Semiconductors and Band Gaps Introduction

Semiconductors are crystalline solids whose electronic conductivity lies between that of metals and insulators, controlled by a small band gap

E_{g}

(typically 0.1–4 eV) separating filled valence states from empty conduction states. Key result: the number of mobile charge carriers depends exponentially on temperature and gap,

\propto e^{- E_{g} /2 k_{B} T}

, making conductivity tunable by heat, light, and doping. Where: transistors, diodes, solar cells, LEDs, photodetectors. Who/when: band theory developed by Bloch, Wilson, and others (1928–1931); silicon devices from the 1940s onward.

A semiconductor is a material in which a continuous band of occupied electronic states (the valence band) is separated from an empty band of allowed states (the conduction band) by a forbidden energy region — the band gap — that is small enough for thermal energy, light, or impurities to promote electrons across it. The band-gap concept emerges directly from quantum mechanics applied to periodic crystals: the periodic potential of the lattice splits the allowed electron energies into bands separated by gaps. This matters because controlling carrier concentration via temperature, doping, and illumination is the foundation of all modern electronics and optoelectronics. A surprising fact: a perfectly pure semiconductor at absolute zero is a perfect insulator — its "semiconducting" behaviour arises entirely from thermal excitation and impurities, not from any intrinsic conductivity of the pure crystal.

The Core Insight

When atoms assemble into a periodic crystal, the discrete atomic energy levels broaden into continuous bands of allowed states separated by gaps of forbidden energies. Whether a solid is a metal, insulator, or semiconductor is decided by where the electrons stop filling: if the highest occupied band is only partly full, the material is a metal; if a band is exactly filled and the next empty band lies a small energy $E_{g}$ above it, the material is a semiconductor. The exponential sensitivity of carrier number to $E_{g} / k_{B} T$ is what makes semiconductors uniquely controllable.

Formal Statement

Band Gap and Carrier Statistics

The band gap $E_{g}$ is the energy difference between the top of the valence band $E_{v}$ and the bottom of the conduction band $E_{c}$ :

E_{g} = E_{c} - E_{v}

⟳read directly off the band-structure diagram

E (k)

as the minimum vertical separation between the lowest unoccupied and highest occupied bands.

The density of allowed states per unit volume near a parabolic band edge (effective mass $m^{*}$ ) is

g (E) = \frac{1}{2 π ^{2}} (\frac{2 m ^{*}}{ℏ ^{2}})^{3/2} E - E_{c}, E \geq E_{c}

⟳count

k

-states in a sphere of radius

k

in reciprocal space,

N = 2 \cdot \frac{( 4/3 ) π k ^{3}}{( 2 π / L ) ^{3}}

, use

E - E_{c} = ℏ^{2} k^{2} /2 m^{*}

, and differentiate.

The occupation probability of a state at energy $E$ is the Fermi–Dirac distribution:

f (E) = \frac{1}{e ^{(E - E_{F}) / k_{B} T} + 1}

⟳maximize entropy of a fermion gas under fixed particle number and energy;

E_{F}

is the Lagrange multiplier (chemical potential).

Symbols: $E_{g}$ band gap (J or eV), $E_{c}$ / $E_{v}$ conduction/valence band edges, $m^{*}$ carrier effective mass (kg), $ℏ$ reduced Planck constant, $E_{F}$ Fermi level, $k_{B}$ Boltzmann constant, $T$ absolute temperature.

Intrinsic Carrier Concentration

For an intrinsic (undoped) semiconductor with $E_{g} ≫ k_{B} T$ , the equilibrium concentration of conduction electrons $n$ equals the hole concentration $p$ , and

n_{i} = N_{c} N_{v} e^{- E_{g} /2 k_{B} T}

where the effective densities of states are

N_{c} = 2 (\frac{m _{e}^{*} k _{B} T}{2 π ℏ ^{2}})^{3/2}, N_{v} = 2 (\frac{m _{h}^{*} k _{B} T}{2 π ℏ ^{2}})^{3/2}

⟳integrate

g (E) f (E)

over the conduction band using the Boltzmann tail

f \approx e^{- (E - E_{F}) / k_{B} T}

; the Gaussian-type integral

\int_{0}^{\infty} x e^{- x / k_{B} T} d x

yields

N_{c}

. Multiply

n \cdot p

and impose

n = p

Why It Holds / Derivation

Band Formation and the Exponential Carrier Law

Step 1 — Periodic potential gives Bloch states. In a crystal the potential satisfies

V (r + R) = V (r)

for lattice vectors

R

. Bloch's theorem (see Bloch's Theorem) forces solutions of the form

ψ_{k} (r) = e^{i k \cdot r} u_{k} (r)

, with

u_{k}

lattice-periodic. Step 2 — Gaps open at zone boundaries. Near a Brillouin-zone boundary, degenerate plane waves

e^{ik x}

and

e^{i (k - G) x}

are coupled by the Fourier component

V_{G}

of the potential. Diagonalizing the

2 \times 2

degenerate-perturbation matrix splits the energy by

2∣ V_{G} ∣

E_{\pm} = \frac{ℏ ^{2}}{2 m} (k^{2} + (k - G)^{2}) /2 \pm ∣ V_{G} ∣

This split is the band gap: $E_{g} \approx 2∣ V_{G} ∣$ in the nearly-free-electron limit.

Step 3 — Filling determines material class. Each band holds

2 N

electrons (

N

unit cells, factor 2 for spin). With an even number of valence electrons per cell, bands fill exactly: the valence band is full, the conduction band empty. The Fermi level sits in the gap. Step 4 — Thermal excitation. Electrons in the Boltzmann tail of

f (E)

are promoted across

E_{g}

. Integrating

g (E) f (E)

over conduction states and

g (E) [1 - f (E)]

over valence states (Step 1 of equation [4]) gives

n = N_{c} e^{- (E_{c} - E_{F}) / k_{B} T}

and

p = N_{v} e^{- (E_{F} - E_{v}) / k_{B} T}

. Step 5 — Mass-action and the gap. Multiplying:

n p = N_{c} N_{v} e^{- E_{g} / k_{B} T}

independent of $E_{F}$ . For intrinsic material $n = p = n_{i}$ , taking the square root yields equation [4]. The factor of 2 in the exponent ( $E_{g} /2$ ) reflects that exciting one electron creates two carriers (an electron and a hole). $■$

Alan Herries Wilson | 1906–1995

In two landmark 1931 papers, The Theory of Electronic Semi-Conductors, Wilson applied the new band theory to explain the distinction between metals, insulators, and semiconductors. He recognized that conductivity arises from electrons thermally excited across a gap and from impurity levels lying within it — the conceptual birth of the band-gap picture of semiconductors. His work built on Bloch's 1928 theory of electrons in periodic lattices and resolved the puzzle of why some solids with filled bands could still conduct weakly.

Building Intuition

Think of energy levels as parking floors in a garage. In a metal, the top occupied floor is half-empty — cars (electrons) can shuffle around freely, so current flows. In an insulator, every floor is either completely full or completely empty, with a huge wall ( $E_{g} \sim 5$ eV) to the next empty floor; no shuffling is possible. A semiconductor is the same as an insulator but the wall is short ( $E_{g} \sim 1$ eV) — at room temperature a few thermally energetic electrons can hop over the wall, leaving empty spaces ("holes") behind on the lower floor. Now both floors can have movement: electrons on top, holes below. The exponential $e^{- E_{g} /2 k_{B} T}$ is just the Boltzmann odds of finding an electron with enough energy to make the jump.

Band structure of a metal, semiconductor, and insulator. The conduction band (top), forbidden gap (middle), and valence band (bottom) are shown; the gold curve traces the Fermi–Dirac occupation.

Worked Examples

Intrinsic carrier concentration of silicon

Given: Silicon at

T = 300

E_{g} = 1.12

eV,

m_{e}^{*} = 1.08 m_{0}

m_{h}^{*} = 0.81 m_{0}

k_{B} T = 0.02585

eV,

m_{0} = 9.11 \times 1 0^{- 31}

kg. Step 1 — Compute $N_{c}$ . Using equation [5] with the prefactor

2 (m_{0} k_{B} T /2 π ℏ^{2})^{3/2} = 2.51 \times 1 0^{19} cm^{- 3}

at 300 K:

N_{c} = 2.51 \times 1 0^{19} (1.08)^{3/2} = 2.82 \times 1 0^{19} cm^{- 3} .

Step 2 — Compute $N_{v}$ .

N_{v} = 2.51 \times 1 0^{19} (0.81)^{3/2} = 1.83 \times 1 0^{19} cm^{- 3} .

Step 3 — Exponential factor.

\frac{E _{g}}{2 k _{B} T} = \frac{1.12}{2 \times 0.02585} = 21.66, e^{- 21.66} = 3.9 \times 1 0^{- 10} .

Step 4 — Combine (equation [4]).

n_{i} = (2.82 \times 1 0^{19}) (1.83 \times 1 0^{19}) \times 3.9 \times 1 0^{- 10} .

N_{c} N_{v} = 2.27 \times 1 0^{19} cm^{- 3} .

n_{i} \approx 9.0 \times 1 0^{9} cm^{- 3}

(The accepted value is $\sim 1.0 \times 1 0^{10} cm^{- 3}$ ; the small difference comes from temperature-dependence of $E_{g}$ and rounding.)

Sensitivity to band gap

Given: Compare intrinsic carrier concentrations of germanium (

E_{g} = 0.66

eV) and diamond (

E_{g} = 5.47

eV) at 300 K, taking equal prefactors

N_{c} N_{v} \approx 2 \times 1 0^{19} cm^{- 3}

. Step 1 — Germanium exponent.

\frac{0.66}{2 ( 0.02585 )} = 12.77, e^{- 12.77} = 2.85 \times 1 0^{- 6} .

n_{i} (Ge) \approx 2 \times 1 0^{19} \times 2.85 \times 1 0^{- 6} = 5.7 \times 1 0^{13} cm^{- 3} .

Step 2 — Diamond exponent.

\frac{5.47}{2 ( 0.02585 )} = 105.8, e^{- 105.8} \approx 1 0^{- 46} .

n_{i} (diamond) \approx 2 \times 1 0^{19} \times 1 0^{- 46} = 2 \times 1 0^{- 27} cm^{- 3} .

Step 3 — Interpret. Diamond's predicted intrinsic carrier density is below one carrier in the entire observable universe of crystal — it behaves as a perfect insulator.

n_{i} (Ge) / n_{i} (diamond) \sim 1 0^{40}

A factor of 8 in $E_{g}$ produces 40 orders of magnitude in conductivity — the exponential law at work.

Common Misconceptions

"Semiconductors conduct better than insulators because they have more electrons." They do not have more electrons — a pure semiconductor at

T = 0

has exactly the same filled-band structure as an insulator and conducts nothing. The difference is purely the size of the gap, which sets the exponential probability of thermal excitation. Likewise, "holes" are not real particles but the absence of an electron in the valence band, behaving as positive charge carriers. A small gap does not guarantee semiconducting behaviour. Consider a hypothetical material with

E_{g} = 0

where the conduction-band minimum touches the valence-band maximum — a semimetal (e.g. bismuth, graphene). Here carriers exist at all temperatures and the

e^{- E_{g} /2 k_{B} T}

law degenerates; conductivity does not vanish at

T \to 0

. The clean exponential carrier law strictly requires

E_{g} ≫ k_{B} T > 0

. Conversely, an "insulator" like diamond is structurally a wide-gap semiconductor — the metal/semiconductor/insulator boundary is one of degree, not of kind.

Connections

The band-gap picture rests on Bloch's Theorem, which guarantees that electrons in a periodic potential occupy extended states labelled by crystal momentum $k$ . The detailed shape $E (k)$ comes from the Nearly Free Electron Model and Tight Binding Model, two complementary approximations. Carrier statistics use Fermi-Dirac Statistics and the concept of the Fermi Level. The exponential law is a special case of the Boltzmann Distribution applied to a two-level (valence/conduction) system, and doping introduces ideas from Donor and Acceptor States. Optical absorption across the gap connects to Photon Absorption and Emission.

Real-World Applications

Silicon transistors / MOSFETs ( $E_{g} = 1.12$ eV): the workhorse of all digital logic; the gap ensures a controllable off-state.
Solar cells (Si, GaAs, perovskites): photons with $h ν > E_{g}$ create electron–hole pairs; gap is optimized near 1.3 eV (Shockley–Queisser limit) for the solar spectrum.
Light-emitting diodes (LEDs): GaN ( $E_{g} = 3.4$ eV, blue), GaAs (1.42 eV, infrared) — emission wavelength set directly by $λ = h c / E_{g}$ .
Laser diodes in fibre-optic communication (InGaAsP tuned to 1.55 µm, the silica low-loss window).
Photodetectors and image sensors (CCD/CMOS): gap determines the cutoff wavelength; HgCdTe is engineered for infrared night vision.
Thermoelectric generators (Bi₂Te₃): narrow-gap semiconductors convert heat gradients to voltage.
Power electronics: wide-gap SiC ( $E_{g} = 3.3$ eV) and GaN devices handle high voltages and temperatures in electric vehicles and grid inverters.

Historical Development

1821 — Thomas Seebeck observes thermoelectric effects in semiconducting compounds (unexplained at the time).
1833 — Michael Faraday notes that silver sulphide's resistance decreases with temperature, opposite to metals — the first hint of semiconducting behaviour.
1874 — Ferdinand Braun discovers rectification at metal–semiconductor contacts (the point-contact diode).
1928 — Felix Bloch proves Bloch's theorem, establishing electron bands in periodic crystals.
1930–1931 — Rudolf Peierls and Léon Brillouin develop the zone scheme and explain gap formation at zone boundaries.
1931 — Alan Wilson publishes the band theory of semiconductors, defining intrinsic vs. extrinsic conduction.
1939 — Boris Davydov and Walter Schottky develop the theory of the p–n junction barrier.
1947 — Bardeen, Brattain, and Shockley demonstrate the point-contact transistor at Bell Labs (Nobel 1956).
1954 — First practical silicon solar cell (Chapin, Fuller, Pearson); the integrated circuit follows in 1958–59 (Kilby, Noyce).

Subtleties & Limits

Real band gaps are more subtle than the simple two-band picture suggests. Direct vs. indirect gaps: in silicon the conduction-band minimum lies at a different $k$ than the valence maximum, so optical transitions require a phonon — making silicon a poor light emitter despite decades of effort to build a silicon laser. Temperature dependence: $E_{g}$ itself shrinks with $T$ (Varshni's empirical law), correcting the naive exponential. Many-body effects: the single-particle band gap measured by photoemission (the "quasiparticle gap") differs from the optical gap by the exciton binding energy, and standard density-functional theory systematically underestimates gaps — the "band-gap problem" still drives research in computational materials science (GW approximation, hybrid functionals). Topological materials and Mott insulators further show that band-filling alone does not determine conductivity once strong correlations or topology enter.