Tight-Binding Model

The tight-binding model describes the electronic band structure by starting from atomic orbitals and allowing them to overlap weakly between neighbouring sites.

1D Chain Dispersion

For a one-dimensional chain with lattice constant a and nearest-neighbour hopping integral t, the band energy is:

$E(k) = E_0 - 2t \cos(ka)$

• At k = 0 (band bottom): E = E₀ − 2t (minimum energy)
• At k = π/a (zone boundary): E = E₀ + 2t (maximum energy)
• Bandwidth: W = 4t

The hopping integral t > 0 represents electron delocalisation — larger t means wider bands and more metallic behaviour.

Effective Mass

Near the band bottom (k ≈ 0), the dispersion is parabolic and defines an effective mass:

$m^* = \frac{\hbar^2}{2 t a^2}$

As a ratio to the free-electron mass:

$\frac{m^*}{m_e} = \frac{\hbar^2}{2 t a^2 m_e}$

Group Velocity

The electron group velocity is:

$v_g = \frac{1}{\hbar}\frac{dE}{dk} = \frac{2ta}{\hbar}\sin(ka)$

Maximum velocity occurs at k = π/(2a): v_max = 2ta/ħ.

Implement

def tight_binding_energy_eV(k_m, t_eV, a_m, E0_eV=0):
    # Returns band energy in eV
    ...

def effective_mass_ratio(t_eV, a_m):
    # Returns m*/m_e (dimensionless)
    ...

def group_velocity_m_s(k_m, t_eV, a_m):
    # Returns group velocity in m/s
    ...