Impact of Charge Transitions at Atomic Defect Sites on Electronic Device Performance

Chapter 6 Optical Properties of Vacancies in Corundum ($\alpha$-Al${}_2$O${}_3$)

In this chapter, optical transitions at vacancies in corundum ($\alpha$-Al${}_2$O${}_3$) are analyzed within the framework of effective phonon modes as described in section 2.1.3. The cell parameters of the pristine $\alpha$-Al${}_2$O${}_3$ bulk were relaxed until the residual stress was below 1 MPa, resulting in a mass density of 4.0 g cm${}^{-3}$, which agrees with experimentally determined mass densities of corundum (3.984 g cm${}^{-3}$ [261]). All Al atoms are bonded to 6 O atoms, while all O are bonded to 4 Al. Three of the six Al–O bonds have a bond length of 1.97 Å, the other three 1.86 Å. This is in perfect agreement with experimental data from x-ray measurements (1.97 and 1.86 Å [262]). All defect calculations were performed in a 240-atom 2x4x1 supercell built from the conventional cell of $\alpha$-Al${}_2$O${}_3$ with space group $R\bar{3}c$. The defect supercell containing a single vacancy was created by removing one atom and optimizing the geometry of the structure in different charge states. More details about the computational setup can be found in Appendix E.

6.1 Charge transition levels

An accurate calculation of the CTL is crucial for theoretically predicting optical properties of defects. As can be seen in Fig 2.1, the energy of the emitted or absorbed photon at the vacancy is related to the energetic position of the CTL relative to the band edges. The results for CTLs as calculated according to section 2.1.3 are listed in Table 6.1 and compared with previous results. While other details of the calculations may also play a role, the differences are mainly attributed to the use of different functionals employed in the DFT calculations.

6.1.1 Oxygen vacancy

The results for the formation energy are shown in Fig. 6.1 and the corresponding CTLs are given in Table 6.1 . For the $V_{\mathrm{O}}$, $\epsilon (0/+1)$ and $\epsilon (+1/+2)$ are located near midgap and follow a +2/+1/0 progression with increasing Fermi energy.

6.1.2 Aluminum vacancy

For both the $V_{\mathrm{Al}}$ and $V_{\mathrm{Al},\mathrm{s}}$, $\epsilon (0/-1)$, $\epsilon (-1/-2)$ and $\epsilon (2/-3)$ are inside the band gap. For Fermi energies above 3.26 eV (2.79 eV), $V_{\mathrm{Al}}^{-3}$ ($V_{\mathrm{Al},\mathrm{s}}^{-3}$) is the most stable charge state. The formation energy of the $V_{\mathrm{Al}}^0$ configuration is 57 meV lower than for the $V_{\mathrm{Al},\mathrm{s}}^0$. In contrast, the energy of $V_{\mathrm{Al}}$ is 16 meV higher for $q=-1$, 280 meV higher for $q=-2$ and 0.75 eV higher in energy for $q=-3$ compared to $V_{\mathrm{Al},\mathrm{s}}$. These results qualitatively agree with the results of previous calculations, while quantitatively the calculated defect levels are considerably higher than with PBE and LDA due to the improved evaluation of the electronic band gap. As shown in Fig. 6.1(b), the $V_{\mathrm{Al},\mathrm{s}}$ can be regarded as a complex consisting of an interstitial Al atom situated between two $V_{\mathrm{Al}}$ sites along the $c$-axis [98, 99, 100, 101].

The migration path for the transition of a $V_{\mathrm{Al}}$ to a $V_{\mathrm{Al},\mathrm{s}}$ configuration was calculated with the CI-NEB method for $q=0$ and $q=-3$ and the transition states along the migration path further optimized with the dimer method. The Al atom that moves towards the vacancy is originally at an adjacent lattice site along the $c$-axis. As this Al atom moves towards $V_{\mathrm{Al}}$, it localizes at a new minimum-energy configuration corresponding to the $V_{\mathrm{Al},\mathrm{s}}$ site. The energy barrier $E_a$ therefore occurs for a position of the Al atom in between the $V_{\mathrm{Al}}$ and $V_{\mathrm{Al},\mathrm{s}}$ configurations. $E_a$ in different charge states are given in Table 6.2.

These results indicate that the $V_{\mathrm{Al},\mathrm{s}}$ is the most prevalent cation deficiency configuration in $\alpha$-Al${}_2$O${}_3$. The large energy difference of 0.75 eV for $q=-3$ indicates that most vacancies in the system occupy the split state compared to $V_{\mathrm{Al}}$, since the occupation ratios are given by the Boltzmann factor $e^{-(\mathrm{d}E/k_{\mathrm{B}}T)}$. Furthermore, once a $V_{\mathrm{Al},\mathrm{s}}$ defect forms, it is less likely to transition to a $V_{\mathrm{Al}}$ site compared to the reverse process.