is a double well potential energy surface that is typical of pseudo-cubic perovskite structures. The tilting correction is only needed when using a high-symmetry pseudo-cubic perovskite phase to calculate the pristine bulk energy.
Carrier capture rate: The procedure follows static coupling theory as implemented by Alkauskas et al. S7 and recently extended to anharmonic potential energy surfaces by Kim et al. in the carriercapture.jl package. S8,S9 The electron capture coefficient C n determines the electron capture rate R n at a neutral defect via the equation where N 0 is the neutral defect density and n is the electron density. The capture coefficient is derived by considering electron phonon coupling at first order and using a one-dimensional approximation so that the problem reduces to a single phonon mode Q. As outlined in the main text, for electron capture from an initial state i to a final state f, the carrier capture S-5 Figure S3: Charge transition levels of the iodine interstitial defect in MAPI, calculated using the hybrid HSE06 funtional with spin-orbit coupling. This assumes iodine rich conditions, a crystal in equilibrium with I 2 (g), where interstitial formation is energetically favoured. IP indicates that the defect is lying in the ab-plane and OP indicates that the defect is lying along the c-axis. coefficient is given by where V is the supercell volume, g is the energetic degeneracy of the final state, W if is the electron-phonon coupling matrix element, χ im |Q − Q 0 |χ f n is the overlap of the vibrational wavefunctions χ and the Dirac δ(∆E + mhω i − nhω f ) ensures that there is conservation of energy. In practice the Dirac δ term is replaced by a smearing function; for the calculations in this study this is a gaussian function of width 0.01 eV. Θ m in Equation S3 is the thermal S-6 Figure S4: Charge transition levels of the higher energy iodine interstitial defects in MAPI. The defect geometries correspond to those in the solid boxes in Figure S1. Defects with a lower energy were later found and the charge transition diagram for these are reported above.
occupation of the vibrational state m: The anharmonic potential energy surface was generated from fitting a spline of order four, as implemented in the dierckx.jl package, S10 to the DFT total energies. We consider electron capture from the neutral charge state to the negative charge state which has a singly energy-degenerate geometry.
The 1D Schrödinger equation for the potential energy surface was solved using a finite difference method implemented in the brooglie package S11 to give the vibrational wave-S-7 functions. Using the one-dimensional approximation discussed above, the electron-phonon coupling can be described with a single matrix element: S7 where Ψ {i,f } are many-electron wavefunctions andĤ is the many-body Hamiltonian of the system. We assume that the many-body Hamiltonian and many-electron wavefunctions can be replaced by their single particle counterparts, so that the electron-phonon coupling matrix element is given by: S7 where the single particle wavefunction of the initial (final) charge state is given by ψ i (ψ f ) and has an eigenstate energy of i ( f ). The pawpyseed package S12 was used to derive the orthogonal wavefunctions from the pseudo wavefunctions and perform the overlap integrals in real space. Further details of the methodology can be found in the literature. S7 There are several approximations built into the methodology outlined above: • Static coupling approximation. This assumes that the timescales of the carrier capture process are longer than the phonon lifetimes and periods of lattice vibrations. For this system the iodine interstitial in the neutral charge state has an effective frequency of 1.4 THz, corresponding to a lattice vibration period of the order 10 −12 s. Using our calculated capture coefficient of 1 × 10 −10 cm 3 s −1 , and assuming an electron density of 1 × 10 15 cm −3 (typical for a solar cell under one sun illumination) gives a carrier capture timescale of 1 × 10 −5 s (at a single point defect).
• The description of electron-phonon coupling to first order in perturbation theory. A perturbative treatment of the charge transition is valid for electron-phonon coupling below a threshold value. With reference to previous results in the literature S7 the value calculated in this work, 0.0036 eVamu -1/2Å -1 , is within that range.
• Linear coupling approximation. In this approximation the electron-phonon coupling S-8 term is Taylor expanded in Q and only the first order terms are retained.
• Single mode approximation. This relates to the use of a single configuration coordinate Q. Please see the following section for further discussion in the context of this work.

S-9
Additional analysis of the carrier capture rate Electron-phonon coupling term: The inner product of the wavefunctions at the conduction band minimum φ i and unoccupied defect level φ f are calculated at various Q-values and substituted into Equation S3 to give a value of 0.0036 eVamu -1/2Å -1 for the electron-phonon coupling matrix element (as reported in the main text).
It is possible to estimate the electron-phonon coupling term using an estimate for the degree of localisation of φ i and φ f . For strong electron phonon coupling W if corresponds to S7 where M b is the number of atoms in the supercell and M d is the number of atoms the defect state is localised around. For our system W if ≈ 0.0048 and so electron-phonon coupling for the single mode Q is strong.
Classical barrier vs tunnelling: As shown in Figure S5, there is significan quantum tunnelling before the barrier, in both the harmonic and anharmonic case. Figure S5: Whether a (a) harmonic or (b) anharmonic description of the PES is used, electron capture by MAPbI 3 :I i is not fully classical as phonon overlap m O im;tn persists before the classical barrier E b , indicated by a red vertical line.

S-10
Additional analysis of defect lattice geometries As discussed in the main text, analysis of the defect lattice geometries in S6 show that the large lattice relaxation after charge capture is associated with rotations of the inorganic octahedral cage. Figure S6: For pristine MAPI in the pseudo-cubic phase(top figure) there are three distinct Pb-I-Pb bond angles. Upon introduction of the neutral iodine interstitial (middle figure) the lattice becomes more disordered and includes a wider distribution of bond angles. After charge capture a more symmetric structure is found ( Figure 3 in the main text) and the bond angles shift to form a bimodal distribution.