4  Tight binding

We introduce here a third perspective, which stands at the opposite limit of the nearly-free electron model of Section 3.2.2: here we will consider localised atomic orbitals brought into interaction across the crystal.

4.1 Basic example

Let us start from a very minimalistic example to grasp the key concepts one by one. A basis built out of atomic orbitals has a major problem: it does not properly take advantage of the great simplification provided by Bloch’s theorem; we thus need a basis change.

Definition 4.1: Bloch sum

Given a localised orbital \(\phi_a(\mathbf{r})\) centred on each lattice site, the Bloch sum of quasi-momentum \(\mathbf{k}\) is

\[\psi_\mathbf{k}(\mathbf{r}) = \frac{1}{\sqrt{N}} \sum_{\mathbf{R}\in\mathcal{BL}} e^{i\mathbf{k}\cdot\mathbf{R}}\, \phi_a(\mathbf{r}-\mathbf{R}), \tag{4.1}\]

where the sum runs over all \(N\) lattice sites of the crystal. For every orbital and for every atom in the basis, a Bloch sum should be defined. The resulting function is a Bloch state, the easy demonstration is left to the reader. In the limit of zero overlap between orbitals localised in different cells, the prefactor \(1/\sqrt{N}\) ensures unit normalisation (if overlap is not negligible, this will require further discussion).

Exercise. Show that the Bloch sum \(\psi_\mathbf{k}\) is a Bloch state with quasi-momentum \(\mathbf{k}\).

Here, we want to consider the following extremely simplified limit:

  • the crystal is 1D, with lattice spacing \(a\) and one atom per cell (trivial basis);
  • a single orbital \(\phi_a(x)\) per atom is included in the calculation;
  • we have \(N\) cells with periodic boundary conditions;
  • the full Hamiltonian only couples the first neighbouring atoms.

Given the localised basis is \(|\phi_a(x-t_n)\rangle\) with \(t_n=na\), the last condition is encoded in the hopping integrals \(\langle\phi_a(x)|\mathcal{H}|\phi_a(x-ma)\rangle\) measuring the coupling between orbitals at sites separated by \(m\) lattice spacings. Here we retain only:

  • \(\langle\phi_a(x)|\mathcal{H}|\phi_a(x)\rangle=\varepsilon_0\): the on-site energy;
  • \(\langle\phi_a(x)|\mathcal{H}|\phi_a(x\pm a)\rangle = \gamma\) for nearest-neighbours (\(\gamma < 0\));
  • zero hopping otherwise.

In this very special limit case, we can only define a single Bloch sum for every \(k\), which is normalised to one in the limit of zero overlap \(\langle\phi_a(x)|\phi_a(x+a)\rangle=0\). The Hamiltonian does not couple Bloch sums with different \(k\) so there is basically nothing to diagonalise: the band dispersion can be calculated as the expectation value \(E(k) = \langle\psi_k|\mathcal{H}|\psi_k\rangle/\langle\psi_k|\psi_k\rangle\):

\[E(k) = \frac{1}{N}\sum_{n,n'} e^{ik(na-n'a)}\,\langle\phi_a(x-n'a)|\mathcal{H}|\phi_a(x-na)\rangle,\]

where we assumed \(\langle\psi_k|\psi_k\rangle=1\), which is valid in the limit of zero overlap between nearby orbitals. Substituting \(m = n-n'\) and summing over \(n'\) (which gives a factor \(N\)) we obtain the dispersion

\[ \begin{aligned} E(k) &= \sum_{m=-\infty}^{+\infty} e^{ikma}\langle\phi_a(x)|\mathcal{H}|\phi_a(x-ma)\rangle\\ &= \sum_{m = 0,\pm 1} e^{ikma}\langle\phi_a(x)|\mathcal{H}|\phi_a(x-ma)\rangle\\ &= \varepsilon_0 + \gamma e^{-ika} + \gamma e^{+ika} = \varepsilon_0 + 2\gamma\cos(ka). \end{aligned} \tag{4.2}\]

This dispersion is displayed in Figure 4.1.

Figure 4.1: Tight-binding model on a mono-atomic mono-orbital 1D chain. Top: cosine band \(E(k)=\varepsilon_0+2\gamma\cos(ka)\). Bottom: the corresponding Bloch sum is illustrated; drag the bar on the top panel to modify the value of \(k\).

4.2 Semi-empirical tight binding

Let us now move to a generic case where many missing ingredients will need to be added with respect to Section 4.1: the presence of a non-trivial basis and the expansion over multiple orbitals.

Before diving into the formalism, it is worth stating the philosophy of what follows: we will make a chain of drastic simplifications, with the explicit goal of reducing the band problem to an analytic model containing only a handful of parameters (a few on-site energies and hopping integrals). Those parameters are then not computed from first principles but fitted to data — measured gaps and effective masses, or reference ab-initio bands. The approximations are individually questionable, but their errors are absorbed by the fit: what survives is a cheap, transparent model that reproduces and interpolates the fitted features. The empirical pseudopotential method of Section 5.3 will follow exactly the same strategy, with Fourier components of the potential in place of hopping integrals.

Definition 4.2: Bloch sums (with explicit basis)

For a crystal with a basis \(\{\mathbf{d}_\nu\}\) and atomic orbitals \(\phi_i\) (e.g. \(s, p_x, p_y, p_z, \dots\)) localised at the origin, one Bloch sum is constructed per orbital per basis atom:

\[\psi_{i\nu}(\mathbf{k},\mathbf{r}) = \frac{1}{\sqrt{N}}\sum_{\mathbf{R}\in\mathcal{BL}} e^{i\mathbf{k}\cdot\mathbf{R}}\,\phi_i(\mathbf{r}-\mathbf{R}-\mathbf{d}_\nu). \tag{4.3}\]

This is the same equation stated above, with an explicit indication of the basis vectors. Note again that the sum does not run over the basis vectors: for every basis element and for every orbital, we define a different Bloch sum.

Definition 4.3: Tight binding master equation

The key idea is that crystal eigenstates at fixed \(\mathbf{k}\) are linear combinations \(\psi = \sum_{i\nu}c_{i\nu}\psi_{i\nu}\). After projection the classic eigenvalue equation \(\mathcal{H}\psi = E\psi\) will lead to a generalised eigenvalue problem

\[\sum_{j\mu}\bigl[M_{i\nu,j\mu}(\mathbf{k}) - E\,S_{i\nu,j\mu}(\mathbf{k})\bigr]\,c_{j\mu} = 0, \tag{4.4}\]

with an interaction matrix \(M_{i\nu,j\mu} = \langle\psi_{i\nu}|\mathcal{H}|\psi_{j\mu}\rangle\) and overlap matrix \(S_{i\nu,j\mu} = \langle\psi_{i\nu}|\psi_{j\mu}\rangle\).

The band energies are the roots of \(\det[M(\mathbf{k}) - E\,S(\mathbf{k})] = 0\).

4.2.1 Key approximations

Real TB calculations apply a sequence of simplifications that make Equation 4.4 tractable without losing the essential physics:

  1. Neglect of orbital overlap — assume \(\langle\phi_i(\mathbf{r})|\phi_j(\mathbf{r}-\mathbf{t})\rangle \approx 0\) for \(\mathbf{t}\neq\mathbf{0}\), so \(S = \mathbb{1}\) and the problem reduces to ordinary diagonalisation of \(M(\mathbf{k})\). Overlap corrections can be reintroduced via a non-trivial \(S\) in more refined schemes.
  2. Two-centre approximation — decompose \(V(\mathbf{r}) = V_a(\mathbf{r}) + V'(\mathbf{r})\) with \(V_a\) the on-site atomic potential, then retain in matrix elements only integrals involving two distinct spatial centres (either the two orbitals on distinct atoms with the potential centred on one of them, or both orbitals on the same atom with the potential on another). The neglected three-centre integrals are small when atomic orbitals and potentials decay rapidly.
  3. Diagonal crystal field — for two orbitals centred on the same site, the mean field of the surrounding crystal, \(I_{ij} = \langle\phi_i|V'|\phi_j\rangle\), is assumed diagonal, \(I_{ij} \approx I_i\,\delta_{ij}\). This merely renormalises the on-site energies \(E_i \to E_i + I_i\) and neglects off-diagonal symmetry-breaking contributions.
  4. Nearest-neighbour hopping — restrict the lattice sum to first neighbours. Further shells can be added for higher accuracy.

Under all four, the matrix elements take the compact form

\[M_{i\nu,j\mu}(\mathbf{k}) = E_i\,\delta_{ij}\delta_{\nu\mu} + \sum_{\mathbf{t}_I}\langle\phi_i(\mathbf{r})|V_a(\mathbf{r}-\mathbf{t}_I-\boldsymbol{\delta}_{\mu\nu})|\phi_j(\mathbf{r}-\mathbf{t}_I-\boldsymbol{\delta}_{\mu\nu})\rangle\,e^{i\mathbf{k}\cdot\mathbf{t}_I},\]

with \(\boldsymbol{\delta}_{\mu\nu} = \mathbf{d}_\mu - \mathbf{d}_\nu\) and the sum running over nearest neighbours only.

4.2.2 Slater–Koster two-centre integrals

Assuming a spherically symmetric atomic potential \(V_a(r)\), the two-centre integrals at fixed interatomic distance \(R\) depend only on the orientation of the unit vector \(\mathbf{n} = (n_x,n_y,n_z)\) joining the two sites. The angular dependence is fully determined by the component \(m\) of the orbital angular momentum along \(\mathbf{n}\), which classifies bonds as \(\sigma\) (\(m=0\)), \(\pi\) (\(m=\pm 1\)) or \(\delta\) (\(m=\pm 2\)). For each symmetry class a single Slater–Koster parameter \(V_{\ell\ell' m}\) encodes the radial magnitude; all geometric information is absorbed into angular prefactors. For \(s,p\) orbitals the four independent parameters are

\[V_{ss\sigma}, \quad V_{sp\sigma}, \quad V_{pp\sigma}, \quad V_{pp\pi},\]

and the hopping integrals follow from the Slater–Koster table:

\[\begin{aligned} \langle s|V_a|s\rangle &= V_{ss\sigma}, \\ \langle s|V_a|p_x\rangle &= n_x\,V_{sp\sigma}, \\ \langle p_x|V_a|p_x\rangle &= n_x^2\,V_{pp\sigma} + (1-n_x^2)\,V_{pp\pi}, \\ \langle p_x|V_a|p_y\rangle &= n_x n_y\,(V_{pp\sigma} - V_{pp\pi}), \\ \langle p_x|V_a|p_z\rangle &= n_x n_z\,(V_{pp\sigma} - V_{pp\pi}). \end{aligned} \tag{4.5}\]

Extensions to \(d\) orbitals (and the \(\delta\) bond) add a handful more parameters. With these few numbers — typically fit to reference first-principles calculations or to experimental band gaps and effective masses — one can build the full TB Hamiltonian of any \(sp\)- (or \(spd\)-) bonded crystal, for any bond geometry.

The angular factors of Equation 4.5 can be explored interactively in Figure 4.2: as the second site moves around the first, each pair of \(s\), \(p_x\), \(p_y\) orbitals traces the characteristic \(n_x\), \(n_y\) combination of the table.

Figure 4.2: Angular structure of the Slater–Koster two-centre integrals, restricted to the \(xy\) plane. Left: orbital 1 fixed at the centre, orbital 2 draggable along the dashed circle; the colorplot shows the signed product \(\psi_1\psi_2\) (red \(+\), blue \(-\)), dashed outlines sketch the orbital lobes, and the direction cosines \(n_x=\cos\theta\), \(n_y=\sin\theta\) are displayed live. Right: the integral \(\int \psi_1\psi_2\,d^2r\) versus \(\theta\), normalised to its largest value, with min/max guides — the marker is draggable and stays in sync with the atom. The purple label is the corresponding Slater–Koster entry of Equation 4.5: hopping integrals carry the same angular factors, with radial prefactors of their own sign.

4.3 Graphene: a case study in tight binding

Graphene — a single layer of carbon atoms arranged in a honeycomb lattice — is the most celebrated application of the tight-binding method. It incorporates all the key ingredients (a basis of two atoms, multiple orbitals, symmetry-enforced band crossings) while remaining analytically tractable in 2D.

4.3.1 Crystal structure and Brillouin zone

The honeycomb lattice is not a Bravais lattice. It is described by a triangular Bravais lattice with a two-atom basis (\(A\) and \(B\) sublattices). The primitive vectors are

\[\mathbf{t}_1 = \frac{a}{2}(1,\sqrt{3}), \qquad \mathbf{t}_2 = \frac{a}{2}(-1,\sqrt{3}), \tag{4.6}\]

with \(a = 2.46\,\text{Å}\) the lattice constant (the C–C bond length is \(a_0 = a/\sqrt{3} = 1.42\,\text{Å}\)). The basis vectors are \(\mathbf{d}_A = (0,0)\) and \(\mathbf{d}_B = (0, a/\sqrt{3})\).

The reciprocal lattice vectors are \(\mathbf{g}_{1,2} = (2\pi/a)(\pm 1, 1/\sqrt{3})\), generating a hexagonal Brillouin zone. The two physically inequivalent corner points are

\[K = \frac{2\pi}{a}\!\left(\frac{2}{3},0\right), \qquad K' = \frac{2\pi}{a}\!\left(-\frac{2}{3},0\right), \tag{4.7}\]

known as the Dirac points. Among the six corners of the BZ, three \(K\)-type and three \(K'\)-type points alternate; those related by a reciprocal lattice vector are equivalent. The midpoints of the BZ edges are saddle points \(Q\).

4.3.2 \(\sigma\) and \(\pi\) bands

Each carbon atom has four valence electrons in the \(2s\), \(2p_x\), \(2p_y\), \(2p_z\) orbitals. With two atoms per cell, the Bloch-sum basis has dimension \(8\) and the Hamiltonian matrix is \(8\times 8\). However, the graphene plane is a mirror symmetry plane: \(s\), \(p_x\), \(p_y\) are even under reflection, while \(p_z\) is odd. Since the Hamiltonian respects this symmetry, the matrix decouples into:

  • A \(6\times 6\) block for the even (\(\sigma\)) orbitals — these form deeply bonding \(\sigma\) bands (the backbone of the C–C covalent bonds) and their antibonding \(\sigma^*\) counterparts. All \(\sigma\) bands are fully occupied and lie well below the Fermi level.
  • A \(2\times 2\) block for the odd (\(\pi\)) orbitals — the \(p_z\) Bloch sums on sublattices \(A\) and \(B\). These produce the \(\pi\) (bonding) and \(\pi^*\) (antibonding) bands that control the electronic properties near the Fermi level.

The parity argument is illustrated in Figure 4.3 (a): since a Bloch sum inherits the \(z\)-parity of its atomic orbital, and the Hamiltonian commutes with \(\sigma_h\), matrix elements between even and odd Bloch sums vanish identically.

Figure 4.3: (a) \(z\)-parity of the carbon valence orbitals on the graphene plane (translucent disc = the mirror plane \(\sigma_h\); lobe signs: red \(+\), blue \(-\)). Reflection through the plane maps the \(s\), \(p_x\), \(p_y\) lobes onto themselves — even — while \(p_z\) swaps its signed lobes — odd. Bloch sums built from these orbitals split accordingly into the \(\sigma\) (\(6\times6\)) and \(\pi\) (\(2\times2\)) blocks. Drag to rotate. (b) Tight-binding neighbour shells of graphene around a central carbon atom: the concentric circles mark the three nearest neighbours on the opposite sublattice (hopping \(t\), green), the six second neighbours on the same sublattice (\(t'\), purple) and the three third neighbours (\(t''\), black). Toggle each shell independently; \(\mathbf{a}_1\), \(\mathbf{a}_2\) are the primitive lattice vectors and \(A\)/\(B\) label the two sublattices.

4.3.3 \(\pi\)-band dispersion

The range of the retained hopping integrals sets the level of approximation. Figure 4.3 (b) shows the first three neighbour shells around a carbon atom — the three nearest neighbours on the opposite sublattice (hopping \(t\)), the six second neighbours on the same sublattice (\(t'\)) and the three third neighbours (\(t''\)) — the shells that successively refine the \(\pi\)-band model.

In the nearest-neighbour tight-binding approximation, the on-site energies of the two sublattices are equal (we set them to zero) and the only non-zero hopping integral is \(V_{pp\pi}\) between nearest \(p_z\) orbitals. Each \(A\) atom has three \(B\) neighbours, connected by the vectors \(\boldsymbol{\tau}_j\). The off-diagonal matrix element of the \(2\times 2\) Hamiltonian is

\[M_{AB}(\mathbf{k}) = V_{pp\pi}\,F(\mathbf{k}), \qquad F(\mathbf{k}) = 1 + 2\cos\!\left(\frac{k_x a}{2}\right)e^{-ik_y a\sqrt{3}/2}, \tag{4.8}\]

and the dispersion is simply \(E(\mathbf{k}) = \pm |V_{pp\pi}|\,|F(\mathbf{k})|\). The function \(|F|\) is maximal at \(\Gamma\) (\(|F| = 3\), largest band separation), equals unity at the saddle points \(Q\), and vanishes at \(K\) and \(K'\) — where the \(\pi\) and \(\pi^*\) bands touch. With two \(\pi\) electrons per cell filling half of the four available \(\pi\) states, the Fermi level sits exactly at the Dirac points: graphene is a semimetal with zero density of states at \(E_F\).

Figure 4.4: Electronic structure of graphene. Left: the \(\pi\) (red) and \(\pi^*\) (blue) bands growing out of the \(k\)-plane, which carries the first Brillouin zone, the reciprocal primitive vectors \(\mathbf{g}_{1,2}\) and the in-plane labels \(\Gamma\), \(K\), \(K'\); the Dirac cones sit at the hexagon corners (drag to rotate). Right: band structure along \(\Gamma\)–\(K\)–\(Q\)–\(\Gamma\) and density of states, with the van Hove singularity at the saddle-point energy; the dashed line marks the Fermi level of neutral graphene, pinned at the Dirac point. The checkbox switches on the second-neighbour hopping \(t'\), which breaks particle-hole symmetry: the Dirac point (and \(E_F\) with it) shifts away from zero and the \(\pi\) and \(\pi^*\) bandwidths become different.

To make the real-space meaning of a Bloch state more vivid, Figure 4.5 shows the phase of the actual \(\pi\) eigenstate over the honeycomb lattice for any chosen \(\mathbf{k}\) in the BZ. Each \(A\) site carries the cell factor \(e^{i\mathbf{k}\cdot\mathbf{R}}\); each \(B\) site carries in addition the pseudospin phase of the selected band, \(c_B/c_A = \pm e^{-i\theta_\mathbf{k}}\) with \(\theta_\mathbf{k} = \arg F(\mathbf{k})\) — in phase with \(A\) at \(\Gamma\) for the bonding \(\pi\) band, in antiphase for the antibonding \(\pi^*\). The phase is encoded as a cyclic colour wheel (hue \(= \arg\psi\), full saturation). Faint grey stripes across the lattice mark the wavefronts — surfaces of constant cell phase, perpendicular to \(\mathbf{k}\) and spaced by \(2\pi/|\mathbf{k}|\).

Figure 4.5: Bloch-state phase pattern on graphene. Left: hexagonal first Brillouin zone with \(\Gamma\), \(K\), \(K'\) and \(Q\) marked; click anywhere or use the go to buttons to choose \(\mathbf{k}\) (blue marker). Right: the honeycomb lattice, each atom coloured by the phase of the \(\pi\) eigenstate at its site (hue cycles 0–2π): the \(B\) sublattice carries the extra pseudospin phase of the band chosen in the selector — switch between VB and CB at \(\Gamma\) to see the bonding/antibonding character flip the \(B\) colours. Semi-transparent grey stripes are the wavefronts, perpendicular to \(\mathbf{k}\) and spaced by \(2\pi/|\mathbf{k}|\). At \(\Gamma\) each sublattice is uniform; at the zone boundary (\(Q\), \(K\)) the cell phase rotates by \(\pi\) or \(2\pi/3\) between adjacent cells.

4.3.4 Dirac cones and massless fermions

Expanding \(F(\mathbf{k})\) near \(K\) with \(\mathbf{k} = \mathbf{K} + \mathbf{q}\) and \(|\mathbf{q}| \ll |\mathbf{K}|\):

\[F(\mathbf{K}+\mathbf{q}) \approx -\frac{a\sqrt{3}}{2}(q_x - iq_y), \tag{4.9}\]

so the effective Hamiltonian becomes

\[\mathcal{H}_K = \hbar v_F\,\boldsymbol{\sigma}\cdot\mathbf{q} = \hbar v_F\begin{pmatrix} 0 & q_x - iq_y \\ q_x + iq_y & 0 \end{pmatrix}, \tag{4.10}\]

where \(\hbar v_F = |V_{pp\pi}|\,a\sqrt{3}/2\) defines the Fermi velocity \(v_F \approx 10^6\,\text{m/s} \approx c/300\), and \(\boldsymbol{\sigma} = (\sigma_x, \sigma_y)\) are Pauli matrices acting in the sublattice space. The eigenvalues are

\[\boxed{E(\mathbf{q}) = \pm\hbar v_F|\mathbf{q}|,} \tag{4.11}\]

a linear (conical) dispersion — identical in form to the relativistic energy–momentum relation \(E = cp\) for a massless particle, with \(v_F\) playing the role of the speed of light. Near the Dirac points, the electrons in graphene behave as massless Dirac fermions.

The two-component (sublattice) structure of these states hides more physics — a pseudospin locked to the momentum direction, a \(\pi\) Berry phase for loops encircling the Dirac point, and a mass gap opening as soon as the two sublattices are made inequivalent. These developments are collected in Section A.1.

4.4 Density of states of tight-binding toys

The tight-binding dispersions constructed in this chapter are the cleanest playground for the density-of-states machinery of Section 3.3.2: each model below is solvable in closed (or nearly closed) form and displays, in its dimension, the van Hove singularities classified there. As the dimension grows the full \(E(\mathbf{k})\) cannot be drawn, and the conventional workaround is to display the dispersion along a one-dimensional path through high-symmetry points and lines of the FBZ.

The simplest non-trivial DOS belongs to the 1D tight-binding chain of Section 4.1: a single \(s\)-wave orbital per site with nearest-neighbour hopping \(\gamma<0\) (on-site energy set to zero). The dispersion is \[E(k) = 2\gamma\cos(ka), \qquad k\in[-\pi/a, \pi/a],\] so the band spans \([-2|\gamma|, +2|\gamma|]\). The two stationary points (a minimum at \(\Gamma\) and a maximum at the zone boundary \(X = \pi/a\)) generate the canonical 1D van Hove signature: a \(1/\sqrt{|E - E_c|}\) divergence at each band edge. The DOS admits the closed form \[D(E) = \frac{1}{\pi\sqrt{4\gamma^2 - E^2}}, \qquad |E| < 2|\gamma|.\] Drag the iso-energy line in Figure 4.6 to see how the filled (orange) and empty (blue) parts of the band track the cumulative integral of \(D(E)\).

Figure 4.6: 1D tight-binding chain. Left: the cosine band \(E(k) = 2\gamma\cos(ka)\), with the orange/blue split tracking the iso-energy line. Right: analytical DOS \(D(E)\), with \(1/\sqrt{|E - E_c|}\) singularities at both band edges.

The squarium is a pedagogical 2D tight-binding model — an \(s\)-wave orbital on a square lattice with nearest-neighbour hopping \(\gamma<0\):

\[E(k_x,k_y) = 2\gamma(\cos k_x a + \cos k_y a). \tag{4.12}\]

The band spans \([-4|\gamma|,\,+4|\gamma|]\) with:

  • A minimum at \(\Gamma = (0,0)\) — the iso-energetic contour is approximately circular (parabolic dispersion), and \(D(E)\) jumps from zero to a finite value (step, an \(M_0\) singularity).
  • Saddle points at \(X = (\pi/a,0)\) and \(Y = (0,\pi/a)\) — the Fermi contour touches the zone boundary and changes topology (Lifshitz transition). The DOS shows a logarithmic van Hove divergence (\(M_1\) singularity).
  • A maximum at \(M = (\pi/a,\pi/a)\) — the Fermi contour is again circular (but now an empty contour enclosing unoccupied states), and \(D(E)\) drops back to zero (step, an \(M_2\) singularity).
Figure 4.7: Density of states of the squarium. Left: band dispersion \(E(\mathbf{k})\) along the high-symmetry path \(\Gamma\)–\(M\)–\(X\)–\(\Gamma\). Centre: Brillouin zone map showing occupied (orange) and empty (blue) states; the thick black contour is the Fermi line. Right: density of states \(D(E)\) with the logarithmic van Hove singularity at \(E = 0\). Drag the Fermi energy slider to observe the evolution of the Fermi contour topology.

The same analysis extends naturally to three dimensions. The cubium — a simple-cubic lattice with nearest-neighbour hopping \(\gamma<0\) — has dispersion \(E(\mathbf{k}) = 2\gamma(\cos k_x a + \cos k_y a + \cos k_z a)\), with three independent cosines instead of two. The density of states can again be expressed in closed form, as a convolution of the 2D result with a one-dimensional density of states. The resulting \(D(E)\) displays two square-root van Hove cusps at \(E = \pm 2|\gamma|\) — in 3D the DOS stays finite at the saddle points, with a divergent one-sided slope — and square-root onsets at the band edges \(E = \pm 6|\gamma|\).

Figure 4.8: Density of states of the cubium. Left: 3D Brillouin zone with the Fermi iso-surface (drag to rotate). Right: band dispersion \(E(\mathbf{k})\) along \(\Gamma\)–\(X\)–\(M\)–\(R\)–\(\Gamma\) and density of states \(D(E)\). Drag the energy to explore the evolution of the Fermi surface topology.
ImportantTake-home message
  • The Bloch sum turns any set of localised orbitals into Bloch-compliant basis states; the crystal Hamiltonian is then a small \(\mathbf{k}\)-dependent matrix, one row per orbital per basis atom.
  • Slater–Koster reduces every two-centre hopping integral to a few radial parameters times purely geometric direction-cosine factors.
  • Graphene is the payoff: two equivalent sublattices force a symmetry-protected band touching — the Dirac cones — with a \(\pi\) Berry phase and pseudospin-suppressed backscattering. Break the sublattice symmetry and a mass gap opens.