Intrinsic Semiconductors

A semiconductor is a material whose electrical conductivity sits between that of a conductor (like copper) and an insulator (like rubber). An intrinsic semiconductor is a perfectly pure sample of such a material — no foreign atoms added, no defects intentionally introduced. Silicon (Si) and germanium (Ge) are the classic examples. At absolute zero (0 K), an intrinsic semiconductor behaves like a perfect insulator: every electron is locked in a covalent bond and there are no free charge carriers. Raise the temperature, however, and something remarkable happens: thermal energy breaks some of those bonds, liberating electrons and allowing electrical conduction. The key idea is that semiconductors possess a band gap — an energy range where no electron states exist — separating the filled valence band from the empty conduction band. In silicon this gap is about 1.1 eV, much smaller than the ~9 eV gap in a typical insulator like diamond, but still far larger than the zero gap in metals. Because the gap is small enough for thermal energy (or absorbed photons) to bridge it, the conductivity of an intrinsic semiconductor increases dramatically with temperature — the exact opposite of a metal's behaviour. When a bond breaks, two distinct charge carriers are created simultaneously: a free electron (negative charge –e) and a hole (an absence of an electron that behaves as a positive charge +e). Both carriers contribute to current flow. In intrinsic semiconductors these two populations are always equal in number, a relationship enforced by the physics of electron–hole pair generation and recombination. Understanding intrinsic semiconductors is the essential foundation before studying doped (extrinsic) semiconductors and the p–n junction devices that underpin virtually all modern electronics.

Key takeaways

Covalent Bonding and the Crystal Lattice
Thermal Generation of Electron–Hole Pairs
Charge Neutrality and the Intrinsic Carrier Concentration
Temperature Dependence of Conductivity
Current Flow: Drift and Diffusion

Covalent Bonding and the Crystal Lattice

Silicon has four valence electrons. In a crystal, each silicon atom forms four covalent bonds with its neighbours, sharing one electron per bond. At 0 K every electron is committed to a bond; the valence band is completely full and the conduction band is completely empty. No current can flow — the material is an insulator.

Thermal Generation of Electron–Hole Pairs

At temperatures above 0 K, lattice vibrations carry thermal energy. Occasionally a vibration delivers energy greater than the band-gap energy E_g to a bonded electron, breaking the bond. The electron is promoted to the conduction band, where it is free to move under an electric field. The vacancy it leaves behind — a 'hole' — can be filled by an adjacent bonded electron, which then leaves its own vacancy. The hole therefore migrates through the lattice in the direction opposite to electron flow, behaving exactly like a positive charge carrier.

Charge Neutrality and the Intrinsic Carrier Concentration

Because electrons and holes are always generated in pairs (and recombine in pairs), the electron concentration n always equals the hole concentration p in an intrinsic semiconductor: n = p = n_i, where n_i is the intrinsic carrier concentration. For silicon at 300 K, n_i ≈ 1.5 × 10¹⁰ cm⁻³ — tiny compared with the ~5 × 10²² atoms cm⁻³ in the crystal, which is why pure silicon is still a poor conductor at room temperature.

Temperature Dependence of Conductivity

The intrinsic carrier concentration grows exponentially with temperature: n_i ∝ T^(3/2) · exp(−E_g / 2k_B T), where k_B is Boltzmann's constant. As T rises, exponentially more bonds break, flooding the crystal with carriers and dramatically increasing conductivity. A simple analogy: the band gap is like a locked door — at low temperature no one has enough energy to open it, but as the 'temperature' (crowd energy) rises, more and more people can. This extreme sensitivity to temperature is both a challenge (devices must be kept cool) and an opportunity (thermistors and photodetectors exploit it).

Current Flow: Drift and Diffusion

When an electric field E is applied, free electrons drift opposite to the field and holes drift along the field, both contributing to conventional current in the same direction. The total current density is J = (n·e·μ_e + p·e·μ_h)·E, where μ_e and μ_h are the electron and hole mobilities respectively. In silicon, μ_e ≈ 1400 cm²/V·s and μ_h ≈ 450 cm²/V·s — electrons are about three times more mobile than holes because they travel through the conduction band with less scattering.

The Key Equation

n_i = \sqrt{N_C \, N_V} \; \exp\!\left(\frac{-E_g}{2k_B T}\right)

n_i — Intrinsic carrier concentration (cm⁻³): Number of free electrons (or holes) per unit volume in the pure semiconductor.
N_C — Effective density of states (conduction band) (cm⁻³): The equivalent number of electron states at the conduction band edge; depends on T^(3/2) and the effective electron mass.
N_V — Effective density of states (valence band) (cm⁻³): The equivalent number of hole states at the valence band edge; depends on T^(3/2) and the effective hole mass.
E_g — Band-gap energy (eV or J): Energy separation between the top of the valence band and the bottom of the conduction band. For Si: 1.12 eV at 300 K.
k_B — Boltzmann constant (J/K): k_B = 1.381 × 10⁻²³ J/K; at 300 K the thermal energy k_BT ≈ 0.026 eV.
T — Absolute temperature (K): Temperature measured from absolute zero. Room temperature ≈ 300 K.

Worked Example

Problem. The band gap of germanium (Ge) is E_g = 0.67 eV. At T = 300 K, use the simplified exponential factor to find the ratio of n_i(Ge) to n_i(Si), given E_g(Si) = 1.12 eV. Assume N_C·N_V is the same for both materials and k_BT = 0.02585 eV at 300 K.

\frac{n_i(\text{Ge})}{n_i(\text{Si})} = e^{8.70} \approx 6.0 \times 10^{3}

Answer. Germanium has an intrinsic carrier concentration roughly 6000 times larger than silicon at 300 K, entirely because its smaller band gap makes electron–hole pair generation exponentially easier.

Figure

Diagram

Origins · 1 minWilliam Shockley and how this was discovered

William Shockley

1910–1989 · American (British-born)

Shockley, together with John Bardeen and Walter Brattain, invented the transistor at Bell Labs in 1947–1948, an achievement built directly on the physics of semiconductors. He developed the first detailed theoretical framework explaining how electron and hole currents flow in semiconducting materials and pioneered the concept of minority carrier injection. The trio shared the 1956 Nobel Prize in Physics for this work, which launched the entire semiconductor electronics industry. Shockley was so driven to commercialise semiconductor technology that he founded Shockley Semiconductor Laboratory in Silicon Valley in 1956 — directly inspiring the name of the region. His notoriously difficult management style caused eight of his best researchers to resign and found Fairchild Semiconductor, a split that seeded dozens of future technology companies.

Extrinsic Semiconductors (Doping) — Adding impurity atoms (dopants) to an intrinsic semiconductor deliberately unbalances the electron and hole populations, creating n-type or p-type materials that are far more conducting and technologically useful.
The p–n Junction — Joining p-type and n-type semiconductors creates a junction whose asymmetric carrier concentrations produce the diode effect — the foundation of LEDs, solar cells, and transistors.
Energy Bands in Solids — The band theory of solids explains why band gaps exist and distinguishes conductors, semiconductors, and insulators at the quantum mechanical level.
Photoelectric Effect — Just as photons can eject electrons from metals, photons with energy above E_g can generate electron–hole pairs in a semiconductor — the principle behind photodetectors and solar cells.