Band theory of solids: the Bloch theorem, the tight-binding model, and conductors versus insulators

Intuition Beginner

A lithium atom has one valence electron in a 2s orbital. Two lithium atoms form Li2 with a bonding and an antibonding MO, as you saw in 14.05.01. But a piece of lithium metal contains roughly $6\times 10^{23}$ atoms, not two. What happens to the molecular orbital picture when $N$ is Avogadro's number?

Each atom contributes one atomic orbital. With $N$ atoms, there are $N$ atomic orbitals that combine into $N$ molecular orbitals. The bonding orbital sits at the lowest energy and the antibonding orbital at the highest, but now there are $N-2$ orbitals packed in between. Because $N$ is astronomically large, these $N$ energy levels form a quasi-continuous energy band — a range of allowed energies so densely packed that it acts like a continuum. This is the central idea of band theory.

For lithium, the $N$ valence electrons fill the lowest $N/2$ orbitals (two electrons per orbital). The band is exactly half full. The highest occupied energy is called the Fermi level. Because there are empty energy levels arbitrarily close above the Fermi level, an applied electric field can easily promote electrons into slightly higher-energy states — and these promoted electrons carry current. Lithium is a conductor (metal).

Now consider diamond (carbon). Each carbon atom has four valence electrons (2s${}^2$2p${}^2$). In the diamond crystal, the 2s and 2p orbitals form two bands separated by a large energy gap — the band gap. The lower band (the valence band) is completely full; the upper band (the conduction band) is completely empty. With no empty states nearby, electrons cannot be promoted by an ordinary electric field. Diamond is an insulator.

Silicon has the same crystal structure as diamond but a smaller band gap (about 1.1 eV versus 5.5 eV for diamond). At room temperature, thermal energy promotes a small but measurable number of electrons across the gap. Silicon is a semiconductor — it conducts poorly at low temperature and better at higher temperature, the opposite of a metal.

Visual Beginner

Picture a long row of identical atoms, each spaced a distance $a$ apart. Each atom has one orbital. When the atoms are far apart, every atom has the same energy and the orbitals do not interact — $N$ degenerate levels. As the atoms approach, the orbitals begin to overlap. The $N$-fold degeneracy splits into $N$ closely spaced levels spanning a range of energies called a band. The width of the band depends on how strongly the orbitals overlap: large overlap gives a wide band, small overlap gives a narrow band.

For a conductor, the Fermi level cuts through the middle of a band. Electrons near the Fermi level have empty states immediately above and can move in response to a field. For an insulator, the Fermi level sits in the band gap. The valence band below is full; the conduction band above is empty; no states are available for conduction.

Worked example Beginner

Problem. A one-dimensional chain of $N$ hydrogen atoms (distance $a$ between neighbours) has one 1s orbital per atom. The $N$ orbitals form a band. Each hydrogen contributes one electron. Classify this chain as a conductor or insulator.

Step 1. $N$ atoms give $N$ orbitals, forming one band. The band holds $2N$ electrons (two per orbital).

Step 2. $N$ hydrogen atoms contribute $N$ electrons. The band is half full ($N$ out of $2N$ possible).

Step 3. A half-filled band means the Fermi level lies inside the band. Empty states are available at arbitrarily small energy cost.

Answer. The 1D hydrogen chain is a conductor. (In practice, a real 1D chain of hydrogen atoms undergoes a Peierls distortion — it dimerises, opening a gap. But the undistorted chain is metallic. This subtlety is taken up in the Master tier.)

Check your understanding Beginner

Formal definition Intermediate+

The periodic potential and the Bloch theorem

A crystalline solid has atoms arranged on a Bravais lattice — an infinite set of points $\mathbf{R}=n_1{\mathbf{a}}_1+n_2{\mathbf{a}}_2+n_3{\mathbf{a}}_3$ with integer $n_i$ and primitive vectors ${\mathbf{a}}_i$. The crystal potential $V(\mathbf{r})$ satisfies $V(\mathbf{r}+\mathbf{R})=V(\mathbf{r})$ for all lattice vectors $\mathbf{R}$.

A single electron in this periodic potential obeys the Schrodinger equation $\hat{H}\psi =E\psi$ with $\hat{H}=-\frac{{\hslash }^2}{2m}{\nabla }^2+V(\mathbf{r})$. Because $V$ is periodic, $\hat{H}$ commutes with every lattice translation ${\hat{T}}_{\mathbf{R}}$, defined by ${\hat{T}}_{\mathbf{R}}f(\mathbf{r})=f(\mathbf{r}+\mathbf{R})$. The simultaneous eigenstates of $\hat{H}$ and all ${\hat{T}}_{\mathbf{R}}$ are the Bloch waves.

Theorem (Bloch). If $V(\mathbf{r}+\mathbf{R})=V(\mathbf{r})$ for all lattice vectors $\mathbf{R}$, then the eigenstates of $\hat{H}=-\frac{{\hslash }^2}{2m}{\nabla }^2+V(\mathbf{r})$ can be chosen to have the form

$${\psi }_{n\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot \mathbf{r}}{u}_{n\mathbf{k}}(\mathbf{r}),$$

where $u_{n\mathbf{k}}(\mathbf{r}+\mathbf{R})=u_{n\mathbf{k}}(\mathbf{r})$ has the periodicity of the lattice, $n$ is the band index, and $\mathbf{k}$ is the Bloch wavevector.

The wavevector $\mathbf{k}$ lives in the reciprocal lattice. The primitive reciprocal vectors are ${\mathbf{b}}_j=2\pi {\mathbf{a}}_i\times {\mathbf{a}}_k/({\mathbf{a}}_1\cdot {\mathbf{a}}_2\times {\mathbf{a}}_3)$ with cyclic permutations. The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice — the set of points closer to the origin than to any other reciprocal lattice point. Because ${\psi }_{n\mathbf{k}+\mathbf{G}}={\psi }_{n\mathbf{k}}$ for any reciprocal lattice vector $\mathbf{G}$ (they differ only by a phase), all physically distinct states are labelled by $\mathbf{k}$ within the first Brillouin zone.

For a finite crystal of $N$ unit cells with periodic boundary conditions (the Born-von Karman boundary condition), the allowed $\mathbf{k}$-values form a discrete mesh with spacing $2\pi /L$ where $L$ is the crystal dimension. In the thermodynamic limit $N\to \infty$, the spectrum $E_n(\mathbf{k})$ for each band index $n$ traces out a continuous band as $\mathbf{k}$ ranges over the Brillouin zone.

Tight-binding model for a 1D chain

The tight-binding model is the solid-state analogue of the LCAO method from 14.05.01. Place $N$ atoms on a one-dimensional chain with lattice constant $a$. Each atom has one orbital ${\varphi }_j(\mathbf{r})=\varphi (\mathbf{r}-ja)$ centred at site $j$. The tight-binding Hamiltonian is

$$\hat{H}=\sum _{j=1}^N\alpha |j⟩⟨j|+\sum _{j=1}^{N-1}\beta (|j⟩⟨j+1|+|j+1⟩⟨j|),$$

where $\alpha =⟨j|\hat{H}|j⟩$ is the on-site energy and $\beta =⟨j|\hat{H}|j+1⟩$ is the hopping integral (nearest-neighbour coupling). This is the direct generalisation of the $2\times 2$ secular equation for H2 to an $N\times N$ tridiagonal matrix.

By Bloch's theorem, the eigenstates are labelled by wavevector $k$. The Bloch ansatz for a 1D chain is

$${\psi }_k(\mathbf{r})=\frac{1}{\sqrt{N}}\sum _{j=1}^N{e}^{ikja}\varphi (\mathbf{r}-ja),$$

where $k$ ranges over the first Brillouin zone $-\pi /a\le k\le \pi /a$. Substituting into the Schrodinger equation gives the dispersion relation (band energy as a function of $k$):

$$E(k)=\alpha +2\beta \cos (ka).$$

With $\beta <0$ (the usual sign for bonding overlap), the band minimum is at $k=0$ with energy $E_{\min }=\alpha +2\beta$ (most bonding) and the band maximum is at $k=\pm \pi /a$ with energy $E_{\max }=\alpha -2\beta$ (most antibonding). The bandwidth is $W=E_{\max }-E_{\min }=4|\beta |$. Larger hopping integral $|\beta |$ (stronger orbital overlap) gives a wider band.

This is the $N$-atom generalisation of the two-orbital splitting $\Delta E=2(\alpha S-\beta )/(1-S^2)$ from 14.05.01: instead of two discrete levels separated by $\Delta E$, the $N$ levels form a continuous band of width $4|\beta |$.

Valence band, conduction band, and band gap

In a real solid, multiple atomic orbitals per atom (2s, 2p, 3s, 3p, ...) each generate their own bands. When interatomic spacing is large, these bands are narrow and well separated. As the atoms move closer, the bands widen and may overlap or develop gaps.

• The valence band is the highest energy band that is fully or partially occupied at zero temperature.
• The conduction band is the lowest energy band that is empty or partially occupied at zero temperature.
• The band gap $E_g$ is the energy difference between the top of the valence band and the bottom of the conduction band.
• The Fermi level $E_F$ is the chemical potential of the electrons — the energy at which the occupation probability is exactly 1/2 in the Fermi-Dirac distribution.

Classification of solids:

Type	Band filling	Band gap	Conductivity
Metal (conductor)	Partially filled band, or overlapping bands	$E_g=0$	High
Semiconductor	Full valence band, empty conduction band	$0<E_g\lesssim 4$ eV	Moderate, increases with T
Insulator	Full valence band, empty conduction band	$E_g\gtrsim 4$ eV	Very low

Effective mass and density of states

Near a band extremum, the dispersion can be expanded as $E(k)\approx E_0+\frac{{\hslash }^2}{2m^{\ast }}(k-k_0{)}^2$, where $m^{\ast }$ is the effective mass:

$$\frac{1}{m^{\ast }}=\frac{1}{{\hslash }^2}\frac{d^{2}E}{d{k}^2}.$$

For the 1D tight-binding band $E(k)=\alpha +2\beta \cos (ka)$, the effective mass at the band bottom ($k=0$) is $m^{\ast }=-{\hslash }^2/(2\beta a^2)$. With $\beta <0$, $m^{\ast }>0$. Near the band top ($k=\pi /a$), $m^{\ast }<0$ — holes (absence of electrons) behave as positively charged particles with positive mass.

The density of states $g(E)$ counts the number of available states per unit energy per unit volume. In 1D:

$$g(E)=\frac{1}{\pi }\frac{dk}{dE}=\frac{1}{\pi a}\frac{1}{|dE/dk|}.$$

For the tight-binding band, $dE/dk=-2\beta a\sin (ka)$, so $g(E)$ diverges at the band edges ($k=0$ and $k=\pi /a$ where $\sin (ka)=0$). These divergences are the Van Hove singularities — a hallmark of one-dimensional systems.

Counterexamples to common slips

• "A band is just a molecular orbital." A band contains $N$ molecular orbitals (one per $\mathbf{k}$-point in the Brillouin zone). The band is the collection of all these MOs and their energy range. A single Bloch state ${\psi }_{n\mathbf{k}}$ is one MO; the band is the full set $\{E_n(\mathbf{k}):\mathbf{k}\in \text{BZ}\}$.

• "Insulators have no electrons in the conduction band at any temperature." At finite temperature, the Fermi-Dirac distribution $f(E)=1/(e^{(E-E_F)/kT}+1)$ guarantees a nonzero probability of electrons being thermally excited across the gap. For insulators the number is exponentially suppressed as $e^{-E_g/2kT}$ and is negligible for practical purposes, but it is not identically zero.

• "Semiconductors and insulators are fundamentally different materials." The distinction is quantitative (band gap size), not qualitative. Diamond and silicon have the same crystal structure and bonding type; diamond's larger gap makes it insulating while silicon's smaller gap makes it semiconducting. There is no sharp boundary.

• "The tight-binding model with only nearest-neighbour hopping is physically accurate." It captures the qualitative physics (band formation, bandwidth, filling) but quantitatively it neglects longer-range hopping, orbital overlap beyond nearest neighbours, and electron-electron interactions. Real band-structure calculations use density functional theory 14.04.04 with a plane-wave basis.

Key theorem with proof Intermediate+

Theorem (1D tight-binding dispersion). Consider $N$ sites on a 1D chain with lattice constant $a$, on-site energy $\alpha$, and nearest-neighbour hopping $\beta$. With periodic boundary conditions (site $N+1$ identified with site 1), the eigenvalues of the $N\times N$ tight-binding Hamiltonian are

$$E(k)=\alpha +2\beta \cos (ka),k=\frac{2\pi m}{Na},m=0,1,\dots ,N-1.$$

The corresponding eigenvectors have components $c_j=\frac{1}{\sqrt{N}}{e}^{ikja}$ for $j=1,\dots ,N$.

Proof. The Hamiltonian matrix is the $N\times N$ tridiagonal matrix with $\alpha$ on the diagonal and $\beta$ on the sub- and super-diagonals, with the periodic boundary condition adding $\beta$ in the $(1,N)$ and $(N,1)$ entries:

$$H=\left(\begin{array}{ccccc}\alpha & \beta & 0& \cdots & \beta \\ \beta & \alpha & \beta & \cdots & 0\\ 0& \beta & \alpha & \cdots & 0\\ ⋮& & & \ddots & \beta \\ \beta & 0& 0& \beta & \alpha \end{array}\right).$$

The eigenvalue equation $H\mathbf{c}=E\mathbf{c}$ gives, for interior sites $j=2,\dots ,N-1$:

$$\beta c_{j-1}+\alpha c_j+\beta c_{j+1}=E{c}_j.$$

For $j=1$: $\alpha c_1+\beta c_2+\beta c_N=E{c}_1$ (the last term is the periodic wrap-around). For $j=N$: analogous.

Try the Bloch ansatz $c_j=e^{ikja}/\sqrt{N}$. The interior equation becomes:

$$\beta e^{ik(j-1)a}+\alpha e^{ikja}+\beta e^{ik(j+1)a}=E{e}^{ikja},$$ $$E=\alpha +\beta e^{-ika}+\beta e^{ika}=\alpha +2\beta \cos (ka).$$

The boundary equation for $j=1$ requires $\beta e^{ikNa}=\beta$, hence $e^{ikNa}=1$, so $kNa=2\pi m$ for integer $m$. The $N$ distinct values are $m=0,1,\dots ,N-1$, giving $k=2\pi m/Na$. Normalisation ${\sum }_j|c_j{|}^2=1$ is satisfied by the factor $1/\sqrt{N}$. $\square$

Corollary (Bandwidth). The energy range spanned by the band is

$$W=E_{\max }-E_{\min }=\alpha -2\beta -(\alpha +2\beta )=-4\beta =4|\beta |.$$

The bandwidth is determined entirely by the hopping integral $\beta$ and is independent of $N$. Adding more atoms increases the density of states within the band but does not change its width.

Worked example: Mg versus Ne

Magnesium has the valence configuration 3s${}^2$. Neon has the configuration 2s${}^2$2p${}^6$. Both have filled subshells. Why is Mg a metal and Ne an insulator?

In solid Mg, the 3s band overlaps with the 3p band — the two bands merge into one large band with capacity $2N$ (from 3s) $+6N$ (from 3p) $=8N$ electrons. The $2N$ valence electrons from the 3s${}^2$ configuration partially fill this combined band. The Fermi level lies inside the band: Mg is a metal.

In solid Ne, the 2s and 2p bands are both full (2s${}^2$ gives $2N$ electrons filling the 2s band; 2p${}^6$ gives $6N$ electrons filling the 2p band). Above the 2p band is the 3s band, separated by a large gap (roughly 20 eV). All bands below the gap are full, and the next band is empty and far away. Solid neon is an insulator — a noble-gas solid with no mobile carriers.

The key difference is not the filled subshell but whether a partially filled band exists. In Mg, band overlap creates a partially filled band from nominally full subshells. In Ne, the bands remain well separated.

Exercises Intermediate+

From molecular orbitals to crystal orbitals: the large-N limit Master

The tight-binding model is the direct descendant of the LCAO-MO theory of 14.05.01 and 14.05.02. For H2, two 1s orbitals produce a bonding and antibonding pair separated by $\Delta E$. For a linear H${}_N$ chain, $N$ orbitals produce a band of $N$ levels spanning $4|\beta |$. The band is the large-$N$ limit of the MO diagram.

The Bloch wavevector $k$ plays the role of the LCAO coefficient pattern. In H2, the bonding MO has $c_A=c_B=1/\sqrt{2}$ (both in phase) and the antibonding has $c_A=-c_B=1/\sqrt{2}$ (opposite phase). In the chain, $c_j=e^{ikja}/\sqrt{N}$ — the coefficient at site $j$ has phase $kja$. The bonding state ($k=0$) has all coefficients in phase; the antibonding state ($k=\pi /a$) has alternating signs. Intermediate $k$ values interpolate continuously between these extremes.

The dispersion relation $E(k)=\alpha +2\beta \cos (ka)$ encodes the same physics as the secular equation in 14.05.01. For H2 ($N=2$, $a=R$), the allowed $k$ values are $k=0$ and $k=\pi /R$, giving energies $E(0)=\alpha +2\beta$ and $E(\pi /R)=\alpha -2\beta$. In the overlap-neglected Hückel limit ($S=0$), the H2 bonding and antibonding energies from 14.05.01 are $\alpha +\beta$ and $\alpha -\beta$, differing by a factor of 2 in the splitting. The discrepancy arises because the $N=2$ chain with periodic boundary conditions has each atom coupled to the other by two paths (direct hopping plus the periodic wrap-around), doubling the effective hopping. For $N\ge 3$ with open boundary conditions, this artifact disappears.

The transition from discrete levels to continuous bands has a precise mathematical statement. The spectral theorem for the infinite one-dimensional tight-binding Hamiltonian (the discrete Laplacian on $\mathbb{Z}$) gives a purely absolutely continuous spectrum on $[\alpha -2|\beta |,\alpha +2|\beta |]$ — a perfect band with no gaps. Introducing a periodic modulation of the on-site energies (a superlattice with two atoms per unit cell) opens a gap in the spectrum, producing the band structure of a binary compound like GaAs.

The Bloch theorem: proof and the reciprocal lattice Master

Bloch's theorem in one dimension states that if $V(x+a)=V(x)$, then the eigenfunctions of $\hat{H}=-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}+V(x)$ satisfy ${\psi }_k(x+a)=e^{ika}{\psi }_k(x)$. Writing ${\psi }_k(x)=e^{ikx}{u}_k(x)$ with $u_k(x+a)=u_k(x)$ gives the Bloch form.

The proof relies on the commutation $[\hat{H},{\hat{T}}_a]=0$ where ${\hat{T}}_{a}f(x)=f(x+a)$. Since ${\hat{T}}_a$ is unitary, its eigenvalues are pure phases: ${\hat{T}}_a\psi =e^{ika}\psi$. The wavevector $k$ labels the irreducible representations of the translation group $\mathbb{Z}$ (or ${\mathbb{Z}}_N$ for finite $N$ with periodic boundary conditions). This is a direct application of the representation theory developed in 14.05.02 for the $D_{\infty h}$ point group — here the symmetry group is the translation group of the lattice, and the irreducible representations are one-dimensional, labelled by $k$.

The reciprocal lattice $\{\mathbf{G}\}$ is the Fourier dual of the direct lattice $\{\mathbf{R}\}$: $\mathbf{G}\cdot \mathbf{R}=2\pi \times \text{integer}$. The Brillouin zone is the fundamental domain for $\mathbf{k}$-space: any $\mathbf{k}$ outside it can be reduced to a $\mathbf{k}$ inside by subtracting a reciprocal lattice vector $\mathbf{G}$ (Umklapp process). In 1D, the Brillouin zone is $[-\pi /a,\pi /a]$.

The number of distinct $k$-points in the Brillouin zone equals the number of unit cells $N$. For each $k$, the Schrodinger equation yields a discrete set of energies $E_1(k),E_2(k),\dots$ — the band indices $n$. The collection $\{E_n(k):k\in \text{BZ}\}$ for each fixed $n$ is one energy band. The Bloch theorem guarantees that these bands are continuous functions of $k$ (for a finite $N$ they are discrete but converge to continuous as $N\to \infty$).

The Peierls instability and the Mott insulator Master

The 1D hydrogen chain with one electron per atom is predicted to be a metal by the simple tight-binding model. In reality, a 1D chain with a half-filled band undergoes the Peierls transition: the chain dimerises, with alternating long and short bonds (atoms pair up). This doubles the unit cell, folding the Brillouin zone from $[-\pi /a,\pi /a]$ to $[-\pi /2a,\pi /2a]$, and opens a gap at the Fermi level $k_F=\pi /2a$. The electronic energy gained by opening the gap exceeds the elastic energy cost of the distortion for any nonzero electron-phonon coupling. The Peierls instability is universal in 1D: no 1D half-filled band is stable against dimerisation.

This was shown by Peierls (1930, Ann. Phys. 4, 121) and independently by Fröhlich. The modern treatment uses the Su-Schrieffer-Heeger (SSH) model of polyacetylene, where the dimerisation opens a gap of about 1.4 eV. Polyacetylene is therefore a semiconductor, not a metal, despite having one electron per site.

A different failure mode of the simple band picture is the Mott insulator. The band theory prediction for NiO is a metal: the Ni 3d band is partially filled. Experimentally, NiO is an insulator. The reason is strong electron-electron repulsion (the Hubbard $U$) on the Ni sites. When $U$ exceeds the bandwidth $W$, the electrons localise to avoid each other, and the system becomes insulating despite the half-filled band. The criterion is $U>W$ for the Mott transition. Band theory (a single-particle theory) cannot capture this because it treats electrons as independent particles moving in an average potential. The Mott insulator is a many-body phenomenon requiring the Hubbard model or dynamical mean-field theory (DMFT) for its description.

Band structures of real materials: Si, Ge, diamond, and metals Master

Diamond (C). Each carbon atom has four valence electrons (2s${}^2$2p${}^2$). In the diamond crystal structure (two interpenetrating FCC lattices), the 2s and 2p orbitals hybridise into four sp${}^3$ hybrids per atom, forming two bands: the bonding sp${}^3$ band (the valence band, holding 4N electrons from $4N$ valence electrons total, fully occupied) and the antibonding sp${}^3$ band (the conduction band, empty). The gap is 5.5 eV — diamond is an insulator.

Silicon. Same crystal structure and bonding type as diamond. The sp${}^3$ hybridisation produces the same two-band structure, but the larger atomic radius and smaller orbital overlap reduce the band gap to 1.1 eV. Silicon is a semiconductor. The valence band maximum occurs at the $\Gamma$ point ($\mathbf{k}=0$) but the conduction band minimum occurs near the $X$ point ($\mathbf{k}\approx (2\pi /a)(1,0,0)$). Silicon is an indirect-gap semiconductor: the valence band maximum and conduction band minimum are at different $\mathbf{k}$-points. Optical transitions requiring large momentum change are suppressed, making silicon a poor light emitter.

Germanium. Same structure, gap of 0.67 eV, also indirect-gap. Ge has a smaller gap than Si because the 4s and 4p orbitals are more diffuse than the 3s and 3p, giving a smaller sp${}^3$ splitting.

Metals (Na, Cu, Al). Sodium has a half-filled 3s band — a simple metal. Copper has a filled 3d band (narrow, high density of states) overlapping with a partially filled 4s band (wide, low density of states). The 3d band lies just below the Fermi level and is responsible for copper's characteristic optical absorption (the d-to-s interband transition gives copper its reddish colour). Aluminium has overlapping 3s and 3p bands, as discussed in Exercise 5.

Real band-structure calculations use density functional theory 14.04.04 with a plane-wave basis. The tight-binding model gives the correct qualitative picture (band formation, gap opening, classification of metals/semiconductors/insulators) but quantitative accuracy requires the full DFT treatment with exchange-correlation functionals.

Density of states in three dimensions Master

In 3D, the density of states per unit volume for a single band is

$$g(E)=\frac{1}{4{\pi }^3}{\int }_{S(E)}\frac{dS}{|{\nabla }_{\mathbf{k}}E(\mathbf{k})|},$$

where $S(E)$ is the constant-energy surface in $\mathbf{k}$-space and ${\nabla }_{\mathbf{k}}E$ is the group velocity. Near a parabolic band edge, $E\approx E_0+{\hslash }^2{k}^2/(2m^{\ast })$, giving

$$g(E)=\frac{1}{2{\pi }^2}{\left(\frac{2m^{\ast }}{{\hslash }^2}\right)}^{3/2}\sqrt{E-E_0},$$

the standard $\sqrt{E}$ density of states for free electrons, modified by the effective mass $m^{\ast }$.

The Van Hove singularities in 3D occur where ${\nabla }_{\mathbf{k}}E=0$ — at band extrema (maxima, minima) and saddle points. In 3D these produce kinks (discontinuities in $dg/dE$) rather than true divergences. The Van Hove theorem (1953) classifies these singularities by the number of negative eigenvalues of the Hessian ${\partial }^{2}E/\partial k_i\partial k_j$.

The total number of electrons in the crystal is obtained by integrating the density of states weighted by the Fermi-Dirac occupation:

$$N_e={\int }_{-\infty }^{\infty }g(E)f(E)dE={\int }_{-\infty }^{\infty }\frac{g(E)}{e^{(E-E_F)/k_{B}T}+1}dE.$$

At $T=0$, $f(E)=1$ for $E<E_F$ and $f(E)=0$ for $E>E_F$, so $N_e={\int }_{-\infty }^{E_F}g(E)dE$. The Fermi level is determined self-consistently by fixing $N_e$ (the number of valence electrons) and solving for $E_F$. For a metal with a partially filled band, $E_F$ lies inside the band. For an insulator with a gap, $E_F$ lies in the gap.

Connections Master

• Molecular orbital theory [14.05.01, 14.05.02]. Band theory is the direct extension of LCAO-MO theory to the $N\to \infty$ limit of a periodic array. The $2\times 2$ secular equation of 14.05.01 becomes the tight-binding matrix on an infinite lattice; the bonding/antibonding pair becomes the full energy band; the LCAO coefficients become the Bloch wavevector phases. Every concept in this unit — on-site energy $\alpha$, hopping integral $\beta$, bandwidth, band gap — has its molecular precursor in 14.05.01.

• Density functional theory 14.04.04. Quantitative band structures are computed by solving the Kohn-Sham equations on a plane-wave basis with periodic boundary conditions. The tight-binding model is the qualitative skeleton that DFT fleshes out numerically. The band gap predicted by DFT with standard functionals (LDA, GGA) is systematically underestimated compared to experiment — the "band gap problem" — because the Kohn-Sham eigenvalues are not quasiparticle energies. GW corrections and hybrid functionals (HSE06, PBE0) repair this.

• Statistical mechanics 14.07.01. The Fermi-Dirac distribution used to determine the temperature-dependent occupation of band states is derived from the grand canonical ensemble for fermions. The electron gas in a metal is the canonical application of Fermi-Dirac statistics. The specific heat of electrons in a metal (proportional to $T$, not $T^3$ as for phonons) follows from the linearised Fermi-Dirac distribution near $E_F$.

• Spectroscopy 14.12.01. Photoemission spectroscopy (XPS, UPS) measures the occupied density of states by ejecting electrons from below $E_F$. Inverse photoemission measures the unoccupied states above $E_F$. Optical absorption in semiconductors measures the band gap via interband transitions. ARPES (angle-resolved photoemission) directly maps the band dispersion $E(\mathbf{k})$.

• Acid-base and redox chemistry [14.10.01, 14.11.01]. The Fermi level of a semiconductor is the electrochemical potential of its electrons. When two semiconductors (or a semiconductor and an electrolyte) are brought into contact, the Fermi levels equilibrate, producing band bending at the interface. This is the physical basis of the semiconductor-electrolyte junction in photoelectrochemical cells and the p-n junction in diodes and transistors.

Historical & philosophical context Master

Felix Bloch's 1928 paper "Uber die Quantenmechanik der Elektronen in Kristallgittern" (Z. Phys. 52, 555) established that electrons in a periodic potential are described by the wave form that bears his name. Bloch was a PhD student of Heisenberg in Leipzig; his insight was to treat the crystal lattice as a periodic boundary condition on the electron wavefunction, converting the solid-state problem from an impossibly large molecular-orbital calculation into a band-structure problem parametrised by the wavevector $\mathbf{k}$.

Rudolf Peierls (1929, Z. Phys. 53, 243) showed that magnetic fields quantise electron orbits in a periodic potential, explaining the de Haas-van Alphen effect. The Peierls instability of 1D metals came later (1930). The Kronig-Penney model (1931, Proc. R. Soc. A 130, 499) provided an exactly solvable periodic potential (delta-function barriers) demonstrating band formation and gap opening analytically.

Alan Wilson's 1931 paper (Proc. R. Soc. A 133, 458) gave the first band-theoretic explanation of the distinction between metals, semiconductors, and insulators. Wilson identified the band gap as the key parameter and correctly classified diamond as an insulator and silicon/germanium as semiconductors. The Wilson model replaced the older Drude free-electron theory (which treated all solids as metals) and established band theory as the correct framework for understanding electronic properties of solids.

The invention of the transistor at Bell Labs in 1947 (Bardeen, Brattain, Shockley) was a direct application of semiconductor band theory. Shockley's 1950 book Electrons and Holes in Semiconductors laid out the band-theoretic treatment of p-n junctions, carrier injection, and transistor action. The subsequent development of integrated circuits, microprocessors, and the entire digital revolution rests on the band-theoretic understanding of semiconductors developed in this unit.

The tight-binding method was formalised by Slater and Koster (1954, Phys. Rev. 94, 1498), who tabulated the hopping integrals for all symmetry types of atomic orbital pairs as functions of direction — the Slater-Koster tables remain a standard reference for parameterising tight-binding models.

The philosophical dimension: the Bloch state ${\psi }_{n\mathbf{k}}$ is delocalised across the entire crystal. This is the solid-state analogue of the delocalisation question for molecular orbitals discussed in 14.05.02. An electron in a Bloch state has equal probability of being found in any unit cell — it belongs to the crystal as a whole, not to any particular atom. The localised alternative (Wannier functions, defined as the Fourier transform of Bloch states over the Brillouin zone) provides an orthogonal set of localised orbitals, analogous to the localised molecular orbitals of Boys or Edmiston-Ruedenberg. The choice between Bloch and Wannier representations is a change of basis; the physics is invariant.

Bibliography Master

• Bloch, F., "Uber die Quantenmechanik der Elektronen in Kristallgittern," Z. Phys. 52 (1928), 555. The founding paper of band theory.

• Peierls, R., "Zur Theorie der Elektronenleitung in Kristallen," Ann. Phys. 4 (1930), 121. The Peierls instability.

• Kronig, R. de L. & Penney, W. G., "Quantum mechanics of electrons in crystal lattices," Proc. R. Soc. A 130 (1931), 499. Exactly solvable periodic potential model.

• Wilson, A. H., "The theory of electronic semi-conductors," Proc. R. Soc. A 133 (1931), 458; 134 (1931), 277. Band-theoretic classification of metals, semiconductors, and insulators.

• Slater, J. C. & Koster, G. F., "Simplified LCAO method for the periodic potential problem," Phys. Rev. 94 (1954), 1498. The Slater-Koster tight-binding parameterisation.

• Shockley, W., Electrons and Holes in Semiconductors (Van Nostrand, 1950). The band-theoretic treatment of semiconductors and the transistor.

• Ashcroft, N. W. & Mermin, N. D., Solid State Physics (Holt, Rinehart and Winston, 1976). Ch. 8-9 (Bloch theorem, tight-binding), Ch. 10-11 (semiconductors, Fermi surface).

• Kittel, C., Introduction to Solid State Physics, 8e (Wiley, 2005). Ch. 7 (energy bands), Ch. 8 (semiconductor crystals).

• Zumdahl, S. S. & DeCoste, D. J., Chemical Principles, 8e (Cengage, 2017), Ch. 10.5. Introductory treatment of band theory for chemistry students.

• Kohn, W. & Sham, L. J., "Self-consistent equations including exchange and correlation effects," Phys. Rev. 140 (1965), A1133. Foundation of modern band-structure calculations.

• Van Hove, L., "The occurrence of singularities in the elastic frequency distribution of a crystal," Phys. Rev. 89 (1953), 1189. Classification of singularities in the density of states.