PHY3707 Assignment 6 Solutions 2026

PHY3707 Assignment 6 Solutions 2026

Unique number: 197350

Due Date: 11 September 2026

, PHY3707 — Assignment 6
Solid State Physics

QUESTION 1
Find the energy-wavevector relationship for a one-dimensional monatomic crystal
of lattice constant 𝑎using the tight-binding approximation. Use this relation to
obtain an expression for the effective mass. For which value of 𝑘is the electron
velocity maximum?
Tight-binding wavefunction
Consider a one-dimensional monatomic crystal with atoms separated by a distance 𝑎.
In the tight-binding approximation, the electron wavefunction can be written as a linear
combination of atomic orbitals:
1
𝜓𝑘 (𝑥) = ∑ 𝑒 𝑖𝑘𝑛𝑎 𝜙(𝑥 − 𝑛𝑎)
√𝑁 𝑛

where:
• 𝑁= number of atoms
• 𝑎= lattice constant
• 𝑘= wavevector
• 𝜙(𝑥 − 𝑛𝑎)= atomic orbital centred on the 𝑛-th atom.
For nearest-neighbour interactions, an electron can interact with the atom at 𝑛, as well
as its two nearest neighbours:
𝑛 − 1, 𝑛 + 1
Tight-binding energy expression
The tight-binding energy for a one-dimensional monatomic chain is
𝐸(𝑘) = 𝐸0 − 2Δcos(𝑘𝑎)

where:
• 𝐸0 is the energy of the isolated atomic orbital.
• Δis the nearest-neighbour overlap/resonance integral.
Therefore,
𝐸(𝑘) = 𝐸0 − 2Δcos(𝑘𝑎)

This is the required energy-wavevector relationship.
Depending on the sign convention used for the overlap integral, the same result may
also be written as
𝐸(𝑘) = 𝐸0 + 2𝛾cos(𝑘𝑎).

The physical dispersion is the same apart from the definition/sign of the hopping
parameter.