Re-interpretation of Experimental Data

4.5 Comparison with Experimental Results

4.5.3 Re-interpretation of Experimental Data

4.5. Comparison with Experimental Results 80

Ci

Vc

−2e Vc

−2e

+2e [1 1 0]

(a) S i⁴⁺

Vc

−2e

Ci +1e [1 1 0]

Ci +1e

(b) Al³⁺and Fe³⁺

Figure 4.5: Impurity - vacancy clusters for different impurities

4.5. Comparison with Experimental Results 81

shows the binding energy (g_b) and the values of Schottky formation enthalpy for the differ- ent impurity types taking care of the impurity-vacancy association.

Table 4.7: Experimental Schottky defect enthalpies

Impurity g_b c β degree of h^m(eV) h_S^f(eV)

(eV) (ppm) association

S i⁴⁺ 2.0 400 0.96 high 0.56 5.8

Al³⁺,Fe³⁺ 1.7 200 0.003 low 0.99 3.8

Thus we see that the dominant impurity in the crystal sample of Harding and Price which is S i⁴⁺(400 ppm) suggests the most probable value of h_S^f to be 5.8 eV for MgO in excellent agreement with the theoretical value of the present work with EPPI model (5.88 eV).

Chapter 5

Atomistic Simulation of Low Symmetry β − Ga ₂ O ₃ in MPPI Framework

The atomistic simulations of low symmetry materials are usually difficult for variety of reasons. First, such structures are complex and have large number of ions per unit cell.

Secondly, the number of independent variables becomes very large. More importantly, even in perfect crystals, the electric fields at ionic sites do not vanish. If the ions are highly polarizable, the electric fields polarize the ions. Even though, the net polarization is absent, individual ions have dipole moments. In simulations based on Shell model, ionic polarizations are treated by means of springs connecting the cores and shells. Hence, the Shell model only deals with rigid monopole charges, though the number of monopoles may be double to that of ions. The low symmetry adds to complexity only by increasing the number of ions, but the formulation is exactly same and straight-forward.

In contrast to Shell model, the point polarization ion (PPI) models[50], become very complex. For this reason, the atomistic simulations in various PPI models were restricted

83

range potentials can be obtained by optimizing the crystals with rigid ion approximation.

The defect calculations, then use the same short range potentials. With the low symmetry crystals, we must evolve a method to obtain the short range potentials in presence of ionic dipoles.

Another difficulty arises due to lack of knowledge of electronic polarizabilities. Free ion values of electronic polarizabilities overestimate the polarizations in defect calculations.

The polarizabilities in crystal environment are usually suppressed. The MPPI model[44]

has clearly demonstrated this point. In MPPI, since the short range potentials are already known, displacement polarizabilitities are calculated using the force constants and are used in turn to calculate the electronic polarizabilities. The generalization of this procedure to low symmetry crystals is not straight forward, since the perfect crystal simulations itself must use electronic polarizabilities. Thus the electronic polarizabilities and short range parameters must be simultaneously deduced by perfect crystal optimizations.

The motivation behind this study is two-fold. First, the formalism of MPPI (and later, EPPI) needs be generalized to include low symmetry materials. These models have been very successful in defect simulation of high symmetry materials, mostly cubic. Secondly, to make these calculations accessible to material scientists, we need to develop packages like GULP or HADES. These packages were built on Shell model and are very popular.

There is a lack of such packages for MPPI models. In this work, first we have formulated and built algorithms for modeling of perfect crystals, based on MPPI models. Then we have suggested formulations for the defect calculations. All through this work, we have applied these formulations to monoclinicβ−Ga₂O₃as an example.

β−Ga₂O₃is the most stable phase at room temperature out of the five existing phases (α, β,γ,δand)[115]. It is intrinsically an insulator with a band gap of 4.8eV. However when synthesized under reduced conditions, the material becomes an n-type semiconductor[116].

5.1. Structure and Symmetry ofβ−Ga₂O₃ 84

In Gallium Oxide, the gallium and oxygen atoms are bonded ionically . Like any other metallic oxides, this material has wide applications in the field of material science and optoelectronics. It is used as a component in preparing the anodic oxide on GaAs which has drawn subsequent attention in the semiconductor industry[117] and also acts as an important MASER material when doped with Cr³⁺[118]. Very recently, it has found its application as an ultraviolet transparent conducting oxide in excimer LASER[119] and has created sensations in the environmental technology by being used as a gas sensor. The new gas sensors based on Ga₂O₃films have very stable operating characteristics and are largely insensitive to humidity[120].

β−Ga2O3which exhibits some features like blue luminescence and ultra violet emission [121, 122] implying the presence of grown oxygen vacancies as well as some impurities, stands as a very interesting candidate for studying the point defects and defect induced transport properties. Owing to the low symmetry structure of the material, there is lack of enough theoretical work to describe the complete picture of energetics in the perfect and imperfect crystals ofβ−Ga₂O₃. One common approach for this is the semi-empirical atomistic simulation. In the present work, we present the atomistic simulations using MPPI models.

5.1 Structure and Symmetry of β − Ga

₂

O

₃

Single crystals ofβ−Ga₂O₃ can be grown by several techniques like Verneuil technique, chemical transport method, vapour phase reaction method etc.[123, 124, 125, 126]. In spite of some debate[127], several investigations[128, 129, 130] on these crystals have shown that the crystal is monoclinic in structure with a space group of C2/m. There are four

5.1. Structure and Symmetry ofβ−Ga₂O₃ 85

(a) along the symmetry axis

(b) perpendicular to the symmetry axis

Figure 5.1: Cross sections of the Ga2O3unit cell. Big spheres are galliums, small ones are oxygens.

of this structure is that the Ga³⁺ ions show two different co-ordinations in the unit cell.

Ga_I ions are surrounded by tetrahedra each of four oxygens while Ga_II are surrounded by octahedra each of six oxygens. The oxygens on the other hand do not show any definite co-ordination. The cross sections of the unit cell along and perpendicular to the unique axis b are shown in fig 5.1. The reported experimental values of the cell dimensions, fractional co-ordiantes and average interionic distances[128] are listed in table 5.1. With the space group symmetry of C −2/m, the multiplicity of each atom in the unit cell ofβ−Ga2O3

is 8. The cross-sectional view of the symmetry elements perpendicular to and along the symmetry axis are shown in fig 5.2. These figures are reproduced from the International