Band Theory and Semiconductors

The free-electron gas explains conduction but treats every metal alike; it cannot say why quartz resists $1 0^{16} Ω \cdot m$ while copper resists $1 0^{- 8} Ω \cdot m$ , a span of twenty-four orders of magnitude. The missing ingredient is the effect of the periodic lattice on the electron energies. Adding it splits the continuous free-electron energies into bands, and the band structure decides everything.

Bands from a periodic potential

An electron in a crystal moves in the periodic potential of the ion lattice. Bloch proved that the solutions of the Schrödinger equation in a potential of period $L$ have the form

ψ (x) = u_{k} (x) e^{ik x}, u_{k} (x) = u_{k} (x + L),

a plane wave modulated by a function with the lattice periodicity. Solving the Schrödinger equation for a periodic array of square wells (the Kronig-Penney model) shows that traveling-wave solutions exist only for certain ranges of energy — the allowed bands — separated by forbidden gaps where no traveling wave can propagate.

The gaps appear at wave numbers satisfying $k a = nπ$ , which is the Bragg condition: there the electron wave is Bragg-reflected by the lattice and forms a standing wave. Two standing waves are possible, $ψ_{1} \propto sin (π x / a)$ concentrating charge between the ions and $ψ_{2} \propto cos (π x / a)$ concentrating it on the ions. They have different potential energies, and that difference is the gap.

Energy versus wave number. A free electron follows the smooth parabola E = ℏ²k²/2m; in the periodic lattice, gaps open at the Brillouin-zone edges k_1 = π/a and k_2 = 2π/a, where Bragg reflection forms standing waves, breaking the parabola into allowed bands.

The same bands emerge from the other direction: as $N$ atoms are brought together, each sharp atomic level splits into $N$ closely spaced levels, exactly as the two atomic levels of H₂ split into bonding and antibonding. For a macroscopic $N$ these merge into a quasi-continuous band.

As the interatomic separation decreases, each discrete atomic level broadens into a band of N closely spaced levels; the equilibrium spacing R_0 fixes which bands overlap and how wide the gap between them is.

Conductors, insulators, semiconductors

The band occupied by the outermost electrons is the valence band; the next higher band is the conduction band. Whether a solid conducts depends on how these are filled.

Four band structures. A partly filled band (a) or overlapping bands (b) leave empty states just above the filled ones, giving a conductor; a wide gap (c) gives an insulator; a narrow gap (d) gives a semiconductor.

Conductor. The valence band is only partly filled (sodium: one $3 s$ electron per atom, so the $3 s$ band is half full), or a filled band overlaps an empty one (magnesium: filled $3 s$ overlapping empty $3 p$ ). Empty states sit just above the filled ones, so a field accelerates the electrons freely.
Insulator. The valence band is completely full and the gap to the empty conduction band exceeds about $2 eV$ . Diamond's gap is $7 eV$ ; ordinary fields cannot lift an electron across it.
Semiconductor. The valence band is full but the gap is small — silicon $1.1 eV$ , germanium $0.7 eV$ — so thermal excitation puts a modest number of electrons into the conduction band.

Every electron promoted to the conduction band leaves a vacancy, or hole, in the valence band. Neighboring valence electrons hop into the hole, so the hole migrates in the direction of the field and acts like a positive charge carrier.

The number of carriers is set by the Fermi-Dirac distribution evaluated a half-gap above the Fermi level, which sits near the middle of the gap:

f_{F D} (E) = \frac{1}{e ^{(E - E_{F}) / k T} + 1} .

$E - E_{F}$	multiple of $k T$	$f_{F D}$ at 293 K
$0.05 eV$	2	$1.2 \times 1 0^{- 1}$
$0.10 eV$	4	$1.8 \times 1 0^{- 2}$
$0.25 eV$	10	$5.1 \times 1 0^{- 5}$
$1.0 eV$	40	$6.5 \times 1 0^{- 18}$

The occupation collapses as the gap widens: raising the gap from $0.25$ to $1.0 eV$ drops the carrier fraction by twelve orders of magnitude. This also explains the negative temperature coefficient of a semiconductor's resistivity — heating creates far more carriers than it adds scattering, so resistance falls as temperature rises, opposite to a metal.

Near a band edge the $E$ -versus- $k$ curvature differs from a free electron, and it is captured by an effective mass $m^{*}$ defined through $1/ m^{*} = (1/ ℏ^{2}) d^{2} E / d k^{2}$ . For silicon $m^{*} \approx 0.2 m_{e}$ ; effective masses can be a small fraction of $m_{e}$ and can even be negative near the top of a band.

Doping

Adding a controlled trace of impurity — doping — creates carriers without thermal excitation across the full gap. Replacing a silicon atom (four valence electrons) with arsenic (five) leaves one weakly bound extra electron. The electron plus the arsenic ion core form a hydrogen-like system, but embedded in a medium of dielectric constant $κ$ and with the electron's effective mass, so its binding energy is scaled far down:

E_{1} = - \frac{1}{2} (\frac{k e ^{2}}{ℏ})^{2} \frac{m ^{*}}{κ ^{2}} \frac{1}{n ^{2}}, ⟨ r_{1} ⟩ = a_{0} \frac{κ}{m ^{*} / m _{e}} .

For silicon ( $m^{*} = 0.2 m_{e}$ , $κ = 11.8$ ) the ground state lies only $0.020 eV$ below the conduction band, with orbit radius $\sim 3 nm$ — sixty Bohr radii. These donor levels sit just under the conduction band and give up their electrons easily, making an n-type semiconductor (negative carriers).

Replacing silicon with gallium (three valence electrons) does the opposite: the gallium accepts an electron from the valence band, creating a hole. Its acceptor levels sit just above the valence band, making a p-type semiconductor (positive carriers).

Doping introduces levels inside the gap. Donor levels (n-type) lie just below the conduction band and release electrons upward; acceptor levels (p-type) lie just above the valence band and capture electrons, leaving holes.

Doping is potent: one impurity atom per million can raise the conductivity by several orders of magnitude.

The p-n junction

Join an n-type and a p-type region — in practice a single silicon crystal doped differently on its two sides. Electrons diffuse from the n side to the p side and holes the other way, until the resulting charge double layer builds a contact potential that stops further diffusion. The junction region, swept clear of carriers, is the high-resistance depletion region.

The p-n junction in equilibrium: diffusion of electrons and holes leaves a depletion region of fixed charge and a contact potential that halts further diffusion; the n side sits at higher potential than the p side.

Apply an external voltage. Forward bias (battery's positive terminal on the p side) lowers the barrier and floods the junction with diffusing carriers, giving a large current. Reverse bias raises the barrier and blocks diffusion; only a tiny minority-carrier current flows. The junction conducts essentially one way — it is a rectifier. Treating the carriers with a Boltzmann tail gives the diode equation,

I = I_{0} (e^{e V_{b} / k T} - 1),

where $I_{0}$ is the small reverse saturation current.

The diode current-voltage curve. Forward bias gives an exponentially rising current; reverse bias gives only the tiny saturation current I_0 until breakdown, so the junction passes current in one direction.

Variations on the junction give a family of devices:

Tunnel diode. Doping both sides so heavily that the bands overlap lets electrons tunnel across the thin depletion region; over part of the curve raising the voltage decreases the current (negative resistance), useful for fast switching.
Solar cell. Photons with energy above the gap create electron-hole pairs; the junction field sweeps them apart, producing a photovoltage of about $0.6 V$ .
Light-emitting diode. Under large forward bias, electrons and holes recombine across the gap and emit photons; a resonant cavity makes it a diode laser, with light-out to power-in efficiency above 50%.
Transistor. Two junctions in series (npn or pnp) with a thin base between emitter and collector; a small base signal controls a large collector current, the basis of amplification and logic.

The Hall effect

Which sign of carrier actually moves is settled by the Hall effect. Pass a current through a thin doped strip in a perpendicular magnetic field $B$ . The magnetic force $q v_{d} \times B$ pushes carriers to one edge regardless of their sign, building a transverse Hall voltage until the electric force balances the magnetic one:

V_{H} = v_{d} B w,

for strip width $w$ . The sign of $V_{H}$ reveals whether the carriers are negative (n-type) or positive (p-type), and its magnitude gives $v_{d}$ , hence the carrier density $n$ from $j = n q v_{d}$ .

Hall geometry: a current along the strip in a perpendicular field B deflects the carriers to one edge, and the resulting transverse Hall voltage V_H reveals the sign and density of the charge carriers.

A closing note on magnetism in solids: it too comes from electron spins. Atoms with unpaired spins carry magnetic moments; when neighboring moments align spontaneously the solid is ferromagnetic, when they merely respond to an applied field it is paramagnetic, and when the response opposes the field — as in a superconductor — it is diamagnetic. The next lessons of this module first make the carrier count quantitative, then follow the p-n junction into the devices it builds.