The electronic density of states (DOS) is a fundamental measure used to describe the electronic state of a material.
ρ(E)dE = number of electron states with energies in interval (E, E + dE).
After conducting a DFT calculation, the electronic density of states (DOS) can be computed by integrating the electron density obtained in k-space. Having a large number of k points is crucial for calculating the Density of States (DOS) since the details of the DOS arise from integrals in k space.
The energy displayed in the DOS image is not represented in absolute terms, but rather in relation to the Fermi energy, Ef. The Fermi energy refers to the energy level of the most energetically inhabited electronic state at absolute zero temperature (T = 0 K). A simple definition of a metal is that metals are substances that exhibit a non-zero density of states (DOS) at the Fermi level. An effective approach to conceptualize this is to examine the effects of applying an electric field to the material. When there is an electric field, electrons move faster and have more energy than when there is no field. There are electronic states in metals just above the Fermi energy that these fast-moving electrons can fill. This makes the metal a good conductor of electricity.
The valence band and the conduction band are two separate parts of the DOS in a different type of material. At T = 0 K, all the electronic states that are occupied are in the valence band. At this point, all the states in the conduction band are empty. The band gap is the area of energy between the VBM and CBM that doesn't have any electric states. It is
16 clear that when an electric field is given to these materials, electron conduction will not happen as easily as it does in metals. As long as the band gap is "small," a material is a semiconductor. If the band gap is "wide," the material is an insulator.
It is often helpful to know what states are important near certain atoms in order to understand the electronic structure of a material. The local (projected) density of states (TDOS/PDOS) is a common way to do this. It is the number of electronic states at a certain energy weighted by the fraction of the total electron density for those states that appears in a certain space around a nucleus [34].
The density of state (DOS) calculation is performed with Gaussian smearing model and Gamma-centered 4x4x4 grid. The DOS and PDOS figures used in this thesis were mostly made with the Vaspkit [35] code, which set the Fermi energy to 0 eV automatically.
By looking up the eigenvalues, we can find the minimum distance between the highest occupied and the lowest unoccupied for all k points (indirect bandgap), and for one k point (direct bandgap).
𝐼𝑛𝑑𝑖𝑟𝑒𝑐𝑡 = 𝑚𝑖𝑛𝑘,𝑘′[𝐸𝑛+1(𝑘′) − 𝐸𝑛(𝑘)] (2.4) 𝐷𝑖𝑟𝑒𝑐𝑡 = 𝑚𝑖𝑛𝑘[𝐸𝑛+1(𝑘) − 𝐸𝑛(𝑘)] (2.5)
Figure 2.6. Indirect and Direct band gap.
2.2.1.3. Formation energy of different defects and impurity level calculation
17 The notation for defect using in this thesis follow Krửger–Vink Point Defect Notation[36] 𝑋𝑆𝑞
Where X is the species (element or vacancy)
S is the lattice site (Element symbol of original state)
The variable q represents the electrical charge of the species in relation to the specific location it occupies.
The formation energy diagram is a crucial tool for evaluating the properties of defects. It provides information on both the probability of a defect occurring and the level at which the charge state of the defect makes a transition. The formation energy of a defect is influenced by the chemical potential of all the components involved in its production.
Although the many chemical potentials are interrelated, it is theoretically simpler to categorize them into two groups: the chemical potential of the elements and the chemical potential of the electron. The Fermi level, also known as the chemical potential of the electron, takes into consideration the influence of external factors, such as the presence of other defects, on the thermodynamics of adding or removing an electron from the defect.
Given that the electrons in the system may be controlled once the defect is created, the Fermi level is commonly regarded as an independent variable and is shown as the x-axis in the formation energy diagram. Once the defect forms, the chemical potentials of the atoms involved in the defect become less variable and are said to be constant. The equation that represents the formation energy of a defect is:
𝐸𝑓[𝑋𝑞]= 𝐸𝑡𝑜𝑡𝑎𝑙[𝑋𝑞]− 𝐸𝑡𝑜𝑡𝑎𝑙[𝑏𝑢𝑙𝑘]−∑ 𝑛𝑖 𝑖𝜇𝑖 + 𝑞𝐸𝐹 (2.6) Where
𝐸𝑡𝑜𝑡𝑎𝑙[𝑋𝑞]: the total energy of the defect state with charge q
𝐸𝑡𝑜𝑡𝑎𝑙[𝑏𝑢𝑙𝑘]: the total energy of a perfect unit cell (the same size with the defected unit cell)
18 𝑛𝑖𝜇𝑖: the reference energy for 𝑛𝑖 atoms of element 𝑖 that were added at chemical potential 𝜇𝑖.
To determine the formation energy, it is necessary to calculate each of the terms that have been specified previously. The procedures are outlined as follows:
• Choose the compound and defect to be examined, as well as the computational model to be employed.
• Employ the selected computational model to optimize the pure bulk system.
• Calculate the total energy of the bulk for the elements present in the defect.
• Calculate 𝐸𝑡𝑜𝑡𝑎𝑙[𝑏𝑢𝑙𝑘] and EVBM for the perfect structure.
• Begin by introducing the charged defect into the system and then proceed to relax the system in order to obtain 𝐸𝑡𝑜𝑡𝑎𝑙[𝑋𝑞].
After got the formation energy of defect, we made the defect formation energy diagram. The example of defect formation energy diagram is shown in Figure 2.7.
Figure 2.7. Defect formation energy diagram.
19 Electronic carriers, like electrons and holes, are made by charged defects. It is not possible for a neutral defect (isoelectronic doping or alloying) to make electronic carriers.
A defect with a positive charge has a positive slope in a defect diagram and corresponds to a defect with donor-like characteristics. Basically, the defect becomes ionized and gains a positive charge by giving electrons, which is why it is referred to be a donor. Donors typically provide an n-type characteristic to the material. Similarly, defects with a negative charge have properties similar to acceptors and have a tendency to create a p-type. The determination of whether a material is doped as n-type or p-type is depend on various other parameters.
The first step of generating defect formation energy diagram is finding the VBM energy (Ev) and CBM energy (Ec). We determine two energy levels by calculating the difference in total energy for the perfect structure of three cases: N(neutral), N+1 (adding one electron to the system), and N-1 (removing one electron from the system). The Ev and Ec then calculated by:
𝐸𝑣 = 𝐸𝑡𝑜𝑡𝑎𝑙[𝑁] − 𝐸𝑡𝑜𝑡𝑎𝑙[𝑁 − 1] (2.7)
𝐸𝑐 = 𝐸𝑡𝑜𝑡𝑎𝑙[𝑁 + 1] − 𝐸𝑡𝑜𝑡𝑎𝑙[𝑁] (2.8)
In a defect diagram, the plot shows the relationship between the energy required to generate a defect and the Fermi energy, which is usually measured relative to the maximum energy level in the valence band, Ev = 0.
Next, we need to calculate the donor and acceptor levels of defect. The charge transition level (q/q') of a defect refers to the Fermi energy at which the formation energy of charge states q and q' are in equilibrium. In a defect diagram, charge transition levels are determined by the sites where the slope of the defect lines changes. A defect may exhibit several charge transfer levels or may not have any within the band gap.
20 Refer to above description, for donor, the transition from 0 to +1 charge states occurs when the formation of charge state 0 and + are equal: Hf(D+1) = Hf(D+1). Acceptor case is similar, the transition from 0 to -1 charge states occurs when Hf(A-1) = Hf(A0).
Using the same method for calculating Ec and Ev, we found donor and acceptor levels. In this time, a defected cell is made, and the total energy of 3 cases are obtained:
N(neutral), N+1 (adding one electron to the system), and N-1 (removing one electron from the system). Then the donor level (Ed) and acceptor level (Ea) are calculated by: