Solid State Physics — Electronic Properties of Materials
Energy Band Theory
What moves? Electrons occupy allowed states. A hole is an unoccupied valence-band state, not a separate positive particle. When a neighbouring electron fills it, the vacancy shifts in the opposite direction. The row below the bands demonstrates this process.
Filled blue dots are electrons; hollow pink markers show vacancies. Without an applied field, thermal motion gives no net drift. The f(E) curve shows occupation probability, not the number of states.
Teaching model: complete dopant ionisation and non-degenerate statistics; counts are amplified for visibility. Freeze-out at low temperature and very heavy doping are outside its quantitative accuracy. The gap is held constant. Read about electrons and holes.
Calculated
Key Formula
ni = √(NcNv) · exp(−Eg/2kT)
ni ≈ 1.5e+10 cm⁻³
———————
n · p = ni²
n = 1.5e+10
p = 1.5e+10
Curriculum
Semiconductors — The Tunable Middle Ground
- •The band gap Eg ≈ 0.5–2 eV is small enough that thermal excitation creates a measurable number of electron-hole pairs at room temperature, but large enough that conductivity is controllable.
- •Intrinsic semiconductors: ni = √(NcNv) exp(−Eg/2kT). For Si at 300 K, ni is of order 10¹⁰ cm⁻³ — tiny compared to metals (~10²² cm⁻³), but enough for electronics.
- •Doping introduces donor (n-type) or acceptor (p-type) impurities at ~10¹⁵–10¹⁹ cm⁻³, overwhelming intrinsic carriers and shifting the Fermi level toward Ec or Ev.
- •The Fermi level is the energy at which the probability of occupation is 50%. In n-type material it rises toward Ec; in p-type it falls toward Ev.
- •Temperature dependence: at low T, carriers freeze out onto dopants. At moderate T, extrinsic conduction dominates. At high T, intrinsic carriers dominate again — this limits device operating temperatures.
General Principles
- •Energy bands arise from the periodic potential of the crystal lattice. The Schrödinger equation in a periodic potential (Kronig-Penney model) produces allowed energy bands separated by forbidden gaps.
- •The Pauli exclusion principle prevents electrons from occupying the same quantum state. A filled band cannot conduct because there are no adjacent empty states for electrons to move into.
- •The Fermi-Dirac distribution f(E) = 1/(exp((E−Ef)/kT) + 1) gives the probability that a state at energy E is occupied. At T = 0, it is a step function; at T > 0, the step smooths over ~kT.
- •Direct bandgap (GaAs): the conduction band minimum and valence band maximum occur at the same k-point. An electron can drop directly, emitting a photon — ideal for LEDs and lasers.
- •Indirect bandgap (Si, Ge): the band edges occur at different k-points. Recombination requires a phonon to conserve momentum, making photon emission far less efficient.
- •The density of states g(E) ∝ √(E−Ec) in the conduction band and ∝ √(Ev−E) in the valence band. This parabolic shape arises from the free-electron-like dispersion near band edges.