In some of the investigated jumps, the saddle point remained the same even though the reaction was different since the initial and final states were different. This is normally indicative of a higher order saddle point, i.e., a common saddle point for different reactions. In the case of a first order saddle point, all the modes or eigenvalues of the
Hessian matrix should be positive except one. The negative eigenvalue corresponds to an eigenvector that points towards the direction of the migration path. More negative eigenvalues, indicate the existence of multiple different paths leading to the same transition state or saddle point. The order of the saddle point can be verified by performing vibrational frequency calculations at the saddle point. In all cases a single negative eigenvalue was observed, confirming that the saddle point was a first order saddle point. Instead, the common saddle point can be explained by a more detailed NEB calculation, i.e., adding more images in order to describe the path with increased resolution. Adding more images to the NEB calculations revealed that the mechanisms of rotation, discussed earlier, were taking place before the jump in these cases. In other words, an on-site rotation occurs before the main jump, resulting in the same saddle point, in the cases where a common saddle point was observed.
An estimate for the annealing temperatures can be obtained using the HTST and Eq. (3.33). The static barrier in Eq. (3.33) refers to the calculated migration barriers.
Using a prefactor of Γ0 = 1013Hz and a success rate of Γ = 1 Hz, the estimated anneal-
ing temperature for a barrier of 2.3 eV is close to 900 K. This temperature is relevant for the growth of GaN both with the MBE and the MOCVD technique. Additionally, the gallium interstitials are expected to be mobile even at room temperatures due to their significantly lower migration barriers. This is in excellent agreement with experimental observations by Chow et al. (2000).
Overall, standard DFT calculations were used to investigate the migration barriers of native and carbon related point defects in GaN. In the case of carbon interstitials, the two most stable configurations were considered as possible initial and final states. Regarding the native defects, the obtained results are in good agreement with ex- perimental observations. Most of the charge states are found to exhibit a preference over one of the first nearest neighbor mechanisms. In general, mechanism B is more
favorable but the barrier differences compared to mechanism A are small. Gallium interstitials exhibit very low barriers and are expected to be mobile even at room temperature. In the case of carbon, only the neutral interstitial favors mechanism A. In the +1 and −1 charge states, mechanisms A and B exhibit similar barriers, while
there is a preference for mechanism B in the case of Ci2+. The on-site transformations
were found to play a significant role in the migration of carbon because they are used as secondary mechanisms that properly align the defects before or after the main jumps. Their migration barriers indicate that these mechanisms are readily available at the temperatures that are relevant for the migration of the defects. Except for the gallium interstitials and the on-site mechanisms of carbon, the rest of the defects are expected to anneal at temperatures well above 600 K.
Chapter 5
Vacancies and acceptor dopants in Ga2O3
5.1
Introduction
Ga2O3 crystallizes in five different phases, commonly called α, β, γ, δ, and . Among
them, the monoclinic β-Ga2O3 is the most stable and technologically relevant phase
(Yoshioka et al., 2007). Therefore, this phase is the one investigated in this work.
Ga2O3 is a material with various applications. The characteristic dependence of the
conductivity of the material on the ambient concentration of certain gases, such as oxygen and hydrogen, makes it an excellent sensing material in gas sensors (Ogita
et al., 1999; Weh et al., 2001; Trinchi et al., 2004). Ga2O3 is also suitable for opto-
electronic applications as a deep UV TCO due to its wide band gap of 4.9 eV (Orita et al., 2000; Ueda et al., 1997). TCOs have the ability to conduct electricity while being transparent in the visible spectrum. Therefore, their main applications are in solar cells, flat panel displays, and LEDs (Ohta et al., 2003; Hosono, 2007).
Angle-resolved photoemission spectroscopy (ARPES) and theoretical calculations
indicate that Ga2O3 is an indirect semiconductor. The CBM is located at the Γ
point while the VBM appears near the M = 12,12,12 point (Yamaguchi, 2004; He
et al., 2006; Janowitz et al., 2011). However, Ga2O3 is considered a direct band gap
material for all practical purposes because the energy difference of the real VBM and the eigenvalue of the highest occupied orbital at the Γ-point is of the order of a few meV. A property that is often overlooked is the anisotropy of the band gap with respect to the polarization of the incident light, which ranges from 4.5 to 4.9 eV
(Matsumoto et al., 1974; Varley and Schleife, 2015; Ricci et al., 2016). As many
other TCOs, Ga2O3 exhibits unintentional n-type conductivity (Hosono, 2007). In
fact, the conductivity of Ga2O3 is an easily tunable property of the material over
many orders of magnitude (10−12–102Ω−1cm−1) using the appropriate dopants and
growth conditions (Ueda et al., 1997; Passlack et al., 1994; M¨uller et al., 2014; Aubay
and Gourier, 1993). In order for the material to be widely utilized in optoelectronic
applications, it is essential to obtain p-type Ga2O3 as well.
Among the native defects, the oxygen vacancies are considered responsible for the n-type conductivity of the material (Harwig et al., 1976; Ueda et al., 1997; V´ıllora et al., 2002). On the other hand, gallium vacancies act as compensating acceptors, de- creasing the n-type conductivity (Kyrtsos et al., 2017; Varley et al., 2011). Therefore, the study of the properties of the vacancies are essential in determining the electrical and optical properties of the material. Additionally, this work explores a number
of potential impurities for effective p-type doping. The realization of p-type Ga2O3
has been proven challenging, making the search for possible p-type dopants crucial. In order for a dopant to contribute to the p-type conductivity, it should introduce a shallow acceptor level. Namely, the defect should be stable in the neutral charge state, and the transition from the neutral to the negative charge state should occur close to or below the VBM. Therefore, the accurate determination of the level ε(0/−) obtained by Eq. (3.20) is imperative for the study of the p-type conductivity of the
material. Specifically, the studied impurities include the substitutional XGa, where
X = {Li, Na, K, Be, Mg, Ca, Cu, Au, Zn}. The selection of these impurities covers a range of groups of the periodic table that are likely to result in acceptor dopants.
5.2
Methods
Both the PBE and the HSE functional with a mixing parameter of a = 0.32 were used. A 120-atom supercell was used to determine the defect levels of the dopants, while supercells of up to 360 atoms were used to study the convergence of the migration barriers results. A 2×2×2 Monkhorst-Pack mesh was used to sample the Brillouin zone in all cases. The energy cutoff of the PW basis set was set at 450 eV in all cases, except
for the case of the substitutional LiGa, in which the cutoff was set at 500 eV. The
gallium 3d electrons were treated as valence electrons and all the calculations were spin polarized. The force convergence criterion for the PBE and HSE calculations
was set at 5 × 10−3 and 5 × 10−2eV/˚A, respectively.
The migration of the vacancies was studied using the PBE functional in order to reduce the computational cost. Both the NEB and the dimer methods were used
with a force convergence criterion of 5 × 10−3 and 10−2eV/˚A, respectively. The
elastic interactions contribute to the finite supercell error. This error can be treated by extrapolating the formation energies of the finite size supercell to an infinite size
supercell based on the known 1/L3 scaling for different supercell sizes (Freysoldt et al.,
2014). Specifically in the case of migrations, the local deformation of the crystal in the transition state is typically large. Hence, this effect should be examined for the calculation of the migration barriers.
The formation energies of the vacancies were studied both in Ga- and O-rich
conditions and were obtained using Eq. (3.15). The Freysoldt correction scheme
was used for the electrostatic corrections, and the migration barriers were obtained employing Eq. (3.22). The electrostatic correction applied to the migration barriers resulted in negligible differences of less than 0.05 eV.
While the ionization levels can be obtained using a local or semilocal approach (LDA or GGA), in practice, this approach suffers from the standard DFT shortcom-
ings, i.e., the artificial electron self-interaction error and the lack of the derivative discontinuities of the exchange-correlation functional with respect to the occupa- tion number, leading to the known band gap underestimation error (Nieminen, 2009; Perdew, 1985). Therefore, while GGA calculations yield reasonable results for valu- able properties such as the lattice parameters, they usually fail to reproduce the experimentally observed ionization levels. Nonetheless, there has been reports where corrections for on-site coulombic interactions (GGA+U ) may improve the GGA re-
sults in the study of defect levels (Varley et al., 2009). Another approach is to
use hybrid functional calculations, where a portion of HF exact exchange is mixed into the exchange-correlation functional, increasing the band gap by mitigating the
self-interaction error and introducing the derivative discontinuities. As a results,
the agreement of the calculated ionization levels with experiments is improved im- mensely. Nevertheless, the improved performance of the hybrid functionals comes with increased computational cost.
Even though GGA calculations are less accurate compared to HSE calculations, their computational efficiency is still attractive for studying problems for which they are known to yield incorrect results. A typical example is the determination of defect levels, where in addition to the underestimation of the band gap, the specific defect levels are also typically incorrect. However, GGA calculations may act as a screener for identifying promising dopants which can be studied at a higher level of theory af- terwards. This approach is justifiable by the way the band edges are aligned between PBE and HSE. Typically, the alignment is performed with respect to a common ref- erence level which is usually the vacuum level. Another option is to use the average electrostatic potential in the bulk as the common reference level. Both ways of align- ing the VBM should yield identical results as long as the charge density produced by the two theoretical schemes (PBE and HSE) is identical (Alkauskas and Pasquarello,
semilocal
hybrid
CBM
CBM
VBM
VBM
common reference level
a a
b
b
Figure 5·1: Deep (a) and shallow (b) ionization levels in a semilocal and a hybrid functional approach.
2011; Alkauskas et al., 2011). Usually, the VBM of the hybrid approach lies lower compared to the VBM of the semilocal approach after alignment. Similarly, the CBM obtained by HSE calculations lies higher.
Figure 5·1 shows a schematic representation of the alignment between a semilocal and a hybrid approach. In general, two different cases of defect levels may be distin- guished. In the case of a deep defect level like (a) in Fig. 5·1, the widening of the band gap usually does not affect the absolute position of the defect level with respect to the common reference level. On the other hand, shallow levels like (b) in Fig. 5·1, tend to follow the band edge to which they are bound (Alkauskas and Pasquarello, 2011; Alkauskas et al., 2011). In other words, if a defect is predicted to yield a deep defect level in the semilocal approach, most likely it will remain deep in the hybrid approach too. Similarly, a shallow defect level obtained by the semilocal approach, will most likely remain shallow in the hybrid approach too. Thus, the GGA approach may be used as an efficient method for screening promising defect levels, which can
Figure 5·2: The 120-atom supercell showing the monoclinic (C2/m)
crystal structure and the unit cell of β-Ga2O3. There are two different
gallium and three different oxygen sites designated as Ga(1), Ga(2) and
O(1), O(2), and O(3) respectively.
be investigated further with more accurate, yet, computationally expensive methods.