The tight-binding formulation of the Kronig-Penney model

Area of Science:

• Condensed Matter Physics
• Quantum Mechanics
• Materials Science

Background:

• Electronic band structure calculations are often approximated using the tight-binding model.
• The tight-binding model is widely applied in studying electron correlations in materials.

Purpose of the Study:

• To derive the tight-binding model from the exact solution of a particle in a periodic array of square wells.
• To analyze the necessity of next-nearest-neighbor hopping and electron-hole asymmetry in this model.

Main Methods:

• Exact solution of a particle bound in a periodic one-dimensional array of square well potentials.
• Derivation of the dispersion relation for the tight-binding model.
• Analysis of transition amplitudes for a two-state effective model of a double-well potential.

Main Results:

• An effective next-nearest-neighbor hopping parameter is required for an accurate description of the derived band structure.
• Electron-hole asymmetry is observed, except in the extreme tight-binding limit, arising from the effective next-nearest-neighbor hopping term.
• The necessity of a next-nearest-neighbor tunneling parameter in a simplified double-well model does not imply actual next-nearest-neighbor tunneling.

Conclusions:

• The derived tight-binding model provides a more accurate description by including effective next-nearest-neighbor hopping.
• The study clarifies the origin of electron-hole asymmetry in such models.
• The findings offer insights into the parametrization and interpretation of tight-binding models in condensed matter physics.