Impact of Charge Transitions at Atomic Defect Sites on Electronic Device Performance
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Chapter D Stability of polarons: PBE vs. PBE0_TC_LRC
The content presented in this section has been published in the Supplementary Material of [53]. The text and the table have been adapted from this publication.
The importance of using the non-local PBE0_TC_LRC hybrid functional over the standard PBE to evaluate the stability of polarons is emphasized here by comparing the electronic and thermodynamic properties obtained from employing both functionals. The shortcomings of using PBE are outlined in the following.
Foremost, the standard PBE cannot correctly predict the experimental band gap. Using only PBE for the DFT calculations results in an underestimation of about 40 % on average when compared to calculations employing the hybrid functional or experimental determined values ranging between 4.5 and 5.3 eV as discussed in section 5.2. This leads to completely different results of the CTLs when calculated with PBE and PBE0_TC_LRC as summarized in Table D.1.
Table D.1: Comparison of average band gaps and CTLs for hole (\(\varepsilon (0/+1)\)) and electron (\(\varepsilon (0/-1)\)) polarons in \(a\)-Si\(_3\)N\(_4\):H obtained from DFT calculations employing PBE and
PBE0_TC_LRC
| Functional | Band gap [eV] | \(\varepsilon (0/+1)\) [eV] | \(\varepsilon (0/-1)\) [eV] |
| PBE | \(2.94\textpm 0.39\) | \(-0.13\textpm 0.06\) | \(2.47\textpm 0.34\) |
| PBE0_TC_LRC | \(4.78\textpm 0.25\) | \(0.46\textpm 0.15\) | \(3.63\textpm 0.35\) |
Furthermore, it was shown in [276] that polaron formation energies as well as the energies of the Kohn-Sham states strongly depend on the Hartree-Fock mixing parameter of the hybrid functional. The calculated CTLs of the hole polarons with PBE are distributed below the VBM of the corresponding \(a\)-Si\(_3\)N\(_4\) structures. This means that according to calculations employing only the standard PBE, the hole polaron would not be stable in \(a\)-Si\(_3\)N\(_4\):H. Employing only the standard PBE would also result in a completely different interpretation of the CTL when considering a Si/Si\(_3\)N\(_4\) system as discussed in the main text. The CTLs for electron polarons are distributed inside the band gap of a Si substrate when calculated with the standard PBE and therefore intrinsic sites could easily trap electrons from the conduction band edge of Si, even without an employed bias. This is in strong contrast to the results of the hybrid functional calculations, where the trap levels are distributed above the CBM of the substrate.