Band theory of solids

Result

Band theory of solids

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Statement

Conductors, insulators and semiconductors from overlapping orbitals.

Why it matters

crystal-lattices-unit-cell and close-packing-efficiency describe how atoms are arranged periodically in a crystal; this result explains what that periodicity does to the electrons themselves. Simple molecular orbital theory, applied to a handful of atoms, gives a few discrete, well-separated orbital energies; extending that same logic to the roughly \(10^{23}\) atoms of a macroscopic solid produces qualitatively new behaviour, and band theory is the standard framework for understanding it.

Above all, band theory is the single unifying explanation for why some solids conduct electricity freely, some do not conduct at all, and some sit in between and can be engineered to do either — a distinction that, without this result, would otherwise look like three unrelated empirical categories rather than one underlying physical picture.

Hypotheses

Electrons are treated in a one-electron (mean-field) approximation, each moving in the averaged potential of the nuclei and all other electrons.This is the same simplifying assumption used throughout molecular orbital theory (lcao-molecular-orbitals); it allows the solid's electronic structure to be built up orbital by orbital rather than solved as one intractable many-body problem, at the cost of neglecting detailed electron-electron correlation. The crystal is treated as effectively infinite and perfectly periodic.Real crystals are finite and contain defects, but for a macroscopic sample the fraction of atoms near a surface or defect is negligible; the idealised infinite, periodic lattice is what allows the atomic orbitals to be combined into continuous bands rather than a finite set of discrete levels. The one-electron approximation breaks down for materials with strong electron-electron correlation (so-called Mott insulators), where a material band theory predicts should be metallic (a partially filled band) is experimentally an insulator, because strong on-site Coulomb repulsion localises electrons that the simple model treats as freely delocalised.

Proof

N \text{ atomic orbitals combine, by the LCAO principle, into } N \text{ molecular orbitals.}

This is the same linear-combination-of-atomic-orbitals logic already used for small molecules; the number of resulting molecular orbitals always equals the number of atomic orbitals combined, regardless of how large \(N\) becomes. A

\text{As } N\to10^{23}, \text{ the } N \text{ orbital energies within one atomic-orbital-derived set become an effectively continuous band.}

For a small molecule the resulting orbitals are separated by an energy gap comparable to the total bandwidth; for a macroscopic crystal that same finite bandwidth is divided among on the order of Avogadro's number of orbitals, so adjacent orbital energies differ by a vanishingly small amount and the discrete set is, for essentially all practical purposes, a continuous band of allowed energies. B

\text{Each band accommodates } 2N \text{ electrons (Pauli exclusion, two spins per orbital); bands from different atomic orbitals may be separated by a gap or may overlap.}

Whether two adjacent bands overlap in energy or are separated by a forbidden gap depends on the atomic orbitals' energies and the strength of their overlap in the solid, exactly as in molecular orbital theory applied to a single pair of atoms, just repeated across the whole lattice. A

\text{A partially filled band, or two overlapping bands with electrons only partly filling the combined levels, permits conduction; a completely filled band separated by a gap from the next empty band does not.}

Electrical conduction requires electrons able to accelerate into nearby, slightly higher-energy available states under an applied field; in a completely full band there are no such nearby empty states within the same band, while in a partially filled (or overlapping) band there are, immediately above the highest occupied level. A

\text{Insulators: gap } E_g \text{ large compared with thermal energy } kT. \quad \text{Semiconductors: } E_g \text{ small enough for appreciable thermal (or photo-) excitation across it.}

Even with a filled valence band and empty conduction band, if the gap is small enough, thermal energy at ordinary temperature promotes a small but non-negligible fraction of electrons across it, populating the conduction band and leaving behind mobile "holes" in the valence band — the origin of intrinsic semiconductor behaviour. A

Result

\text{Metal: partially filled/overlapping bands} \quad\big|\quad \text{Insulator: full band, large } E_g \quad\big|\quad \text{Semiconductor: full band, small } E_g

Reading. Whether a solid conducts is entirely a question of whether there are empty electronic states immediately accessible in energy just above the highest occupied ones — a partially filled or overlapping band supplies them directly, while a full band separated by a large gap does not.

Scope. Applies to crystalline solids describable within the one-electron approximation (Hypotheses); strongly correlated materials, where electron-electron interactions dominate, require methods beyond simple band theory.

Corollaries & converses

Doping a semiconductor with atoms that donate extra electrons (into the conduction band) or accept electrons (creating holes in the valence band) is a direct, controlled application of Step 5: a small, deliberately introduced impurity concentration can change the population of nearly-empty or nearly-full bands by orders of magnitude.
born-lande-lattice-energy treats bonding in ionic solids through localised, point-charge electrostatics; band theory instead treats the delocalised, shared electrons characteristic of metallic bonding — the two results describe the two opposite limits of chemical bonding in the solid state.
Converse: a measured, strongly temperature-dependent conductivity that increases with temperature (rather than the metallic decrease expected from increased lattice vibration scattering) is itself evidence for semiconductor-like thermal carrier excitation across a gap, diagnostic of Step 5's mechanism operating.

Fails without

Apply the one-electron (mean-field) approximation to a strongly correlated material: a material band theory predicts should be metallic (a partially filled band) can experimentally be an insulator (a Mott insulator), because strong on-site Coulomb repulsion localises electrons the simple model treats as freely delocalised (Hypotheses, third point).
Treat a small, finite cluster of atoms as if it already had continuous bands: for only a handful of atoms, the discrete orbital spacing remains comparable to the total bandwidth, so the continuous-band picture of Step 2 (valid only as \(N\to10^{23}\)) does not yet apply, and the cluster instead shows discrete, molecule-like orbital energies.

Common errors

Assuming an insulator and a semiconductor differ in kind rather than in degree; both have a full valence band and a gap, differing only in whether that gap is small enough for practically significant thermal excitation (Step 5).
Thinking a "band" implies electrons spread evenly and independently throughout the crystal with no relation to the original atomic orbitals; each band still traces back to, and inherits much of the character of, a specific atomic orbital or set of orbitals (Step 1).
Forgetting that metallic conductivity can arise either from a single partially filled band or from two adjacent bands overlapping in energy (Step 3), not only from partial filling of one band.
Applying simple band theory uncritically to strongly correlated materials, where it can qualitatively mispredict metal/insulator behaviour (Hypotheses).

Discussion

Band theory emerged from applying the newly developed quantum mechanics of the 1920s to periodic solids, notably through Felix Bloch's 1928 theorem describing how a wavefunction behaves in a periodic potential, providing the mathematical basis for treating an electron's allowed states in a crystal as continuous bands rather than the discrete orbitals of an isolated atom.

The size of the band gap is not a fixed property of a chemical element alone but depends on the specific crystal structure and bonding: carbon crystallises both as diamond, an excellent insulator with a large band gap, and as graphite, an electrical conductor, because the different bonding geometries (localised, fully saturated \(sp^3\) bonds in diamond versus delocalised \(\pi\) electrons across extended sheets in graphite) produce entirely different band structures from the identical element.

Common misconception: that band theory implies every solid's conductivity is fixed and intrinsic to the pure material alone. In practice, semiconductor conductivity is dominated in most technological applications by deliberately introduced doping rather than by the small intrinsic thermal excitation of the pure material, precisely because doping can change carrier concentration far more dramatically and controllably than temperature alone.

Worked examples

\text{Diamond: } E_g\approx5.5\,\text{eV} \quad\text{(insulator)}; \qquad \text{Silicon: } E_g\approx1.1\,\text{eV} \quad\text{(semiconductor)}

Both are group 14 elements with the same tetrahedral, diamond-cubic crystal structure and fully saturated covalent bonding, yet their band gaps differ by a factor of \(5\); diamond's much stronger, shorter carbon-carbon bonding produces a much larger bonding-antibonding energy splitting (a larger gap between the filled, bonding-derived band and the empty, antibonding-derived band) than silicon's longer, weaker bonds. A

\text{Same structure, same bonding type, different } E_g \Rightarrow \text{ insulator versus semiconductor}

Reading. Band gap size, not merely the qualitative type of bonding present, is what separates an insulator from a semiconductor within the same structural family.

Scope. The same trend continues down group 14: germanium (\(E_g\approx0.7\,\text{eV}\)) and grey tin (\(E_g\) close to zero) show progressively smaller gaps and progressively more metallic behaviour, tracking the weakening bond strength down the group.

Problems

Explain, in band-theory terms, why sodium metal (electron configuration \([\text{Ne}]3s^1\)) conducts electricity well.
Solution
Each sodium atom contributes one \(3s\) electron and one \(3s\) atomic orbital; combining \(N\) such orbitals gives a \(3s\)-derived band capable of holding \(2N\) electrons but populated by only \(N\) electrons, exactly half filling the band. A half-filled band has abundant empty states immediately above the highest occupied level, so electrons can accelerate freely under an applied field (Step 4), giving good metallic conductivity.
A material has a full valence band separated by a \(0.2\,\text{eV}\) gap from an empty conduction band, while another has a full valence band separated by a \(4\,\text{eV}\) gap. Which behaves more like a semiconductor at room temperature, and why?
Solution
The \(0.2\,\text{eV}\)-gap material behaves as a semiconductor: room-temperature thermal energy (\(kT\approx0.025\,\text{eV}\) at \(298\,\text{K}\)) is small compared with either gap, but the Boltzmann-distributed fraction of electrons with enough energy to cross a \(0.2\,\text{eV}\) gap is very much larger than the fraction able to cross a \(4\,\text{eV}\) gap, since the excitation probability falls off exponentially with the gap-to-\(kT\) ratio (Step 5); the \(4\,\text{eV}\)-gap material remains an effective insulator at ordinary temperature.
Graphite conducts electricity well within its layered sheets but very poorly perpendicular to them. Suggest a band-theory explanation.
Solution
Within a graphite sheet, delocalised \(\pi\) electrons from unhybridised \(p\) orbitals overlap extensively across the whole two-dimensional sheet, forming a partially filled, highly delocalised band that conducts well in-plane (Step 4). Between sheets, bonding is due only to comparatively weak interlayer interactions with far less orbital overlap, giving a much narrower, more nearly atomic-like band structure in that direction and correspondingly poor out-of-plane conduction.