Band Theory: Conductors, Semiconductors and Insulators

The transistor at the heart of every microprocessor is a direct application of doped-semiconductor band engineering: a metal-oxide-semiconductor field-effect transistor (MOSFET) uses a gate voltage to locally bend the silicon bands, switching a channel between insulating and conducting by moving $E_F$ relative to the band edges — the same physics as Example 3, applied dynamically. Modern chips push feature sizes below 5 nm, where quantum confinement begins to modify the very band structure this article assumes is bulk-like, a frontier area of active semiconductor research. Light-emitting diodes and diode lasers use direct-gap III–V and II–VI semiconductors (GaAs, GaN, InP, and their alloys, many crystallizing in the Zinc Blende, Wurtzite and Rutile Structures already familiar from ionic/covalent solid-state structures) where an electron falling from the conduction band directly to the valence band emits a photon of energy essentially equal to $E_g$. Tuning alloy composition tunes $E_g$ continuously, giving the entire visible spectrum from a handful of materials — red and infrared from GaAs-based alloys, blue and white lighting from GaN, a discovery (Akasaki, Amano, Nakamura) recognized with the 2014 Nobel Prize in Physics. Photovoltaic cells run the LED process in reverse: absorbed photons with energy above $E_g$ promote electrons across the gap, and a built-in p–n junction field separates the resulting electron–hole pairs before they recombine, generating usable current. The efficiency ceiling for a single-junction cell (the Shockley–Queisser limit) is itself a band-gap optimization problem — silicon's 1.1 eV gap is close to ideal for the solar spectrum, which is one reason silicon dominates the photovoltaic market despite indirect-gap inefficiencies that other materials avoid. Thermoelectric materials (used in radioisotope thermoelectric generators for deep-space probes, and in solid-state Peltier coolers) exploit the same $E_g$-controlled carrier statistics: engineers deliberately tune doping level and band gap to maximize the Seebeck coefficient while retaining enough carrier density for reasonable electrical conductivity, a genuine multi-parameter optimization of exactly the equations derived above. Transparent conducting oxides such as indium tin oxide combine a wide band gap (transparent to visible light, $E_g > 3$ eV) with heavy degenerate doping (metallic conductivity) — a materials-design trick that seems paradoxical until you recognize it as deliberately engineering both halves of band theory (gap size and Fermi-level position) independently, enabling touchscreens and solar-cell electrodes alike.

:::quiz[Magnesium metal conducts electricity despite having the atomic configuration [Ne]3s². Why?]

• Its 3s electrons are not real electrons in the solid
• The filled 3s band overlaps in energy with the empty 3p band ✓
• Magnesium has a partially filled 2p band instead
• Band theory does not apply to group 2 metals

Ask the tutor: How does the Fermi level move, quantitatively, when a semiconductor is doped with both donors and acceptors simultaneously (compensation), and what happens to $n_i^2$ in that case?

Start from something already familiar: the MO diagram of $\mathrm{H_2}$. Two $1s$ orbitals combine to give a bonding $\sigma_g$ and an antibonding $\sigma_u^*$ MO, split by an energy that depends on the overlap integral between the atoms. Now imagine a one-dimensional chain of $N$ hydrogen atoms, each contributing one $1s$ orbital. Linear combinations of these $N$ atomic orbitals produce $N$ molecular orbitals, ranging from the fully in-phase combination (most bonding, lowest energy) to the fully out-of-phase alternating combination (most antibonding, highest energy). The orbitals in between have an increasing number of nodes, exactly as in a particle-in-a-box or in the $\pi$ system of a long polyene chain. As $N$ grows from 2 to 10 to $10^{23}$, the total energy spread (bandwidth) stays finite — set by the nearest-neighbor overlap integral, roughly constant regardless of chain length — but the number of orbitals packed into that width grows without bound. The level spacing shrinks as $1/N$, and for a mole of atoms ($N \approx 6\times10^{23}$) neighboring levels differ in energy by about $10^{-23}$ eV, utterly indistinguishable from a continuum at any accessible temperature ($k_BT \approx 0.025$ eV at 300 K is $10^{21}$ times larger). This continuum of allowed orbital energies is a band. Because it derives from one atomic orbital (here, $1s$) it is called the "$1s$ band"; a solid built from atoms with several valence orbitals ($2s$, $2p$, $3d$, …) has one band per atomic orbital type, and these bands can overlap or remain separated in energy. The crucial refinement beyond "a band is just a blurred atomic level" is that bands can have gaps between them even though each individual band is continuous. This happens because a periodic crystal potential Bragg-reflects electron waves whose wavelength matches the lattice spacing. Treat electrons initially as free particles with dispersion $E(k) = \hbar^2k^2/2m$ — a single continuous parabola. Turning on a weak periodic potential $V(x)$ with period $a$ couples plane waves $e^{ikx}$ and $e^{i(k - 2\pi/a)x}$ whenever $k = \pm\pi/a$ (the zone boundary). Degenerate perturbation theory on this pair of states splits their common energy into two: one lowered, one raised, by an amount set by the Fourier component of the potential. This is the origin of the band gap — literally a range of energy forbidden to the electron because no propagating wave solution exists there; the wave is reflected rather than transmitted, exactly as light is reflected by a Bragg mirror.

Where does this leave real solids? Silicon, germanium, and the compound semiconductors from Zinc Blende, Wurtzite and Rutile Structures all have four valence electrons per formula unit (or an average of four, for III–V and II–VI compounds) in $sp^3$-hybridized bonding networks. The bonding combinations of $sp^3$ hybrids form a valence band; the antibonding combinations form a conduction band, separated by a band gap $E_g$. At absolute zero the valence band is completely filled (it holds exactly enough states for all the bonding electrons) and the conduction band is completely empty. Whether the solid conducts at finite temperature is then a question of how large $E_g$ is compared to $k_BT$, and whether any band happens to be only partially filled to begin with (the metallic case).

Overlapping bands matter as much as gapped ones. Magnesium, for instance, has the electron configuration $[\mathrm{Ne}]3s^2$: naively a filled $3s$ band should make it an insulator, just as filled-shell helium is inert. But in the solid, the $3s$ band and the empty $3p$ band overlap in energy (the $3p$ band's bottom lies below the $3s$ band's top), so electrons spill from $3s$ into $3p$, leaving both partially filled. Magnesium conducts because of this overlap, not despite its "closed-shell" atomic configuration — a direct illustration that band structure, not atomic electron count alone, governs conductivity.

The tight-binding (LCAO) dispersion for a one-dimensional chain of identical atoms with lattice spacing $a$, one orbital per site, and nearest-neighbor hopping integral $\beta$ (negative, as is conventional for a bonding interaction) is:

where:

• $\alpha$ is the on-site (Coulomb) energy of the atomic orbital,
• $\beta$ is the resonance (hopping/overlap) integral between adjacent atoms, $\beta<0$,
• $k$ is the electron wavevector, restricted to the first Brillouin zone $-\pi/a < k \le \pi/a$,
• $a$ is the interatomic spacing.

Reading it: at $k=0$ (all orbitals in phase) $E = \alpha + 2\beta$, the bottom of the band (most bonding, since $\beta<0$); at $k = \pi/a$ (alternating phases) $E = \alpha - 2\beta$, the top of the band. The bandwidth is $W = 4|\beta|$ — bigger orbital overlap gives a wider band, exactly as stronger bonding gives a bigger HOMO–LUMO gap in a diatomic. Near the zone boundary, a weak periodic potential opens a gap. The size of that gap is:

where:

• $V_G$ is the Fourier component of the crystal potential at the reciprocal lattice vector $G = 2\pi/a$ connecting the two degenerate free-electron states at the zone edge,
• the factor of 2 reflects that the degeneracy splits symmetrically into a lower and an upper branch.

Reading it: a perfectly flat (structureless) potential has $V_G = 0$ and no gap — free-electron behavior throughout; the more strongly the ion cores scatter electrons at that particular periodicity, the wider the forbidden zone. The carrier concentration in an intrinsic (undoped) semiconductor, obtained by integrating the Fermi–Dirac distribution against the conduction-band density of states in the non-degenerate (Boltzmann) limit, is:

where:

• $N_c, N_v$ are effective densities of states at the conduction- and valence-band edges (each $\propto T^{3/2}$, set by the effective masses $m_e^*, m_h^*$),
• $E_c, E_v$ are the conduction-band-minimum and valence-band-maximum energies,
• $E_F$ is the Fermi level (chemical potential for electrons),
• $T$ is absolute temperature.

Multiplying these gives the law of mass action for intrinsic semiconductors:

where $n_i$ is the intrinsic carrier density and $E_g = E_c - E_v$. This product is independent of where $E_F$ sits — doping shifts $n$ and $p$ in opposite directions but leaves $np$ fixed at fixed $T$, precisely analogous to $K_w$ for water autoionization.

Example 1 — Foundational: bandwidth from a Hückel-like tight-binding chain. Consider a hypothetical 1D chain of atoms with $\alpha = -8.00$ eV and hopping integral $\beta = -1.35$ eV (asymmetric, chemically reasonable numbers — not a round or symmetric choice). Find the band-edge energies and the bandwidth. Using $E(k) = \alpha + 2\beta\cos(ka)$: At $k=0$: $\cos(0) = 1$, so

At $k = \pi/a$: $\cos(\pi) = -1$, so

Bandwidth:

Limiting check: if the atoms were pulled infinitely far apart, $\beta \to 0$ (no overlap), giving $W \to 0$ — the band collapses back to the sharp atomic level at $\alpha = -8.00$ eV, exactly as required for isolated atoms.

Example 2 — Applied: intrinsic carrier density of germanium at 300 K and 400 K. Germanium has $E_g = 0.67$ eV, and at $T = 300$ K the effective densities of states are $N_c = 1.04\times10^{19}\ \text{cm}^{-3}$, $N_v = 6.0\times10^{18}\ \text{cm}^{-3}$ (asymmetric, realistic values). Estimate $n_i$ at 300 K and 400 K, and the ratio. At 300 K, $k_BT = 0.02585$ eV, so $E_g/k_BT = 0.67/0.02585 = 25.9$.

in good agreement with the accepted experimental value for germanium ($\sim 2.4\times10^{13}\ \text{cm}^{-3}$; the small discrepancy comes from using room-temperature literature $N_c,N_v$ rather than fully $T$-dependent effective masses). At 400 K, $k_BT = 0.0345$ eV and $N_c, N_v$ scale as $T^{3/2}$, i.e. by a factor $(400/300)^{3/2} = 1.54$ each, so $N_cN_v$ scales by $1.54^2 = 2.37$.

Ratio: $n_i(400)/n_i(300) \approx 39$ — a nearly 40-fold rise in intrinsic carrier density for only a 100 K increase, which is exactly why unregulated semiconductor devices can thermally runaway: more carriers raise the current, which raises the temperature, which raises the carrier density further. Limiting check: as $T \to 0$, $e^{-E_g/k_BT} \to 0$ faster than any power of $T$ grows, so $n_i \to 0$ — a perfect insulator at absolute zero, as required for any solid with a nonzero gap.

Example 3 — Exam/competition level: doping and the shift of the Fermi level in silicon. Silicon ($E_g = 1.11$ eV, $N_c = 2.8\times10^{19}\ \text{cm}^{-3}$, $N_v = 1.04\times10^{19}\ \text{cm}^{-3}$ at 300 K) is doped with phosphorus at a concentration $N_D = 1\times10^{16}\ \text{cm}^{-3}$, fully ionized. First compute $n_i$ for pure silicon, then find $E_F$ relative to $E_c$ for the doped crystal, then find $E_F$ relative to the intrinsic Fermi level $E_i$. Step A — intrinsic carrier density. $E_g/k_BT = 1.11/0.02585 = 42.9$.

(matches the accepted textbook value $\approx 1.0\times10^{10}\ \text{cm}^{-3}$ within the precision of the input constants — small differences arise from which $N_c,N_v$ literature values are used). Step B — carrier density with full donor ionization. Since $N_D = 10^{16}\ \text{cm}^{-3} \gg n_i \approx 10^{10}\ \text{cm}^{-3}$, essentially every donor electron dominates charge neutrality and

Step C — Fermi level from $E_c$. Invert $n = N_c e^{-(E_c-E_F)/k_BT}$:

So the Fermi level sits $0.205$ eV below the conduction band — far closer to $E_c$ than to mid-gap ($E_g/2 = 0.555$ eV), confirming this is a heavily n-type extrinsic semiconductor, not intrinsic. Step D — shift relative to the intrinsic level. With $m_h^*$ slightly larger than $m_e^*$ for silicon, $E_i$ sits close to but not exactly at mid-gap; taking $E_i \approx E_c - 0.56$ eV (standard result for Si),

The Fermi level has moved up by 0.355 eV from its intrinsic position — a huge shift on the scale of $k_BT = 0.026$ eV (about 14 $k_BT$), which is exactly why $n/n_i = e^{(E_F-E_i)/k_BT} = e^{13.7} \approx 8.9\times10^5$, consistent with $n/n_i = 10^{16}/7.7\times10^9 \approx 1.3\times10^6$ found directly (the two independent routes agree to the precision of the rounding used). Limiting check: as $N_D \to 0$, Step B would require solving the full charge-neutrality equation rather than $n\approx N_D$, and the correct limit smoothly recovers $E_F \to E_i$ and $n \to n_i$ — the intrinsic result of Example 2's method.

Wilson's theoretical breakthrough sits alongside experimental and technological milestones that made band theory indispensable to modern chemistry and physics. The timeline below traces the arc from the foundational quantum mechanics of solids to the transistor that made the theory a matter of daily practical consequence.