1. What is the Band Structure?
All electrons belonging to a single atom occupy each atomic orbital having a different energy level. When two or more atoms are combined and molecules are formed, there will be an overlapping of atomic orbitals.
As electrons cannot have the same quantum numbers according to Pauli's exclusion principle, the overlapped atomic orbitals are separated into different energy levels by forming molecular orbitals. Given this, if the more atoms combined, the more atomic orbitals are overlapped and generate more molecular orbitals. It is causing the gap between energy levels to be more narrow.
The 'Band structure' describes the electron energy level of a crystal structure using two quantum numbers: the Bloch vector k and the band index n. It also explains the electrical, optical, and even, magnetic characteristics of crystals. To understand how it changes with regard to the Fermi level in real space, a band structure can be simply plotted through a “band diagram” for energy versus space. Initially, it was essential to determine the location of the Fermi level (a point where E-EF is 0). If the EF is located within the band gap, the material is an insulator (or a semiconductor). If not, it is a metallic material.
2. Electronic properties from Band Structure
The next figures show the band structures of a hexagonal diamond and an aluminum.
In first, the hexagonal diamond is the insulator. In general, a material that has a band gap of over 2 eV is considered an insulator. Given this, a hexagonal diamond is considered an insulator because its calculated band gap is 3.15520 eV, and the Fermi level is observed within the band gap. On the other hand, aluminum, the metal, can be considered a conductor because the conduction and valence bands are overlapped without gaps near the Fermi level in the band structure.
As the band structure and DOS are the methods expressing the information on the energy level, we can overlap the axis of the two graphs. If the x-axis of DOS and the y-axis of the band structure are combined, the band gap is located in the same position. Thus, we can find out the two graphs describe the same property. Although they contain similar information, and their band structure is advantageous to visualize electronic structures or identify the band transition.
3. Doping Affect on the Electronic structure
In addition, silicon is a semiconductor. If it is doped with a different element, then its conductivity increases. To find the change of the electronic structure, we can observe the band structure and DOS together.
When you observe the band structure of silicon, the band gap and the Fermi level existing between them can be seen. However, if P is doped, the number of electrons increases by one, and the band gap moves below the Fermi level, thus causing it to be conductive. On the other hand, if B is doped, the number of electrons decreases by one, and the band gap moves above the Fermi level, thus causing it to be conductive, as well. In a similar method, you can identify the change of the band gap in the DOS graph located on the right side.
In this way, you can easily find data on an electronic structure using a band structure and receive information on electrical conductivity. In addition, the combination with DOS becomes a stronger tool to interpret an electronic structure.
This weekly tip looks into the band structure and its various examples, depending on the material’s electrical conductivity.
The next weekly tip will deal with obtaining a band structure using Materials Square.