Questions: Tight-Binding Model

Question 1

In the tight-binding model, the bandwidth (energy range of a band) is directly proportional to the hopping integral t. What physical quantity determines the magnitude of t?

• A. The atomic number of the element
• B. The temperature of the crystal
• C. The overlap between atomic orbitals on neighboring sites — larger overlap means larger |t| and wider bands
• D. The number of electrons per atom

Answer: (C)

The hopping integral t = <phi(r - R_i)|H|phi(r - R_j)> measures how much an electron's energy changes when it tunnels from one atomic site to a neighbor. This depends directly on the spatial overlap between orbitals on adjacent sites. Tightly bound core electrons have small overlap and narrow bands; diffuse s and p orbitals have large overlap and wide bands. This is why d-bands in transition metals are narrower than sp-bands, and why bandwidth generally increases with pressure (atoms closer together means more overlap).

Question 2

The tight-binding model and the nearly free electron model are opposite limits of the same physics — they both produce band structures with gaps, just starting from different assumptions.

• T. True
• F. False

Answer: True

The nearly free electron model starts from delocalized plane waves and treats the crystal potential as a weak perturbation — appropriate for wide bands and small gaps. The tight-binding model starts from localized atomic orbitals and treats inter-site tunneling as the perturbation — appropriate for narrow bands formed from localized states. Real materials fall somewhere between these limits. For simple metals (Na, Al), the NFE picture is more natural. For transition metal d-bands and rare earth f-bands, tight-binding is more natural. Both produce the same qualitative result: discrete energy bands separated by gaps.

Question 3

In a one-dimensional tight-binding chain with lattice constant a and nearest-neighbor hopping t, the dispersion is E(k) = ε₀ - 2t cos(ka). At what k-values does the group velocity v = (1/ħ)(dE/dk) vanish, and what does this imply physically?

Model answer: The group velocity vanishes at k = 0 and k = ±π/a (the zone center and zone boundaries). At these k-values, dE/dk = 2ta sin(ka)/ħ = 0. Physically, k = 0 corresponds to all atomic orbitals in phase (a bonding-like state), and k = π/a corresponds to alternating phases (an antibonding-like state). Both are standing waves with zero net velocity. Near these points the electron behaves as if it has a large effective mass. The density of states diverges at the band edges (van Hove singularities in 1D), reflecting the accumulation of states where the dispersion is flat.

The vanishing group velocity at zone boundaries connects to the nearly-free-electron picture: these are precisely the standing waves created by Bragg reflection. In tight-binding language, it's where constructive or destructive interference between hopping paths produces a stationary state.

Question 4

Why do d-electron bands in transition metals tend to be much narrower than s- or p-electron bands?

Model answer: d orbitals are more spatially localized than s or p orbitals at the same principal quantum number, because their angular momentum keeps them closer to the nucleus. This means the overlap integral between d orbitals on neighboring atoms is smaller, giving a smaller hopping integral t. Since bandwidth is proportional to t (and to the coordination number), d-bands are narrow — typically 3-5 eV wide compared to 10-15 eV for sp-bands. This narrowness has profound consequences: narrow bands mean high density of states, strong correlation effects, and magnetism (as in iron, cobalt, nickel).

The same logic applies even more strongly to f-electrons in rare earths and actinides. Their 4f/5f orbitals are so localized that hopping is tiny, producing extremely narrow bands (~0.1 eV) where electron correlations dominate completely.