The polarizabilities of the participating ions are significantly lower in the crystal environment than their free ion values. 1.1 (a) Two main types of point defects, vacancy and interstitial. b) The relaxation of surrounding ions as a point defect (vacancy here) is introduced.
Point Defects and Thermodynamics
Finally, intrinsic type defects exist due to the increase in the configurational entropy of a crystal upon the formation of point defects. It involves an increase in the enthalpy of the crystal, as well as an increase in entropy due to increasing disorder in the system.
Study of Point Defects : Experimental and Theoretical Aspects
Techniques and Physical Models based on Atomistic Theory
Here the whole defect crystal is divided into two regions, Region1 and Region2. This is not true and the extremely high field near the defect results in a non-zero field gradient.
Atomistic Studies on Thermodynamic Parameters
These calculations were followed by a detailed survey of the defect energetics inα−Al2O3and T iO2[73]. In ionic crystals, the displacements of the nearest neighbors to the vacancy are expected to be outward.
The Present Work
Plan of the Thesis
The Short Range Potential Model
Modeling a perfect crystal is incomplete unless we take into account the quantum effects. the electrons in the ions behave as independent entities and interact directly with each other, increasing the energy of the system. The most commonly used function for the repulsive interaction is the Born-Mayer function, defined as. 2.2) The Born-Mayer potential together with the r−6van der Waal dispersive term is known as the Buckingham potential with the form.
The Polarization Model
To avoid the translational mode in the crystal with the applied electric field, we fix one ion in the unit cell and allow all the other ions to move. Ions in a crystal will behave differently when subjected to a high frequency and a low frequency field.
Dipole Moment of Ions in a Perfect Crystal
Lattice Energy of a Perfect Crystal
Electrostatic Energy
In the absence of induced dipole moments, the electrostatic energy of a system of ions can be written as So the total electrostatic energy of a crystal with polarizable punctures is going to be.
The Second Derivative or Hessian Matrix of Energy
The second derivative of the energy is easy and straightforward to calculate only when the ions in the perfect medium are in electrostatically neutral positions. Once the dipole term comes into the picture, it involves a calculation of the second derivative of the dipole moment, which is quite complicated and expensive.
Energetics of Point Defects
ML Scheme
The most relevant and important question at this stage concerns the size of Region1. However, the MPPI model plays a very optimistic role here by saying that if we simulate the interionic forces in the crystal in such a way that it shows the electrical response to the static field as appropriate for the experimental observables, Region1 certainly will be small enough consisting of only a few scales around the defect.
Splitting the Energy Terms with ML Scheme
Region2 is treated harmoniously under the assumption that most of the effects of the defect are already consumed by Region1. Let us now find an explicit representation of the energy by considering the interaction potential between two bodies.
Energy Minimization
Stationary Points: Maxima, Minima and Saddle Points
So any negative eigenvalue will correspond to an imaginary frequency and will identify the system to be unstable with respect to the distortion given by the eigenvector of the imaginary mode. Due to the translational symmetry of the lattice, the first three vibrational frequencies must be equal to zero.
Search Techniques
First, with n number of atoms in the system, n components of the gradient must be calculated at r = rk. In this approach, instead of explicitly computing the second derivative matrix, the inverse of it is constructed and updated in each iteration through the Newton-Raphson algorithm so that it becomes the exact inverse second derivative matrix when the function is close to the minimum. For large systems, storing the Hessian matrix can cause problems, as the storage requirement increases with the square of the number of atoms.
The first approach lies in the model itself in simulating the microcosmics of the problem.
Scheme of Calculation
In this way we avoid the complications that arise due to the interaction between Region1 and the surface of the crystal. Then the entire procedure is performed for a crystal with and without a vacancy. If R is the radius of Region1, the defect energy of the crystal of size L is given by.
It is necessary to keep R, the radius of Region 1 small compared to L, the size of the crystal.
Dipole Moment and Energy Terms
Q is the central charge and X and Y are the position vectors of the representative ions of the shells. Here WCenis is the contribution of the central charge to the energy expression and will be absent for a defective (vacant) crystal where the central charge has been removed. The other terms Wcoul, WP and WS R represent Coulomb, Polarization and Short Range terms respectively.
Results and Discussions
It can be seen that the induced dipole moments near the defect and those at the surface are clearly separated. The induced dipole moments in the presence of the defect are an order of magnitude larger than that without the defect. Therefore, it is important to explore again the nature of Schottky disorder in these systems.
Ec and Ea are the second excitation energies of the cation and that of anion, respectively.
The Physical Model
Outline of the EPPI Model
The quadrupole moment is a second rank, symmetric, traceless tensor and has six independent components. But in the crystal environment, the quadrupole polarizability tensor satisfying the isotropy condition has the following relations[60]. In EPPI model, the contribution of the quadrupoles to the defect enthalpy must be taken into account.
For an anionic vacancy, the equations for the dipole and quadrupole moments of one of the nearest neighbors of a defect with a symmetry axis along z0.
Energy Terms
Here the first term is the Madelung term which involves an infinite sum over the lattice represented by the MadelungαM constant. The second term is the interaction between the cations of the first region with their first and second neighbors coming from Region2 and the third term is the harmonic approximation term. When the first term is the monopole-dipole term, the second term is the interaction of the displacement dipoles of Region1 with the dipole field of Region2, and the last term is the contribution from the dipoles of Region2.
Here Vp2 is the potential of a dipole γc,a of an ion i in Region2 at the new position from the origin, qdis the defect charge, F1mand F2d, includes the monopole and dipole fields as given in section 4.2.1.
Results and Discussions
Comparison with Experimental Results
Self Diffusion in Ionic Crystals
By dividing the free energies into entropy and formation energies, the self-diffusion coefficient D for inner and outer regions comes to be. An Arrhenius plot of ln(D) vs. KT )−1 will give straight lines with slopes 12hsf +hmc for the inner region and hmc for the outer.
Impurity-Vacancy Association Effect
These pairs behave like electric dipoles and do not contribute to the charge transport process. For the trivalent impurity in a divalent crystal (such as MgO), we have an impurity-vacancy dyad as
Re-interpretation of Experimental Data
Due to the low symmetry structure of the material, enough theoretical work is lacking to describe the complete picture of energy in the perfect and imperfect crystals of β−Ga2O3. The cross-sections of the unit cell along and perpendicular to the unique axis b are shown in Fig. 5.1. With the space group symmetry of C −2/m, the multiplicity of each atom in the unit cell of β−Ga2O3.
The cross-sectional view of the symmetry elements perpendicular to and along the axis of symmetry is shown in Fig. 5.2.
Perfect Crystal Calculations
- Rigid Ion Calculations
- MPPI Calculations
- Induced Dipole Moments
- Lattice Energy
- Gradient of the Lattice Energy
As discussed in §2.1.1, a non-zero electric field for ions in perfect lattice sites of low-symmetry crystals induces an ion dipole moment. On the other hand, the short-lived term contributes directly to the total energy of the crystal. Thus, we can write the lattice energy of the unit cell of the crystal as where i goes through the ions in the unit cell and j finally through the remaining ions in the crystal.
As mentioned in §2.6, the availability of the first derivative energy is useful for energy minimization.
Results of Perfect Crystal Calculations
Each time the polarizations are changed, the crystal is re-optimized through cycle 2 of the algorithm. Vector plots of the dipole moments for both SET-I and SET-II are shown for a single unit cell in fig 5.5. As the polarizations increase, the oxygen dipole moments increase in SET-II compared to SET-I.
The actual minima lie in a 60-dimensional space defined by 20 ions in the unit cell of the crystal.
Defect Crystal Calculations
Relaxation and polarization of ions
Referring to §2.1.2, the displacement of an ion in the presence of the field F is given by. where W is the jute matrix and q is the charges. To find the dipole moments of Regio1 ions, we solve a series of linear equations, as in the case of the perfect crystal. The field at each Regio1 ion site due to the other Regio1 ions and the defect can be evaluated exactly and without much computational effort.
The last term is the field at the reference point due to the lags of the dipoles of all other points in the grid and converges much faster than the monopole term.
Energy Terms
With the ML scheme, the total monopole energy can be written as ξjo is the equilibrium ion displacement of Region2 for arbitrary ion displacement in Region1. Let us consider the following two terms T1mand T2m which can be calculated by Ewald's method. The first term of Eq. 5.33 which is the interaction of the displacement dipoles of Region I with the displacement of Region 2b. the dipoles will be quite small and can be neglected.
With this, the total monopole-monopole energy in the disordered environment can be written as
Results and Future Plan
The total energy of the system assumes lower and lower values, showing divergence in the energy surface. The defect energies for both sublattices for a fixed Region1 size showed good asymptotic behavior with the increase in crystal size. With the quadrupole moments included in the picture, the EPPI model provides a better understanding of the defect environment.
We observed that the inclusion of the quadrupolar terms had substantial effects on the defect formation energies of all four.
Coulomb Energy
Coulomb Field
Coulomb Force Constant
Rl+Ri −Rj+ri−rj Here Rl is the lattice vector of the lth cell, Ri and Rj are the position vectors of ion i and j with small displacements ri and rj.
- Input parameters for NaCl
- The increasing crystal size with associated shells and number of ions
- Various full and defect energy terms for R1 = 9 and L = 21 with (a) Anion
- Calculated Schottky energies as a function of Region1 size
- Schottky energies from the present and previous works
- Short range overlap repulsion parameters
- van der Waals paramaters and polarizabilities
- Quadrupole polarizabilities
- Dipole and quadrupole moments
- Various energy terms for anion and cation vacancy with MPPI and EPPI
- Theoretical Schottky defect enthalpies
- Experimental Schottky defect enthalpies
- Experimental structural parameters for β − Ga 2 O 3
- Electric field at experimental sites of symmetry unique ions in β − Ga 2 O 3 . 91
- Fitted parameters from GULP
- Fitted values of dielectric constant and polarizabilities with MPPI scheme . 97
- Electric field and dipole moment at symmetry unique ions
- Phonon frequencies
In a dielectric medium, the dipole moment μ of each atom is proportional to the field E causing polarization in the medium[132]. where is called the polarizability of the atom. For liquids and solids, it is the E field that will be replaced by the local Eloc field. The Ei1 field, due to the polarization charges on the outer surface, known as the depolarization field opposes the outer field and can be written as.
On the other hand, the polarization charges on the surface of the unit cell can be written as.
