Carrier-induced transient defect mechanism for non-radiative recombination in InGaN light-emitting devices

Non-radiative recombination (NRR) of excited carriers poses a serious challenge to optoelectronic device efficiency. Understanding the mechanism is thus crucial to defect physics and technological applications. Here, by using first-principles calculations, we propose a new NRR mechanism, where excited carriers recombine via a Frenkel-pair (FP) defect formation. While in the ground state the FP is high in energy and is unlikely to form, in the electronic excited states its formation is enabled by a strong electron-phonon coupling of the excited carriers. This NRR mechanism is expected to be general for wide-gap semiconductors, rather than being limited to InGaN-based light emitting devices.

a pair of anion vacancy and anion interstitial) is created. As the bonds between the anion and cation are broken, an anion-derived bonding state (near the interstitial) and a cation-derived anti-bonding state (near the vacancy) evolve from the VB and CB into the band gap as the interstitial and vacancy levels, respectively, as shown in Fig. 1(b). The interstitial and vacancy levels carry two holes and two electrons, respectively. Once the two levels approach each other inside the band gap, they exchange carriers. After the exchange, the interstitial moves back to annihilate the vacancy and heal the lattice, by which an NRR process is completed [see Fig. 1(c)].
In the ground state, the FP formation is prohibited because it would require a significant amount of energy to break the bonds and to lift up the (doubly-occupied) interstitial level. With excited carriers, however, such a process becomes possible 19 because 1) the electrons in the CB will drop into the vacancy level, which further shifts downward due to FP formation, and 2) the holes in the VB will rise into the interstitial level, which further shifts up due to FP formation. Thus, both excited electrons and holes can offset the energy required in the FP formation process.

Results
Frenkel pair is unstable in ground state. As an alloy, InGaN can have different local structures made of In clusters (with up to four In atoms around a N atom). Since In-N bond is weaker than Ga-N bond, here for simplicity, we consider only the FP formation at a four In cluster (In 4 ). This is reasonable because from the calculated formation energy of 90 meV, In 4 concentration at a growth temperature of about 750 °C is on the order of 1.6 × 10 18 cm −3 for In 0.1 Ga 0.9 N samples. Figure 2 shows the atomic structures before and after the FP formation. The FP is formed by displacing the N at center to a neighboring interstitial site, which leaves an N vacancy (V N ) behind. Figure 2(b) shows that the displaced N forms a split-interstitial with a neighboring N atom. This is consistent with the fact that, among isolated N interstitial configurations in GaN, split-interstitial is most stable 20,21 . The calculated FP formation energy is 4.42 eV in the electronic ground state, indicating that this is a forbidden process at room temperature. It also suggests that any FP generated during the NRR process will eventually recombine.
Carrier relaxation in the NRR processes. Figure 3, light gray lines, shows the change of the energy levels during the FP formation assisted by charge carrier injection resulting in excited electrons near the CBM and excited holes near the VBM. One may notice from the figure that, even in defect-free InGaN, band edge energies deviate from their original positions due to occupation of excited states. This unphysical result is a drawback of the hybrid functional methods [22][23][24] . The errors have been corrected, according to the degree of localization of the states, yielding the red and blue lines in Fig. 3. It shows that when the N is displaced towards the interstitial site, the In-derived V N level (which can host two electrons) decreases and the N-derived N interstitial (N i ) level (which can host two holes) increases. Insets a and b in Fig. 3 show the charge density in the seventh configuration for V N and N i , respectively, from which we can also see that the V N gap state, evolved from the CB, has an anti-bonding character in regarding the broken bonds between the In atoms of the vacancy and the displaced N i . In contrast, the N i gap state, evolved from the VB, has a bonding character between these states.
Once the FP is formed (i.e., the final configuration in Fig. 3), the V N level is further decreased to 0.55 eV above the VBM, whereas the N i level is increased to 3.60 eV above the CBM. This implies that the levels have crossed due to the FP formation. In the crossing region (i.e., the eighth and ninth configurations in Fig. 3), however, it (a) Initial defect-free state with injected carriers, (b) during FP formation, and (c) re-healing process back to defect-free ground-state after two-electron transfer. VB and CB denote the valence and conduction bands, respectively. Shaded blue regions represent occupied states. During the FP formation, an anion vacancy level and an anion interstitial level appear inside the band gap. The injected electrons (filled dot) at the highest occupied state in the CB drop to the vacancy level, whereas the injected holes (empty dot) in the VB raise to the interstitial level. After the electron transfer between the vacancy and interstitial levels, these levels undergo a reverse process to retreat back into the VB and CB, respectively.
happens that the constrained self-consistent field (SCF) calculation dose not converge. Nevertheless, we can see the level crossing indirectly by checking the charge densities at the tenth configuration, as shown in the insets in Fig. 3. In contrast to insets a and b where the lower-energy state has an N lone pair, insets c and d show that the lower-energy state has In dangling bonds, instead. Note that, for clarity, the issue of level crossing was not included in in Fig. 1, as this schematic figure depicts isolated defects for which the cation level is higher than the anion level.
The lack of convergence in the crossing region can be attributed to charge sloshing between the N i and V N states at each SCF step. A similar situation takes place in conventional DFT calculations, when two or more partially occupied levels are nearly degenerate. One may overcome the problem by smearing the occupation over a certain energy range. Here, before entering the crossing region, two electrons occupy the high-lying V N level and two holes occupy the low-lying N i level. After exiting the crossing region, the system is electronically de-excited, so no empty state is below occupied states. To assess the effect of this occupation change, we resort to fractional occupation, in particular, for the two interacting levels, labeled e L and e H , the following occupation ) at the eighth configuration, and ( × 2 2 3 , × ) at the ninth configuration, are used. As expected, converged SCF results are obtained. To connect with the V N and N i levels in the  non-crossing regions in Fig. 3 Potential energy curves in the NRR process. Figure 4 shows the potential energy manifold during the FP formation. By injecting two electrons and two holes in the CBM and VBM, respectively, the total energy of the system at the initial configuration is increased by twice of the band gap energy (6.81 eV). Here solid lines represent non-crossing regions whereas dashed lines represent the crossing region, as discussed above. The barrier for the FP formation, in the presence of carrier injection, is only 0.56 eV, which is significantly reduced from the 4.79 eV in the ground state. One can qualitatively understand the difference as follows: displacing an N from the defect-free structure increases the total energy of the system. However, the increase is substantially smaller than what would be in the ground state without the excitation, because the significant increase of the depleted N i level no longer costs energy and the significant lowering of the occupied V N level also helps to offset the energy required to break the bonds and associated strain. After forming the FP, the system will revert to the original defect-free structure, as it only needs to overcome a barrier of 0.47 eV.
We have analyzed the occupied energy level in the ground state during the reverse process back to the defect-free structure. It was found that the character of the occupied defect level is continuously changed from that of the V N state to that of the N i state. As the occupied N i level and the empty V N level eventually merge into the VB and CB, respectively, the NRR process assisted by carrier injection is completed. One may classify the transient defect mechanism in terms of a strong electron-phonon coupling, i.e., the significant energy level changes during the FP formation, and the reverse process transforms the initial excitation energy into thermal motion of the atoms.

Discussion
To see the dominate process in recombination, we compare the radiative recombination time of the excited carrier with the FP formation time. The FP annihilation process is, on the other hand, unimportant here because after the FP formation, the excited electrons already fall in a level lower than the holes (cf. Fig. 3) so that radiative recombination is no longer possible. We can estimate the FP formation time using the rate equation where k B is the Boltzmann constant, f is the optical phonon frequency, 2.0 × 10 13 s −1 for GaN 25 , E b is the formation energy, 0.56 eV, and T is the junction temperature during the LED operation, 130 °C 26 . The factor 3 stands for the number of equivalent NRR pathways. The calculated NRR relaxation time is about 160 ns, which is about twice the radiative recombination time of 88 ns 27 . Over this time, about a quarter of the excited electrons undergo the NRR processes.
Note that the magnitude of the band gap is crucial for enabling the NRR process. When the band gap is smaller, the energy required to create two electron-hole pairs through carrier injection is also smaller, for example, in GaAs, it is only 2.84 eV, while the FP formation energy is 4.59 eV 28 . Therefore, no transient FP-formation can take place, even under the carrier injection condition.

Conclusion
Using density-functional theory calculations, we demonstrate the possible existence of a new NRR mechanism, where the injected high-energy carriers induce a structural instability, namely, the low-barrier transient defect formation, and the associated NRR through strong electron-phonon coupling. While our study is focused on GaN-based LEDs, the theory, as shown in Fig. 1, is general and could be an important limiting factor for other wide-gap semiconductors for the efficiency of their optoelectronic and photovoltaic devices. In particular, it raises the important question whether new mechanism(s) may exist between the widely-accepted defect-based SRH mechanism and the defect-free AR mechanism to account for some of the most difficult but technically important materials issues regarding excited carriers.

Methods
Our calculations are based on the density functional theory (DFT) with the Heyd-Scuseria-Ernzerhof screened hybrid exchange-correlation functional 29 , as implemented in the VASP code 30 . We use a mixing parameter of 0.3 for the exact exchange, as in previous studies 31 . Projector augmented wave potentials 32,33 are used to represent ion cores. Plane waves with an energy cutoff of 306 eV and 230 eV are used as basis sets for InGaN and GaAs, respectively. We use a periodic supercell that contains 96 and 128 atoms to model the InGaN alloy and GaAs, respectively. Γ point is used for Brillouin zone sum. Atomic structures are relaxed until the residual forces are less than 0.03 eV/Å. To simulate electronic excitation, we perform constrained DFT calculations, in which we remove two electrons from the VB maximum (VBM) and place them at the CB minimum (CBM). Standard reaction-barrier-search algorithms, such as the nudged elastic band method, are not applicable to calculating the energy barrier in the present case, because electron occupation changes during the process. Instead, we generate ten intermediate configurations between the initial and final configurations and then relax all the atoms except for the diffusing N atom and a Ga atom of choice far away from the N.