Energy Band Structures in Solids
When a particular element’s atoms are far apart, each individual atom exhibits some permissible energy levels. But, with atoms closely packed in a solid, nearby atoms’ electric fields deeply perturb the motion of the outer – valence – electrons, modifying the precise energy levels exhibited in an individual atom.
Energy Bands
In a particular element, the magnitude of the neighboring atoms’ influence depends on the spacing and the electron’s location within the group of atoms. Accordingly, when other atoms are close, the single atoms’ discrete electron energy levels change into energy bands. An energy band is an energy range with many allowed adjacent energy levels, very closely spaced.
Figure 1 shows the energy bands as a function of the spacing between atoms for tetravalent crystals (Group 4A of the Periodic Table: nonmetal diamond, the semiconductors silicon and germanium, and the metals tin and lead). Diamond, silicon, and germanium are technologically important.
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Figure 1. Energy bands of tetravalent crystals.
The valence band comprises the highest energy electrons in the solid, and the conduction band is the lowest empty belt where electrons can remain. These bands are the permissible bands. The energy band between the permissible bands is the band gap or forbidden band, where the electrons cannot exist. Energies within the band gap are not accessible for electron occupancy. The energy of the band gap is the difference between the valence and the conduction bands.
The energy bands below the valence band (not shown) are all filled and do not contribute to the material’s electrical characteristics. The electrical properties of a solid strongly depend on the number of electrons in the valence band.
In figure 1, the circles with a negative sign are the actual valence electrons of one atom in a filled energy state. The empty circles represent the available energy states for the electrons. The filled valence band and empty conduction band assume a perfect crystal lattice and a temperature of 0 K (-273°C) – the ground or lowest-energy state.
When the lattice spacing decreases from its initial value, a single band appears as a combination of the valence and conduction bands. Reducing the spacing, the band divides into different valence and conduction bands with a bandgap in between.
Since no more than two interacting atoms can have the same energy level, new levels take shape with infinitesimally different magnitudes. This group of energy levels in a polyatomic material – the energy band – represents an energy level in a single atom. Each band contains as many discrete levels of energy as there are atoms in the crystal.
Pauli’s exclusion principle limits the number of electrons in a certain NL atomic energy level in a single atom. Likewise, the principle limits the maximum number of electrons contained in a lattice’s energy band.
Figure 2 compares atomic energy levels with energy bands in a lattice via the electron energy band structure’s usual representation for solid materials.
Figure 2. Atomic energy levels and energy bands.
The alkali metals (hydrogen, lithium, sodium, potassium, rubidium, cesium, and francium) have a valence of one. For example, a 3s-atomic level in sodium (Z = 11) can accommodate two electrons, but it has only one. A lattice of N atoms of sodium can hold 2N electrons within the 3s energy band, but there are only N electrons; in this case, we say that the band is half-filled. Figure 3 shows a half-filled energy band.
Figure 3. Half-filled energy band.
The alkali earth metals (beryllium, magnesium, calcium, strontium, barium, and radium) have two valence electrons per atom. This number of electrons is enough to fill the first energy band with two electrons at each level. However, there is an overlapping of bands because the lower levels of the second band require less energy than the first band’s upper levels, such that some electrons jump to levels of the second band.
This overlapping of the uppermost bands is the typical situation for most metals or conductors. For example, magnesium (Z = 12) with a 1s² 2s² 2p⁶ 3s² configuration has all the atomic shells filled. But the first excited level, 3p, overlaps 3s, and the uppermost electrons of the 3s band overlap with the lower energy states of the 3p band. Some 3s electrons move to low 3p-levels until reaching an equilibrium energy level for both bands. Figure 4 shows the overlapping of energy bands in magnesium.
Figure 4. Overlapping of energy bands in magnesium.
Generally, we may conclude that the lower energy bands are filled, but the valence band may or may not be filled.
About Energy Band Structures in Solids
An electron in an atom occupies one of a series of allowed orbital patterns, with expressly permitted energies.
Although electrons in an atom can occupy only particular energy levels, in a lattice, the other nearby atoms modify the precise energy levels of the electrons of an individual atom. Consequently, the energy levels change and electrons can move within specific energy bands.
The valence orbitals are only the ground-state orbitals for the valence electrons. There are many other vacant crystalline orbitals of higher energy, where the allowed energy levels also fall in bands.
An excited crystal will boost an electron from a valence orbital to an upper excited orbital.
There is an energy difference between the highest valence energy level and the lowest excited orbital energy level – the energy gap.
