Impact of metastable defect structures on carrier recombination in solar cells

The efficiency of a solar cell is often limited by electron–hole recombination mediated by defect states within the band gap of the photovoltaic (PV) semiconductor. The Shockley–Read–Hall (SRH) model considers a static trap that can successively capture electrons and holes. In reality however, true trap levels vary with both the defect charge state and local structure. Here we consider the role of metastable structural configurations in capturing electrons and holes, taking the tellurium interstitial in CdTe as an illustrative example. Consideration of the defect dynamics, and symmetry-breaking, changes the qualitative behaviour and activates new pathways for carrier capture. Our results reveal the potential importance of metastable defect structures in non-radiative recombination, in particular for semiconductors with anharmonic/ionic–covalent bonding, multinary compositions, low crystal symmetries or highly-mobile defects.

where E D,q is the calculated energy of the defect supercell and E H is the energy of an equivalent pristine host supercell. The second term ( Â i n i µ i ) accounts for the thermodynamic cost of exchanging n i atoms with their reservoir chemical potential(s) µ i , to form the defect D q from the ideal bulk material. Similarly, qE F represents the electron chemical potential contribution, while E Corr (q) is a correction for any finite-size supercell effects.
Defect thermodynamic charge transition levels e(q 1 /q 2 ) are defined as the Fermi level position for which the formation energies of the q 1 and q 2 charge defects are equal: 2 For a Fermi level position above this defect transition level, the more negative charge state will be favoured, whereas the positive state will be favoured for a lower lying Fermi level. Due to the charge-dependent relationship between defect formation energy and Fermi level energy (DH D,q (E F ) µ qE F ; Eq. (S1)), an expression for the charge transition level e(q 1 /q 2 ) can be derived following: Setting DH D, q 1 equal to DH D, q 2 to solve for E F (i.e. the Fermi level position for which the defect formation energies are equal := e(q 1 /q 2 )): where DH D, q (E F = 0) is the formation energy of defect D with charge q (defined in Eq. (S1)), when the Fermi level is at the zero-reference point (the VBM, by convention).
The shifts in charge transition level positions when metastable defect structures are involved are derived in Sections S1.1 to S1.3 and illustrated in Fig. S1.

S1.1 Charge Capture Into Metastable Defects; D q ! D ⇤q±1
Taking q 1 = q and q 2 = q ± 1 (as charge capture is a single-carrier process) and denoting the energy of the metastable defect state DH D ⇤ , q±1 as: Electronic Supplementary Material (ESI) for Faraday Discussions. This journal is © The Royal Society of Chemistry 2022 where DE is the energy of the metastable defect relative to the ground-state structure, the charge transition level e(q 1 /q ⇤ 2 ) can thus be written: Thus we witness that for charge capture into a metastable structure, D q ! D ⇤q±1 , the transition level will move an energy DE closer to the corresponding band-edge (i.e. to higher energy for electron capture or to lower energy for hole capture), assuming a transition level e(q/q ± 1) initially located within the bandgap.
Given the requirement that e(q/q ⇤ ± 1) must lie within the bandgap in order to effectuate non-radiative recombination, we have the constraint: Thus for electron capture we have: e(q/q 1) + DE < E g (S16) ! DE < E g e(q/q 1) (S17) and for hole capture: Therefore, we see that the energetic separation between the ground-state transition level e(q/q ± 1) and the corresponding band-edge sets the energy window within which the metastable structure must lie in order to affect the electron-hole recombination process (as demonstrated in Fig. S1).

S1.2 Charge Capture From Metastable Defects; D ⇤q ! D q±1
In the case of charge capture from metastable structures, the same equations from the previous section hold, just with a reversal of the sign of the DE term: Hence for charge capture from a metastable structure, D ⇤q ! D q±1 , the transition level will move an energy DE further from the corresponding band-edge, assuming a transition level e(q/q ± 1) initially located within the bandgap.
Likewise, we obtain the following reversed constraints on the relative energy of the metastable structure, to impact carrier recombination:

S1.3 Charge Capture Between Two Metastable Defect Structures; D ⇤q ! D ⇤q±1
As demonstrated in Fig. S1, the positions of defect charge transition levels involving metastable configurations for both charge states are dictated by the balance of the relative energies of both metastable structures D ⇤q(±1) with respect to their ground-state counterparts D q(±1) , such that: with the same constraint that e(q ⇤ /q ⇤ ± 1) must lie within the bandgap in order for recombination to be energetically permitted.

S3 Te i in CdTe: Transition State Theory
To estimate the rate of internal structural transformation for point defects, we can invoke Transition State Theory, 7 which gives the rate of reaction k as: where n is the effective (vibrational) attempt frequency, g is the ratio of the degeneracies of the final and initial states and DE is the activation energy barrier.

S3.1 Internal Conversion vs Charge Capture Rates
To estimate the transition barrier height at which internal conversion and charge capture would have comparable speeds, we equate their reaction rates: where s n is the capture cross-section, v th is the carrier thermal velocity and n is the minority carrier concentration (minority carrier capture often represents the rate-limiting step in the electron-hole recombination cycle). This is then rearranged to: Typical minority carrier concentrations are n ⇠ 1 ⇥ 10 14 cm 3 , with velocities v th ⇠ 1 ⇥ 10 7 cm/s. Taking these values along with the ranges s ⇠ 10 15 10 13 cm 2 for 'killer' defect centres [8][9][10] and usual attempt frequencies of n ⇠ 0.5 10 THz, 5,11,12 we obtain the following upper and lower estimates for DE: DE max = (0.0259) ln (10 15 )(10 7 )(10 14 ) (10 13 )(1) = 0.42 eV @ T = 300 K (S32) DE min = (0.0259) ln (10 13 )(10 7 )(10 14 ) (5 ⇥ 10 11 )(1) = 0.22 eV @ T = 300 K (S33) Thus internal conversion is likely to be the rate-limiting step in the recombination cycle of a fast-capturing defect if the transition energy barrier DE is greater than 0.4 eV, while barriers less than 0.2 eV should yield structural conversion rates far quicker than charge capture. For barriers between these extremal limits, which process becomes the rate-limiting step will depend on the specific parameters for that system.