SUMMARY
The relationship between the periodic part of the Bloch function in a primitive cell, denoted as u_nk(r), and that in a supercell is defined by the transformation of the wave vector k. In a supercell that is three times longer than the primitive cell, the wave vector k transforms to k' + K, where K can take on three specific values: 0, 2π/3d, and 4π/3d. This results in the equation u'_{nKk}(x) = u_{nk}(x) exp(iKx), establishing a clear mathematical relationship between the two functions.
PREREQUISITES
- Understanding of Bloch's theorem and wave functions in solid-state physics.
- Familiarity with the concepts of primitive cells and supercells in crystallography.
- Knowledge of reciprocal lattice vectors and their significance in band structure calculations.
- Basic grasp of mathematical notation used in quantum mechanics and solid-state physics.
NEXT STEPS
- Study the implications of supercell calculations in density functional theory (DFT).
- Explore the mathematical derivation of Bloch's theorem in various lattice structures.
- Learn about the effects of periodic boundary conditions on electronic band structure.
- Investigate the role of symmetry in determining the properties of supercells.
USEFUL FOR
Physicists, materials scientists, and computational chemists who are involved in solid-state physics research, particularly those working with electronic band structure and crystal lattice simulations.