Intrinsic Semiconductors

Introduction

An intrinsic semiconductor is neither a metal with a permanently partly filled band nor a perfect insulator with effectively no thermally accessible carriers. Its valence band is essentially filled and its conduction band essentially empty at low temperature, but a finite gap lets heat or light promote electrons across it. Each promotion leaves behind a mobile vacancy in the valence-band occupancy: a hole . That pair makes electrical transport possible and forms the basis of later discussions of doping, junctions and light emission.

Core explanation

The band gap E g separates the valence-band edge E v from the conduction-band edge E c: E g = E c − E v. Promotion of a valence electron to a conduction state creates one conduction electron and one hole. The electron carries charge −e; the hole acts as a mobile carrier with charge +e because a sequence of valence electrons filling neighbouring empty states moves the empty state oppositely. A hole is not a literal proton or a positively charged atom travelling through the solid. In a chemically pure, electrically neutral intrinsic material at thermal equilibrium, n = p = n i , where n and p are electron and hole number concentrations. MIT's semiconductor teaching module describes thermal excitation and pair creation.

At equilibrium, generation and recombination balance. Individual pairs continuously appear and disappear, but their average concentrations remain steady. Heating usually raises the intrinsic carrier concentration strongly, although the gap and density of available states may also vary with temperature. The probability for an electron to occupy a state is described by the Fermi–Dirac distribution. The density of available states differs in the valence and conduction bands. A simplified nondegenerate model gives n i = √(N cN v) exp[−E g/(2k BT)], where N c and N v are effective band-edge densities of states. The square-root and half-gap factor follow because each excitation creates two carriers. MIT's semiconductor fundamentals lecture derives the equilibrium carrier expressions and identifies their assumptions.

Conductivity is σ = e(nμ n + pμ p) for electron and hole mobilities μ n and μ p in the simple drift model. In an intrinsic sample this becomes σ = en i(μ n + μ p). Equal carrier counts do not imply equal contributions to current; electrons and holes generally have different mobilities because their effective masses and scattering differ. Resistivity is ρ = 1/σ for an isotropic homogeneous sample in the relevant linear regime. Optical illumination can create additional electron–hole pairs, making the sample nonequilibrium. After the light is removed, recombination tends to restore the dark equilibrium concentration. Later pages will distinguish equilibrium mass action from driven photocarrier populations.

“Intrinsic” refers to the origin of carriers under the model, not perfect freedom from defects. Real wafers always have some impurities, surfaces and trap states. A material behaves intrinsically when thermally generated pairs dominate over carriers introduced by dopants or defects. The same sample can be extrinsic over one temperature range and intrinsic at a sufficiently high temperature. At very low temperature, carriers can freeze out. Crystal quality, surface depletion and contacts all affect a measured sample, so an observed resistance is not a direct measure of band gap without an appropriate model.

Step-by-step reasoning

1. Locate filled valence and empty conduction bands and label their energy difference E g. 2. Promote one electron across the gap and count one conduction electron plus one valence hole. 3. For an undoped equilibrium sample, apply charge neutrality n = p. 4. Compute current contributions with both mobilities, rather than treating the carriers as identical. 5. Ask whether temperature, light, dopants or defects invalidate the intrinsic-equilibrium assumption.

Visual explanation

Draw an energy axis with a filled valence band beneath an empty conduction band. A vertical arrow shows an electron crossing the gap; mark a negative electron above and a positive hole below. Draw short curved arrows in the valence band showing neighbouring electrons filling the missing state, so the hole's effective motion is clear. In a second panel, show more pairs after heating, without shifting the band edges unless that effect is being modeled separately.

Real-world analogy

Picture every seat in a lower row occupied and an upper row empty. Moving one person upstairs creates both a person who can move among upper seats and an empty lower seat that can propagate as neighbours switch places. The motion of the empty seat is a convenient description of many individual moves. Unlike a real classroom, crystal states have energies and occupancy probabilities fixed by quantum statistics.

Real-world example

A silicon wafer near room temperature has a small intrinsic carrier population compared with its density of lattice atoms. Heating produces more electron–hole pairs and can greatly reduce intrinsic resistivity. A sensor can exploit light-generated carriers too, but an illuminated photoconductor is not in the simple thermal-equilibrium state assumed by n = p = n i if other charges or fields are present. Silicon's usefulness also depends on controlled dopants and a stable oxide surface, which are separate engineering advantages from the intrinsic pair mechanism.

Why?

Why does a valence hole carry positive charge even though no positive particle was added? Removing an electron from a previously filled set of states changes local electronic charge by +e relative to the filled-band reference. As neighbouring electrons occupy that vacancy, the location of the missing electron moves. In transport equations, representing this process as a positive carrier is simpler and yields the correct direction of conventional current.

Common misconception

“One promoted electron creates two electrons” is wrong: it creates one conduction electron and one hole. “Intrinsic means high purity guarantees equal mobilities” is also wrong: equal numbers follow from pair creation and neutrality, while mobility reflects band structure and scattering. The band gap is not a region of forbidden physical space; it is an energy interval without allowed bulk electronic states in the ideal band picture.

Worked example

Suppose an illustrative intrinsic sample has n i = 2.0 × 10¹⁶ m⁻³, μ n = 0.10 m² V⁻¹ s⁻¹ and μ p = 0.040 m² V⁻¹ s⁻¹. Then σ = e n i(μ n + μ p) = (1.602 × 10⁻¹⁹)(2.0 × 10¹⁶)(0.140) = 4.49 × 10⁻⁴ S m⁻¹ . Electrons contribute 0.10/0.14 ≈ 71% of the drift conductivity despite matching hole number. This exercise assumes both mobilities refer to the stated temperature and that other carrier types or contact limitations are negligible.

Quick check

1. At equilibrium in an ideal undoped semiconductor, what is the relation between electron and hole concentrations? Answer: n = p = n i because thermal excitation creates electrons and holes in pairs and charge neutrality is maintained.

Exam focus

Draw the gap and label both carriers. State the conditions under which n = p holds. If calculating conductivity, include both carrier terms and their separate mobilities. Do not use an intrinsic thermal expression for a strongly doped or illuminated nonequilibrium sample without explaining why it would still apply.

Advanced insight

The intrinsic Fermi level need not lie exactly at the geometrical midpoint of the gap when the conduction and valence effective densities of states differ. A temperature-dependent carrier number can also reflect gap narrowing or changed scattering, so a straight-line conductivity plot alone does not always provide the fundamental gap. More detailed analysis separates concentration and mobility, often through Hall measurements combined with conductivity.

Summary

Thermal excitation across the band gap generates electron–hole pairs in an intrinsic semiconductor. Equilibrium and neutrality give equal concentrations of the two carriers, while different mobilities give unequal current contributions. Temperature, light and doping alter the carrier populations and require careful identification of the applicable regime.

Practice questions

1. If 100 electrons are excited across a gap in an initially ideal filled valence band, how many holes form? Answer: 100 holes, one for each promoted electron. 2. Why is a hole not an ion? Answer: It is an empty valence-band electron state whose effective position moves as electrons exchange occupancy, not a migrating atomic species. 3. For n = p = 10¹⁵ m⁻³ and μ n = 3μ p, which carrier supplies more drift conductivity? Answer: Electrons supply three times the hole contribution because their number is equal but mobility is three times larger. 4. Does strong illumination necessarily leave n = p = n i? Answer: No. Light can create excess nonequilibrium pairs, and charge separation or doping can make n and p unequal locally.