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

6.2 Optical properties

6.2.1 Configuration coordinate diagrams

Optical transitions corresponding to the excitation of an electron from the vacancy site to the CBM and the corresponding recombination of an excited electron from the CBM with a hole localized at the vacancy site are investigated. The effective parameters to model the potential energy curves in 1D upon charge trapping as described in section 2.1.2 were calculated for the $V_{O}^{0 / + 1}$ , $V_{O}^{+ 1 / + 2}$ , $V_{Al}^{- 3 / - 2}$ and $V_{Al, s}^{- 3 / - 2}$ transitions. For the $V_{O}$ the investigation of these specific transitions was motivated by the frequent identification of the $F^{+}$ and $F$ centers with singly and doubly charged oxygen vacancies in experimental studies [97, 96, 264]. For both $V_{Al}$ configurations, optical transitions from the vacancy in charge state $q = - 3$ , which is the most prevalent charge state for a wide range of Fermi levels in the band gap (as discussed in Section 6.1.2) to the CBM were calculated.

For each investigated transition, 11 images were linearly interpolated between the atomic configurations of the equilibrium positions in the two different charge states. Effective frequencies $Ω_{g, e}$ were obtained by fitting a parabolic function to the energies of the interpolated images as calculated with DFT. The resulting values are given in Table 6.3 and the 1D CCD for the $V_{Al}^{- 3 / - 2}$ transition is shown in Fig. 2.1.

The removal of an electron from both the $V_{O}^{0}$ and the $V_{O}^{+ 1}$ is accompanied by significant relaxations of the surrounding Al atoms away from the vacancy, resulting in the large $Δ Q$ and Huang-Rhys factors of the $0 / + 1$ and $+ 1 / + 2$ transitions. For the Al vacancies, the atomic relaxations are not quite as large, but still result in significant $Δ Q$ and $S$ values (see Table 6.3).

Table 6.3: Effective parameters of the optical transitions within the 1D approximation for vacancies in

α

-Al

_{2}

_{3}

Defect $^{q_{1} / q_{2}}$	$Δ Q$  [amu $^{1 / 2} Å$ ]	$ℏ Ω_{g}$  [meV]	$ℏ Ω_{e}$  [meV]	$S_{g}$	$S_{e}$
$V_{O}^{0 / + 1}$	2.02	53	54	25.7	26.3
$V_{O}^{+ 1 / + 2}$	2.82	37	37	35.1	34.9
$V_{Al}^{- 2 / - 3}$	1.58	67	60	19.9	17.2
$V_{Al, s}^{- 2 / - 3}$	1.65	64	58	20.5	17.8

6.2.2 Absorption

First, optical absorption processes corresponding to the excitation of a localized electron at the vacancy site to the CBM are studied. The line shapes of the absorption spectra for the $V_{O}^{0 / + 1}$ , $V_{O}^{+ 1 / + 2}$ , $V_{Al}^{- 3 / - 2}$ and $V_{Al, s}^{- 3 / - 2}$ transitions were calculated according to Eq. (2.15) with the effective parameters of the CCDs from Table 6.3. Additionally, the absorption line shapes are calculated for the excitation of an electron from the VBM to the $V_{O}^{+ 1}$ and $V_{O}^{+ 2}$ sites, an absorption mechanism that was predicted for $F$ -type centers in $α$ -Al $_{2}$ O $_{3}$ [265]. The results are shown in Fig. 6.2.

Figure 6.2 also includes experimental spectra from Refs. [264, 96]; the areas of all experimental spectra were normalized to 1 to facilitate comparison with calculated line shapes. Furthermore, a characteristic FWHM was determined for each spectrum.

To allow easier comparison with experimental data the calculated spectra were slightly shifted; magnitudes of these shifts are given below.

For the $V_{O}^{0 / + 1}$ transition the calculations reveal an absorption line shape centered around 6.05 eV, in very good agreement with measurements for the $F$ center [264], as shown in Fig 6.2(a) (shift 0.05 eV). We thus assign the origin of the 6.05 eV absorption band to the $V_{O}^{0}$ defect, in agreement with insights obtained from experiments [96].

The absorption bands for the excitation of an electron from the valence band to $V_{O}^{+ 1}$ or $V_{O}^{+ 2}$ are shown in Fig 6.2(b). The calculated absorption line shape for the excitation of an electron from the VBM to $V_{O}^{+ 1}$ is in good agreement with the 5.4 eV band as measured in [264] (after applying a shift of $- 0.38$  eV)

For excitation of an electron from the VBM to $V_{O}^{+ 2}$ , the width (FWHM $= 0.5$  eV at $T = 77$  K) and position (shift $- 0.11$  eV) of the calculated absorption line shape agrees well with the 4.85 eV band as detected in Ref. [93] with an assigned FWHM of 0.44 eV [Fig. 6.2(b).]. The data point at 5.48 eV was extrapolated on the assumption of a symmetrical Gaussian-shaped spectrum. This assignment would explain the low intensity of this band detected in [264] and [93], as the $V_{O}^{+ 2}$ is expected to have only an ephemeral existence in $α$ -Al $_{2}$ O $_{3}$ [96].

In Fig. 6.2(b), additionally the absorption line shape for optical excitation of an electron localized $V_{O}^{+ 1}$ to the CBM is shown; the calculated line shape is centered at 6.5 eV with at FWHM of 0.55 eV at 77 K. This compares very well with an absorption peak at 6.4 eV with a FWHM of $0.6 p m 0.02$  eV as detected in [264], which was related to the $F^{+}$ center.

These interpretations also align with considerations concerning changes in defect density. In [266] it was shown that optical excitation at 6.1 eV increases the intensity of the 5.4 eV absorption band of the $F^{+}$ center. This increase is consistent with the generation of $V_{O}^{+ 1}$ defects. Likewise, photoexcitations $> 4.1$  eV increased the intensity of the 6.05 eV band of the $F$ center while reducing the intensity of the 5.4 eV and 4.8 eV band [96]. This agrees with the creation of $V_{O}^{0}$ defects according to the $V_{O}^{+ 1 / 0}$ transition and the annihilation of $V_{O}^{+ 2}$ defects according to the $V_{O}^{+ 2 / + 1}$ transition.

Previous experimental works have related the three absorption bands of the $F^{+}$ center to internal transitions at the $V_{O}^{+ 1}$ defect [93, 267]. This was qualitatively based on older calculations with a point-ion model [268] that predicted the splitting of an excited $p$ -like state by the crystal field. Time-dependent DFT calculations [269] showed similar results, but predicted a different highest absorption band at 5.95 eV as detected in [97]. In the hybrid functional calculations, also three different unoccupied Kohn-Sham states are found localized at the $V_{O}$ site which would align with these predicted transitions. This is in agreement with previous DFT works, which also reported unoccupied localized states of the oxygen vacancy [94, 270]. However, these states are above the CBM and energetically too far away from the occupied defect state of the $V_{O}^{+ 1}$ (2.9 eV above the VBM, 6.3 eV below the lowest excited state) to explain the three absorption bands. Furthermore, it was suspected [271] that both the conduction and the valence band may contain defect-bound charge carriers which further highlights the relevance of charge-state transitions between the vacancy sites and the electronic band edges.

Absorption line shapes of the aluminum vacancies are calculated for $T = 77$  K and shown in Fig. 6.2(c). They are centered at 6.66 and 7.12 eV and have a FWHM of 0.60 eV for $V_{Al}^{- 2 / - 3}$ and a FWHM of 0.58 eV for $V_{Al, s}^{- 2 / - 3}$ . The spectra overlap and the peaks separated by 0.46 eV due to the different $ε (- 2$ / $- 3)$ of both configurations.

6.2.3 Emission

The emission line shapes according to Eq. (2.15) are calculated for recombination of an electron from the CBM with a localized hole at the vacancy sites. The results for $V_{O}^{+ 1 / 0}$ , $V_{O}^{+ 2 / + 1}$ , $V_{Al}^{- 2 / - 3}$ and $V_{Al, s}^{- 2 / - 3}$ transitions are shown in Fig. 6.3.

Calculated line shapes in Fig. 6.3(a) are shown for $T$ = $8$  K to compare with experimental data from Ref. [264]. The calculated $V_{O}^{+ 2 / + 1}$ spectrum (for a transition from the CBM to $V_{O}^{2 +}$ ) was shifted by $- 0.17$  eV. The line shape agrees well with experimental luminescence attributed to the $F^{+}$ center [264]. The presented first-principles calculations thus provide further evidence that the $F^{+}$ center in corundum corresponds to $V_{O}^{+ 1}$ . The calculated line shape for $V_{O}^{+ 1 / 0}$ (i.e., for electron capture from the CBM at $V_{O}^{+ 1}$ ) was shifted by $- 0.39$  eV. It compares well with the luminescence spectrum assigned to the $F$ center in Ref. [264]. No experimental data were reported below 2.5 eV.

The calculated emission line shapes for the $- 2 / - 3$ transition at the aluminum vacancy are also included in Fig. 6.3(a), for both the $V_{Al}$ and the $V_{Al, s}$ configurations. The spectra of $V_{Al}$ and $V_{Al, s}$ overlap, with the peak of the split Al vacancy being $0.5$  eV higher in energy compared to the single Al vacancy. To the best of our knowledge, no experimental papers have attempted to assign observed photoluminescence spectra to Al vacancies.

In Fig. 6.3(b) the calculated FWHM are shown [Eq. (2.18)] of the spectra as a function of temperature. Measurements of the FWHM as a function of $T$ allow determining both the Huang-Rhys factor and the vibrational frequencies by fitting to Eq. (2.18). The width of the $V_{Al}^{- 2 / - 3}$ emission spectrum is slightly larger compared to the width of the $V_{Al, s}^{- 2 / - 3}$ transition due to the larger effective frequencies.