Free Electron Model and Energy Bands
Free Electron Model
An alternative approach to understanding the electronic band structure of solids is to consider electron waves in a periodic crystalline potential. The Drude–Lorentz free electron model treats a metallic solid as a lattice of positive ions surrounded by free electrons.
This model explains:
- High electrical conductivity
- High thermal conductivity
- Optical reflectivity
- Ductility
- Alloy formation
In a more accurate quantum mechanical description, electrons are treated as particles confined in a potential well (particle-in-a-box model), which prevents them from leaving the metal.
Particle in a Box Model
Electrons are assumed to move inside a potential well.
Allowed wavelengths:
\[ \lambda_n = \frac{2L}{n} \]Wave vector:
\[ k_n = \frac{2\pi}{\lambda} = \frac{n\pi}{L} \]Wave Function
Allowed wavefunctions:
\[ \psi_n = \left(\frac{2}{L}\right)^{1/2} \sin \left(\frac{n\pi x}{L}\right) \]Energy Levels
Energy of electron:
\[ E_n = \frac{n^2 h^2}{8mL^2} \]Where:
- h = Planck's constant
- m = electron mass
- L = length of box
Energy level diagram showing allowed wavefunctions for an electron in a one-dimensional potential well.
Energy Bands
Since the potential energy inside the box is assumed to be zero, the total energy is purely kinetic. The relation between energy and wave vector is parabolic. Although energy levels are discrete, in real crystals the size (L) is very large, making the spacing between levels extremely small. Hence, the energy spectrum appears quasi-continuous and forms energy bands.
As the electron becomes more confined (smaller L), energy spacing increases. This is important in quantum confinement and nanostructures.
Parabolic E-k relation with occupied energy states.
Crystalline Solids
In three-dimensional crystals, atoms are arranged periodically in space. This periodic arrangement leads to translational symmetry, meaning the structure repeats itself in all directions. Such symmetry leads to conservation laws and introduces the concept of crystal momentum.
Periodicity of Crystal Lattices
The periodic arrangement of atoms produces a periodic potential inside the crystal. Solving the Schrödinger equation under this periodic potential leads to special wavefunctions described by Bloch’s theorem.
Bloch Theorem
Bloch’s theorem states that the wavefunction of an electron in a periodic potential is:
\[ \psi(r) = u_k(r)e^{ikr} \] \[ u_k(r+T)=u_k(r) \]where \(u_k(r)\) has the same periodicity as the lattice.
Boundary condition:
\[ \psi(Na)=\psi(0) \]Allowed values of k:
\[ k=\frac{2n\pi}{Na} = \frac{2n\pi}{L} \]Read More...
