13.3 Band Theory and Electrical Properties of Solids

Electronic Band Structure

Band Theory and Energy Levels

In isolated atoms, electrons occupy discrete energy levels. When billions of atoms pack together in a solid, those discrete levels broaden into near-continuous ranges of allowed energies called bands. Band theory uses this idea to explain why some solids conduct electricity and others don't.

Two bands matter most:

• The valence band is the highest range of energy levels that electrons occupy at absolute zero. These electrons are still bound to their parent atoms.
• The conduction band is the next range of allowed energies above the valence band. Electrons that reach this band can move through the solid and carry current.
• The band gap ($E_g$) is the energy range between these two bands where no allowed electron states exist. Its size controls a material's electrical behavior.
• The Fermi level ($E_F$) is the energy of the highest occupied electron state at absolute zero. In metals, it sits inside a band. In semiconductors and insulators, it falls within the gap.

Types of Band Structures

The size of the band gap sorts solids into three categories:

• Metals have their valence and conduction bands overlapping (or a partially filled band), so electrons move freely with no energy barrier. This is why metals conduct so well.
• Insulators have large band gaps, typically greater than ~4 eV. Thermal energy at room temperature is far too small (only ~0.025 eV) to push electrons across that gap, so virtually no conduction occurs. Diamond, for example, has a band gap of ~5.5 eV.
• Semiconductors have smaller band gaps, roughly 0.1–4 eV. Silicon sits at 1.1 eV and germanium at 0.67 eV. At room temperature, a small fraction of electrons can be thermally excited across the gap, giving modest conductivity that increases with temperature.

Electrical Conductivity

Conductors and Their Properties

Metals conduct electricity because they have a partially filled band (or overlapping bands), providing a large population of electrons that can respond to an applied electric field.

• Copper, aluminum, silver, and gold are among the best metallic conductors. Silver has the highest conductivity of any element, but copper is used more widely because of cost.
• As temperature decreases, lattice vibrations diminish, so electrons scatter less often. This means metallic conductivity increases at lower temperatures.
• Superconductors are an extreme case: below a material-specific critical temperature ($T_c$), electrical resistance drops to exactly zero. For example, $T_c$ for mercury is 4.2 K, while some cuprate ceramics superconduct above 90 K.

Insulators and Semiconductors

Insulators like rubber, glass, and most ceramics have band gaps so large that negligible numbers of electrons reach the conduction band under normal conditions.

Semiconductors sit between conductors and insulators:

• An intrinsic (pure) semiconductor has equal numbers of electrons in the conduction band and holes (vacant electron states) in the valence band. Both contribute to current flow.
• Raising the temperature excites more electrons across the gap, creating more electron-hole pairs. That's why semiconductor conductivity rises with temperature, the opposite trend from metals.
• The conductivity of an intrinsic semiconductor depends exponentially on temperature and band gap: $\sigma \propto e^{-E_g / 2k_BT}$, where $k_B$ is Boltzmann's constant and $T$ is absolute temperature.

Semiconductor Modification

Doping Process and Effects

Intrinsic semiconductors have limited conductivity on their own. Doping deliberately introduces small, controlled amounts of impurity atoms into the crystal lattice to dramatically change its electrical properties.

Even tiny concentrations matter. Adding roughly 1 impurity atom per million host atoms can increase conductivity by several orders of magnitude. The type of impurity determines how the conductivity changes.

N-type and P-type Semiconductors

The two types of doping correspond to adding either extra electrons or extra holes:

• N-type: A Group 15 element like phosphorus or arsenic replaces a silicon atom. Phosphorus has five valence electrons compared to silicon's four, so one electron per dopant atom is left over with no bond to fill. These extra electrons become the majority charge carriers, and the Fermi level shifts upward toward the conduction band.
• P-type: A Group 13 element like boron or gallium replaces a silicon atom. Boron has only three valence electrons, leaving one bond incomplete. This creates a hole that behaves like a positive charge carrier. Holes become the majority carriers, and the Fermi level shifts downward toward the valence band.

When an n-type region meets a p-type region, the result is a p-n junction. At the interface, electrons and holes recombine to form a depletion zone with a built-in electric field. This junction is the basis of diodes, transistors, LEDs, and solar cells.