Band Theory of Solids

Equivalent forms

\psi_{nk}(r)=e^{ik\cdot r}u_{nk}(r)

The entire distinction between a copper wire, a diamond, and a silicon chip comes down to the size of one energy gap.

Symbol	Name	SI unit	Dimension	Range
$E_g$	Band gapoutput Energy between valence-band top and conduction-band bottom	J	ML^2T^-2	—
$E_c$	Conduction-band edge Lowest conduction-band energy	J	ML^2T^-2	—
$E_v$	Valence-band edge Highest valence-band energy	J	ML^2T^-2	—
$T$	Temperature Absolute temperature	K	Θ	100 – 600
$k_B$	Boltzmann constant Links temperature to energy	J/K	ML^2T^-2*Θ^-1	—

Where it holds

Crystalline solids with periodic potential (single-electron / independent-electron approximation). Amorphous solids, strong correlations (Mott insulators), and disorder require extensions.

Dimensional analysis

E_g \to [M*L^2*T^-2]

(energy)

E_c - E_v

→ energy

-

energy

Difference of two energies is an energy; the exponent E_g/2k_BT is dimensionless.

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Discovery

Felix Bloch, Alan Wilson · 1931

Bloch's 1928 theorem showed electron waves in a periodic lattice take the form of plane waves modulated by a lattice-periodic function, producing allowed bands and forbidden gaps. Alan Wilson (1931) used it to define metals, insulators, and semiconductors by band filling — explaining why some solids conduct and others don't, a puzzle classical physics never solved.

Research status: stable

Real-world applications

Choosing semiconductor materials by band gap (LEDs, solar cells)
Engineering heterostructures and quantum wells
Predicting a material's optical color and transparency

Common misconceptions

Insulators and semiconductors differ only in gap size, not in kind (Si 1.1 eV vs diamond 5.5 eV)
Bands are momentum-space, not real-space, structures
A partly filled band — not the presence of free electrons per se — is what makes a metal

Experimental verification

Optical absorption edges, Hall measurements of carrier density vs temperature, and ARPES (angle-resolved photoemission) directly map E(k) band structures.

Derivation

Bloch's theorem: for a periodic potential

V(r+R)=V(r)

, eigenstates

are \psi _{nk}(r)=e^{ik\cdot r}u_{nk}(r)

with u lattice-periodic.

Solving in the reduced zone gives energy branches E_n(k) — the bands — separated by gaps where Bragg reflection forbids propagating states.

At the gap the density of states vanishes.

Thermal excitation across a gap E_g follows Boltzmann statistics, so intrinsic carrier density

n \propto \exp (-E_g/2k_BT)

.

What if…

What if the band gap were zero?

Valence and conduction bands touch — a semimetal like graphene, which conducts but with vanishing density of states at the Fermi point.

What if you add donor impurities (doping)?

New levels appear inside the gap, so carriers are freed at room temperature — the basis of every transistor.

1

Carrier boost with temperature

Given ·

E g:: 1.1 eV (Si)
T₁:: 300 K
T₂:: 350 K

Find · ratio of intrinsic carrier densities

Steps

Exponent $= (1.1/2)/(8.617e-5) \times (1/300 - 1/350)$
$= 6383 \times 4.76e-4 = 3.04$
$n_{2}/n_{1} = e^{2.04} \approx 8\times$ more carriers

Result ·

n_{2}/n_{1} = \exp [-E_g/2k_B (1/T_{2} - 1/T_{1})] \approx 8

Drude Free-Electron Model→Fermi-Dirac Distribution→doping n p type