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 and, Alloy formation
In a more accurate description electrons are treated quantum mechanically. Here the free valence electrons are assumed to be constrained within a potential well which essentially stops them from leaving the metal (the ‘particle-in-a-box’ model).
Particle in a Box Model
Electrons are assumed to move inside a potential well (particle-in-a-box model).
Allowed wavelengths:
\[ \lambda_n = \frac{2L}{n} \]
Wave vector:
\[ k_n = \frac{2\pi}{\lambda} \]
\[ k_n = \frac{n\pi}{L} \]
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 also showing the form of some of the allowed wavefunctions for an electron confined to a one-dimensional potential well.
Energy Bands
En represents solely kinetic energy since the potential energy is assumed to be zero within the box. Thus there is a parabolic relationship between En and n, and therefore between En and k since k depends directly on n as described above. The permitted energy levels on this parabola are discrete (i.e., quantized): however in principle the size of L for most metal crystals (ranging from microns to millimetres or even centimetres) means that the separation between levels is very small compared with the thermal energy kBTand we can regard the energy distribution as almost continuous (quasi-continuous) so that the levels form a band of allowed energies as shown in figure 1.7. Note that as the electron becomes more localized (i.e., L decreases), the energy of a particular electron state (and more importantly the spacing between energy states) increases; this has important implications for bonding and also for reduced-dimensionality or quantum-confined systems which are discussed later. For very large crystals the spacing between energy levels becomes extremely small. Thus discrete levels merge to form continuous energy bands.
Schematic version of the parabolic relationship between the allowed electron wave vectors and the their energy for electrons confined to a one-dimensional potential well. Shaded energy regions represent those occupied with electrons.
Crystalline solids
The above arguments may be extended from one to three dimensions to consider the electronic properties of bulk crystalline solids. For a perfectly ordered three dimensional crystal, the periodic repetition of atoms (or molecules) along the one dimensional linear chain is replaced by the periodic repetition of a unit cell in all three dimensions. The unit cell contains atoms arranged in the characteristic configuration of the crystal, such that contiguous replication of the unit cell throughout all space is sufficient to generate the entire crystal structure. In otherwords, the crystal has translational symmetry, and the crystal structure may be generated by translations of the unit cell in all three dimensions. Translation symmetry in a periodic structure is a so-called discrete symmetry, because only certain translations – those corresponding to integer multiples of the lattice translation vectors derived from the unit cell – lead to symmetry-equivalent points. Generally, symmetries generate conservation laws; this is known as Noether’s theorem. The continuous translation symmetry of empty space generates the law of momentum conservation; the weaker discrete translation symmetry in crystals leads to a weaker quasi-conservation law for quasi- or crystal momentum. An important consequence of discrete translation symmetry for the electronic properties of crystals is Bloch’s theorem, which is described below.
Bloch Theorem
The three dimensional periodicity of the atomic arrangement in a crystal gives rise to a corresponding periodicity in the internal electric potential due to the ionic cores. Incorporating this periodic potential into the Schro¨ dinger equation results in allowed wavefunctions that are modulated by the lattice periodicity. Bloch’s theorem states that these wavefunctions take the form of a plane wave (given by exp (ik.r)) multiplied by a function which has the same periodicity as the lattice; i.e.,
\[ \psi(r) = u_k(r)e^{ikr} \]
\[ u_k(r+T)=u_k(r) \]
where the potential function uk(r) has the property for any lattice translation vector T. Such wavefunctions are known as Bloch functions, and represent travelling waves passing through the crystal, but with a form modified periodically by the crystal potential due to each atomic site. if we impose periodic 1D boundary conditions at the ends of the chain of atoms of length L=Na:\[ \psi(Na)=\psi(0) \]
Allowed values:
\[ k=\frac{2n\pi}{Na} \]
\[ k=\frac{2n\pi}{L} \]
First Brillouin Zone
For one-dimensional lattice:
\[ -\frac{\pi}{a} \le k \le \frac{\pi}{a} \]
Fermi Energy
Highest occupied energy at 0 K:
\[ E_F = \frac{\hbar^2 k_F^2}{2m} \]
\[ k_F = (3\pi^2 n_e)^{1/3} \]
\[ E_F = \frac{\hbar^2}{2m}(3\pi^2 n_e)^{2/3} \]
Density of States
\[ N(E) = \frac{4\pi(2m)^{3/2}E^{1/2}}{h^3} \]
Fermi–Dirac Distribution
\[ f(E) = \frac{1}{e^{(E-E_F)/k_B T}+1} \]
At 0 K all states below \(E_F\) are filled and states above \(E_F\) are empty.
Electrical Conduction
- Metal: Partially filled band → good conductor
- Semiconductor: Small band gap
- Insulator: Large band gap
