Band Structure

Tight-binding model — E(k) dispersion, Band Gap, material classification

Parameters
Band Structure E(k)
Bloch Theorem

In a periodic potential, the electron wave function is the product of a plane wave and a periodic function:

$$ \psi_{nk}(r) = e^{ikr} u_{nk}(r) $$
Tight-Binding Model (1D)

The simplest model — electron hops between neighboring atoms with energy t:

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

t > 0 — hopping integral (eV). a — lattice parameter. k — wave vector in the first Brillouin zone.

Band Gap

The Δ parameter defines the potential difference between atoms. Band Gap = 2|Δ|:

$$ E_g = 2|\Delta| $$
TypeEg (eV)Example
Metal0Cu, Al, Au
Semiconductor0.1–3.0Si, Ge, GaAs
Insulator> 3.0SiO₂, diamond
First Brillouin Zone

For a 1D lattice, k-space is restricted to the first Brillouin zone:

$$ k \in \left[-\frac{\pi}{a},\, \frac{\pi}{a}\right] $$
Step-by-Step Guide
1
Real Materials
In the Real Materials tab — Si, Ge, GaAs, Cu, SiO₂. View E(k) band diagram and Band Gap.
2
1D tight-binding
With t (hopping) and a (lattice) parameters — E(k) = -2t·cos(ka). Increasing t → band widens.
3
2-Band Model
Δ=0 → metal (bands overlap). Δ>0 → semiconductor/insulator (Band Gap = 2Δ).
4
VBM and CBM
VBM (Valence Band Maximum) and CBM (Conduction Band Minimum). Their difference = Band Gap.
Interpreting Results

Metal: valence and conduction bands overlap — electrons move freely. Semiconductor: Band Gap 0.1–3 eV — conductivity changes with temperature or light. Insulator: Band Gap > 3 eV — electrons cannot reach the conduction band.

Self-Assessment

Q1. What does Band Gap mean?

A Bandwidth
B Energy difference between valence and conduction bands
C Kinetic energy of electron
D Lattice parameter

Q2. What is the approximate Band Gap of Si?

A 0.1 eV
B 3.5 eV
C 1.12 eV
D 9.0 eV

Q3. In the tight-binding model, increasing t causes:

A Bandwidth increases
B Bandwidth decreases
C Band Gap increases
D Nothing changes