5 Band review

This chapter closes the loop with two questions — how are real band structures actually computed, and what do the bands of real materials look like?

5.1 Cellular toy model

Before moving to real bands it is useful to consider a toy solver: the interactive Figure 5.2 treats a single electron in one square unit cell $[0,a]\times[0,a]$, discretises the Schrödinger equation on a finite-difference grid, and sweeps the wavevector $\mathbf{k}$ across the Brillouin zone. It solves the problem only on a single cell, imposing Bloch conditions (here typically indicated as Floquet conditions) on the edges. Bloch’s theorem demands $\psi_\mathbf{k}(\mathbf{r}+\mathbf{R}) = e^{i\mathbf{k}\cdot\mathbf{R}}\,\psi_\mathbf{k}(\mathbf{r})$, so we have

\[ \begin{aligned} \psi(a^-,y) &= e^{i k_x a}\,\psi(0^+,y)\\ \psi(x,a^-) &= e^{i k_y a}\,\psi(x,0^+) \end{aligned} \]

Diagonalising at each $\mathbf{k}$ and following the eigenvalues along $\Gamma$–$M$–$X$–$\Gamma$ traces out the band structure. The natural starting point is $V_0 = 0$: with no potential at all, the “band structure” is nothing but the free-electron paraboloid $E = \hbar^2|\mathbf{k}-\mathbf{G}|^2/2m$ drawn once per reciprocal-lattice vector $\mathbf{G}$ and read back inside the first Brillouin zone — zone folding. Figure 5.1 lets you build interactively the folded spectrum copy by copy.

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Figure 5.1: Zone folding of the free-electron parabola on the square lattice. Left: full $E(k_x,k_y)$ plot, with highlighted cross-sections. Centre: select which translated parabola $E_\mathbf{G}(\mathbf{k})=\hbar^2|\mathbf{k}-\mathbf{G}|^2/2m$ to activate, with $\mathbf{G}=(2\pi/a)(m,n)$. Right: folded band cross-section along $\Gamma$–$M$–$X$–$\Gamma$. This is the natural extension of the classic folded parabola in 1D, perfectly illustrating how much complication in band dispersions comes from basic geometry (conceptually, it is just the same thing).

Figure 5.2 then switches the confining potential on. At $V_0 = 0$ it starts exactly from the folded free-electron spectrum of Figure 5.1; as the well deepens, gaps open at the zone boundary and, in the strong-potential limit, the lowest band flattens into exactly the one-orbital tight-binding cosine of Chapter 4 — the two pictures of the previous chapters, bridged by a single knob.

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Figure 5.2: Band dispersions of a single electron in a square crystal cell: check out how the potential $V(x,y)$ in the single cell lifts degeneracies of the folded parabola.

5.2 Plane-wave expansion

A natural program for computing real bands is to expand the Bloch states on plane waves and diagonalise. Applied directly to a real crystal, however, the program fails: valence wavefunctions live on two utterly different length scales — smooth over the bond, rapidly oscillating near each nucleus to stay orthogonal to the core — and resolving both would require an enormous number of plane waves (see the Herring’s catastrophe box). The classic way out is to expand the valence states in the space orthogonal to the core states, which for their part are practically untouched by the existence of the crystal (OPW approach): the orthogonalisation adds precisely the term that keeps the valence electrons orthogonal to the core electrons in the neighbourhood of the nucleus. Absorbing that term into an effective potential — the pseudopotential — leaves a smooth problem, on which a Fourier expansion with a handful of components finally makes sense (Pseudopotential box). This is ultimately why the NFE-like machinery gives reasonable bands for common crystals (Al, Ca, Si, GaAs…).

Herring’s catastrophe

The main issue is that the problem of eigenstates in a real crystal contains vastly different length scales that make it problematic to carelessly describe it using a plane wave expansion. Consider for instance Si: the $3s/3p$ valence states extend over the bond length, but near each nucleus they must oscillate rapidly to stay orthogonal to the deep $1s$, $2s$, $2p$ core states — two length scales differing by an order of magnitude in the same wavefunction.

The simplest expansion basis for Bloch states is plane waves $\psi_{n\mathbf{k}} = \sum_\mathbf{g} c_{n,\mathbf{k}+\mathbf{g}}\,e^{i(\mathbf{k}+\mathbf{g})\cdot\mathbf{r}}$. However, plane-wave convergence is very slow near atomic nuclei, where the true wavefunctions oscillate rapidly to maintain orthogonality to the core states.

A concrete manifestation of this difficulty is the variational collapse of pure plane-wave expansions: as the plane-wave cutoff is raised, the lowest-energy solutions of the secular equation drift towards tightly bound core states that require an enormous number of Fourier components to represent. The valence and conduction bands of interest are then swamped by poorly converged core states, and the secular equation becomes unreliable across the entire spectrum.

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Figure 5.3: The two scales coexisting in a real crystal. The Coulomb wells of neighbouring nuclei (black) host the $1s$ core level at binding energies of order keV — note the break in the energy axis — with a wavefunction confined within $\sim a_B/Z$ of each nucleus (red). The valence state (blue) lives at eV energies and spans the bond length, but near each nucleus it must oscillate rapidly to stay orthogonal to the core.

OPW approach

Core states in the crystal. The starting observation is that core states are essentially blind to crystal formation: core orbitals on neighbouring atoms do not overlap, all their hopping integrals vanish, and the corresponding bands are flat. The crystalline core states are then simply the Bloch sums (cf. Section 4.1) of the atomic core orbitals $\phi_c$,

\[\psi_{c\mathbf{k}}(\mathbf{r}) = \frac{1}{\sqrt N}\sum_{\mathbf{R}\in\mathcal{BL}} e^{i\mathbf{k}\cdot\mathbf{R}}\,\phi_c(\mathbf{r}-\mathbf{R}), \tag{5.1}\]

and — crucially — they are known eigenstates of the full crystal Hamiltonian,

\[\mathcal{H}\,|\psi_{c\mathbf{k}}\rangle = E_c\,|\psi_{c\mathbf{k}}\rangle,\]

with $E_c$ the corresponding atomic core level.

Orthogonalised plane waves. Herring (1940) proposed to expand the valence states not on the bare plane waves $w_{\mathbf{k}+\mathbf{g}}(\mathbf{r}) = e^{i(\mathbf{k}+\mathbf{g})\cdot\mathbf{r}}/\sqrt{V}$, but on plane waves projected onto the space orthogonal to the core. Introducing the core projector

\[P_{\mathbf k} = \sum_c |\psi_{c\mathbf{k}}\rangle\langle\psi_{c\mathbf{k}}|, \tag{5.2}\]

the orthogonalised plane wave is

\[|\chi_{\mathbf{k}+\mathbf{g}}\rangle = (1 - P_{\mathbf k})\,|w_{\mathbf{k}+\mathbf{g}}\rangle. \tag{5.3}\]

Each $\chi$ is smooth and plane-wave-like far from the nuclei, but near each nucleus the projection equips it with the rapid oscillations required by orthogonality to the core — exactly the feature that bare plane waves struggle to reproduce.

Secular equation. Expand a valence state on the OPWs, $|\psi\rangle = \sum_\mathbf{g} a_\mathbf{g} |\chi_{\mathbf{k}+\mathbf{g}}\rangle = (1-P_\mathbf{k})|\tilde\psi\rangle$, where $|\tilde\psi\rangle = \sum_\mathbf{g} a_\mathbf{g}|w_{\mathbf{k}+\mathbf{g}}\rangle$ is a smooth function. Inserting into $\mathcal{H}|\psi\rangle = E|\psi\rangle$ and using $\mathcal{H}P_\mathbf{k} = \sum_c E_c |\psi_{c\mathbf{k}}\rangle\langle\psi_{c\mathbf{k}}|$, a one-line manipulation yields a closed equation for the smooth part alone:

\[\Bigl[\mathcal{H} + \sum_c (E - E_c)\,|\psi_{c\mathbf{k}}\rangle\langle\psi_{c\mathbf{k}}|\Bigr]\,|\tilde\psi\rangle = E\,|\tilde\psi\rangle. \tag{5.4}\]

For valence energies $E > E_c$ the projector term is repulsive: localised at the nuclei, it raises the energy there and partially cancels the deep attractive potential hiding inside $\mathcal{H}$. The net effective potential felt by $\tilde\psi$ is therefore weak, and a few plane waves suffice where thousands were needed before.

Phillips–Kleinman construction. Reading the square bracket of Equation 5.4 as an effective Hamiltonian, the smooth part $\tilde\psi$ obeys a Schrödinger equation in which the true potential $V$ is replaced by

\[V_\text{PK} = V + \sum_c (E - E_c)\,|\psi_{c\mathbf{k}}\rangle\langle\psi_{c\mathbf{k}}|, \tag{5.5}\]

with the repulsive projector piece cancelling much of the attractive ionic well. This is the pseudopotential in its original form: not an approximation but an exact rewriting of the valence problem — at the price of an energy-dependent, non-local operator. The practical models of the next box trade this exactness for simplicity.

Pseudopotential

5.2.1 The pseudopotential concept

The Phillips–Kleinman rewriting of the OPW box suggests a bolder step: forget the exact (energy-dependent, non-local) $V_\text{PK}$ and directly model a weak, smooth effective potential $V_\text{ps}$ such that the pseudo-wavefunction $|\tilde{\psi}\rangle$ solving

\[\left[-\frac{\hbar^2}{2m}\nabla^2 + V_\text{ps}(\mathbf{r})\right]|\tilde{\psi}\rangle = E\,|\tilde{\psi}\rangle \tag{5.6}\]

reproduces the valence energies of the true problem (and the true wavefunction outside the core region). Two classic realisations:

Ashcroft empty-core. The crudest model: cancellation is taken as complete inside a core radius $r_c$,

\[V_\text{ps}(r) = \begin{cases} 0 & r < r_c \\ -\dfrac{Z e^2}{4\pi\varepsilon_0 r} & r > r_c, \end{cases} \tag{5.7}\]

with a single fitted parameter $r_c$ per element (plus electrostatic screening). Crude as it is, it captures simple metals remarkably well: the aluminium form factors used in Figure 5.5 come exactly from this construction.

Norm-conserving pseudopotentials (Hamann–Schlüter–Chiang, then Troullier–Martins). The modern, ab-initio recipe, built for each angular momentum $\ell$ from an all-electron atomic calculation by requiring that (i) the pseudo eigenvalue equals the all-electron valence eigenvalue; (ii) the pseudo-wavefunction is nodeless and coincides exactly with the all-electron one beyond a chosen cutoff radius; (iii) the charge enclosed within the cutoff is the same — norm conservation. The last condition is the clever one: it guarantees correct electrostatics and, through a classic identity, the correct energy-dependence of the scattering phase shifts — which is what makes the potential transferable from the atom it was built on to molecules and solids. These are the pseudopotentials underlying modern plane-wave DFT codes.

The empirical shortcut. Both constructions still aim at the full potential. The empirical route of the next section is humbler and mirrors semi-empirical tight binding: since only a handful of Fourier components $V(\mathbf{g})$ of the (screened) pseudopotential matter, one can skip their derivation altogether and fit those few numbers to experimental data.

The lesson deserves to be stated plainly. The bare crystal potential is deep and strongly oscillating, yet the effective potential felt by the valence electrons — once the core states have been projected out — is weak. This is why a handful of plane waves already suffices, and it is the belated justification of the nearly-free electron model of Section 3.2.2: the NFE picture is not a lucky starting guess but the correct leading description of valence electrons, provided the deep ionic core is replaced by its smooth pseudopotential. We now look at the two archetypes this enables — the tetrahedral semiconductors, and a simple metal.

5.3 The empirical pseudopotential method

In the empirical pseudopotential method (Cohen & Bergstresser, 1966), one takes the pseudopotential to be a local function, fully characterised by its Fourier components $V(\mathbf{g})$ at the reciprocal lattice vectors $\mathbf{g}$. Since $V_\text{ps}$ is weak, only a few Fourier components — at the shortest reciprocal lattice vectors — are needed. These are treated as adjustable parameters, fitted to experimental data (optical gaps, effective masses, reflectivity).

The philosophy is worth stating explicitly, because it is exactly the one already met in semi-empirical tight binding (Section 4.2). There, a chain of aggressive simplifications (two-centre approximation, nearest neighbours, …) reduced the band problem to an analytic model with a handful of Slater–Koster parameters, which are then fitted to data rather than computed; here, the theory of the previous section is used only to argue that a local, weak potential with a few Fourier components can do the job, and those few numbers are again fitted to experiments. In both cases the sins of the approximations are silently absorbed by the fit — and the payoff is a cheap, analytic model that interpolates reliably between the data points it was tuned on.

The central equation of band theory then becomes a finite matrix eigenvalue problem: at each $\mathbf{k}$, one diagonalises

\[H_{\mathbf{g},\mathbf{g}'} = \frac{\hbar^2|\mathbf{k}+\mathbf{g}|^2}{2m}\,\delta_{\mathbf{g}\mathbf{g}'} + V(\mathbf{g}-\mathbf{g}'), \tag{5.8}\]

truncated to $|\mathbf{g}|^2$ below a cutoff. With $$50 plane waves, the band structure of Si, Ge, GaAs, and many other semiconductors can be computed in excellent agreement with experiment.

5.3.1 Structure factor for diamond and zincblende

For a crystal with a multi-atom basis, the pseudopotential Fourier component is

\[V(\mathbf{g}) = \sum_\alpha v_\alpha(|\mathbf{g}|)\, e^{i\mathbf{g}\cdot\mathbf{d}_\alpha}, \tag{5.9}\]

where $v_\alpha$ is the atomic form factor of atom $\alpha$ at position $\mathbf{d}_\alpha$.

For the diamond structure (Si, Ge) with two identical atoms at $\mathbf{d}_1 = \mathbf{0}$ and $\mathbf{d}_2 = (a/4)(1,1,1)$:

\[V(\mathbf{g}) = v(|\mathbf{g}|)\,\bigl(1 + e^{i\mathbf{g}\cdot\boldsymbol{\tau}}\bigr) = v(|\mathbf{g}|)\,\left[2\cos\!\left(\frac{\pi}{4}(n_1+n_2+n_3)\right)\right] e^{i\mathbf{g}\cdot\boldsymbol{\tau}/2}, \tag{5.10}\]

where $\boldsymbol{\tau} = (a/4)(1,1,1)$ and $\mathbf{g} = (2\pi/a)(n_1,n_2,n_3)$ with $n_i$ all even or all odd (FCC selection rule). Setting $S \equiv n_1+n_2+n_3$:

Table 5.1: Diamond structure factor selection rules.

$S \bmod 4$	$\|\cos(\pi S/4)\|$	Structure factor	Examples ($\mathbf{g}^2$ in $(2\pi/a)^2$)
0	$1$	$2v$	0, 8, 16, 24
1	$1/\sqrt{2}$	$\sqrt{2}\,v$	3, 11, 19
2	$0$	$0$ (extinct)	4, 12, 20
3	$1/\sqrt{2}$	$\sqrt{2}\,v$	3, 11, 19

For zincblende (GaAs, InP) with two different atoms, the potential splits into symmetric and antisymmetric parts:

\[V(\mathbf{g}) = V_S(|\mathbf{g}|)\cos(\mathbf{g}\cdot\boldsymbol{\tau}/2) + i\,V_A(|\mathbf{g}|)\sin(\mathbf{g}\cdot\boldsymbol{\tau}/2), \tag{5.11}\]

where the origin is taken midway between the two atoms (atoms at $\pm\boldsymbol{\tau}/2$, the Cohen–Bergstresser convention), $V_S = (v_\text{cation} + v_\text{anion})/2$ and $V_A = (v_\text{cation} - v_\text{anion})/2$. For elemental semiconductors, $V_A = 0$.

5.3.2 Cohen–Bergstresser form factors

The standard form factors for the lowest reciprocal-lattice shells (in Ry) are:

Table 5.2: EPM form factors adapted from Cohen & Bergstresser (1966). Superscripts denote $|\mathbf{g}|^2$ in units of $(2\pi/a)^2$; these are exactly the values used by Figure 5.4. The only deviation from the published table is Ge $V_S^{8}$, set here to $0$ (C–B give $+0.01$): within the 65-plane-wave truncation used by the figure, the published value would displace the conduction-band minimum away from $L$.

Material	$a$ (Å)	$V_S^3$	$V_S^8$	$V_S^{11}$	$V_A^3$	$V_A^4$	$V_A^{11}$
Si	5.43	$-0.21$	$+0.04$	$+0.08$	$0$	$0$	$0$
Ge	5.66	$-0.23$	$0$	$+0.06$	$0$	$0$	$0$
GaAs	5.64	$-0.23$	$+0.01$	$+0.06$	$+0.07$	$+0.05$	$+0.01$
InAs	6.04	$-0.22$	$0$	$+0.05$	$+0.08$	$+0.05$	$+0.03$

Despite having only 3 adjustable parameters (for Si), the EPM reproduces the full band structure in remarkable agreement with experiment — a testament to how weak the pseudopotential truly is. For zincblende compounds, the antisymmetric form factors $V_A$ break the inversion symmetry of the diamond structure, lifting certain degeneracies and modifying the gap character.

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Figure 5.4: Empirical pseudopotential band structures of the Cohen–Bergstresser workhorses. Right: the bands along the standard FCC path L–Λ–Γ–Δ–X–(U,K)–Σ–Γ, with the filled valence bands overlaid in blue, the $k$-resolved forbidden region in light grey and the absolute gap $[E_v, E_c]$ bounded by dashed lines (energies precomputed with 65 plane waves, $|\mathbf{g}|^2 \le 16$, no spin–orbit). Left: select free (empty lattice), Si, Ge, GaAs or InAs — the bands morph smoothly between materials — and read the Cohen–Bergstresser form factors of the selection in the table; the 3D view below shows the conventional cubic cell of the selected crystal (diamond or zincblende, drag to rotate; the empty red frame for free electrons). Starting from the free-electron case, watch the pseudopotential open the gaps; note how Si and Ge are indirect (CBM on $\Delta$ and at $L$, respectively) while GaAs and InAs are direct at $\Gamma$.

5.4 Fermi energy and surface

Before looking at a real metal we need the vocabulary to describe how its bands are filled.

Definition 5.1: Fermi energy/surface

Given the fermionic nature of electrons, in the $T\to 0$ limit band states of a metal will be occupied up to a so-called Fermi energy $E_F$. In a metal this energy will surely cross one or more partially filled bands: the reciprocal space boundary between occupied and unoccupied states in the FBZ is called Fermi surface:

\[E_n(\mathbf{k})=E_F\]

where $n$ labels the band; in fact, metals can easily have multiple Fermi surfaces. $E_F$ should be regarded as an intrinsic property of a metal.

The Fermi surface is a rather simple concept in an ideal free electron system. It is simply a sphere with radius $k_F$ where $E_F=\hbar^2k_F^2/2m$. The simple addition of periodicity leads to very complex situations, where complexity however mostly derives from the simple geometry of folding — as the case study of the next section makes explicit.

5.5 Aluminium: the textbook nearly-free-electron metal

Aluminium (FCC, three valence electrons per cell) is the poster child of the justified-NFE picture: its pseudopotential is so weak that its bands are, to a good approximation, free-electron parabolas folded into the first Brillouin zone, with small gaps opening only where folded branches cross. Figure 5.5 builds exactly this construction and lets one read off the Fermi surface directly from the band dispersion: each branch that the Fermi level $E_F$ intersects along the high-symmetry path corresponds to a sheet of the Fermi surface. For one electron per cell (the $N_e=N$ snap) the surface is a single sphere inside the zone; by three electrons (aluminium) the first band is full, the second forms a large hole surface and the third opens small electron pockets — the classic multi-sheet Fermi surface of a polyvalent simple metal. Selecting calcium turns on a weak perturbation whose anticrossings separate the sheets more clearly, making the band$\leftrightarrow$surface correspondence explicit.

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Figure 5.5: Nearly-free-electron Fermi surfaces of an fcc lattice with the Ashcroft 1963 aluminium pseudopotential ($V_{111} \simeq 0.244$ eV, $V_{200} \simeq 0.765$ eV). Left: 3D Brillouin zone (truncated octahedron) with the iso-energy surface(s) of the visible band(s); the orange side of each shell is occupied, the light-blue side empty. Right: band-energy ranges as vertical bars, split orange/blue around the iso-energy line. Click on a bar to toggle the corresponding band on/off; drag the iso-energy slider (or the line on the bar plot) to scan the filling — the readout on the right reports the free-electron $N(E)$ in electrons per primitive cell, with snap-equivalents $N \approx 1, 2, 3$ corresponding qualitatively to monovalent, divalent (Ca-like) and trivalent (Al) metals.

5.6 Fermi surfaces of real metals

The machinery of this chapter is enough to compute credible band structures and Fermi surfaces for a small gallery of real metals. Figure 5.6 collects a few representative families:

the alkali metals Li, Na, K (BCC, one valence electron per cell): the half-filled band is nearly free-electron and the Fermi surface is an almost perfect sphere, comfortably inside the first Brillouin zone — the crystalline potential is barely felt;
calcium (FCC, two valence electrons per cell): the textbook divalent metal of Figure 5.5 — the free-electron sphere spills just beyond the zone boundary, and the weak anticrossing separates it into a first-band hole surface and a second-band electron surface;
aluminium (FCC, three electrons per cell): the sphere overflows the zone and folds into the two sheets already met in Section 5.5 — the second-band hole surface and the third-band “monster”;
the noble metals Cu, Ag, Au (FCC, eleven electrons per cell): the filled $d$ complex sits a few eV below $E_F$ and hybridises with the $s$ band; the Fermi surface is a sphere that bulges towards the hexagonal faces and opens necks at the $L$ points — an open Fermi surface, a notion explored in the folded-sphere exercise below;
one true transition metal, palladium (FCC, ten valence electrons): here $E_F$ falls inside the $d$ complex, just below its top — the $d$ bands are only partially filled and the Fermi surface fragments into multiple sheets (a $\Gamma$-centred electron sheet plus open hole sheets), the unmistakable signature of a transition metal.

Li, Na, K, Ca and Al are computed with a weak local pseudopotential (two representative form factors each, same method as Section 5.3; for Ca they are the same Animalu–Heine values used in Figure 5.5); Cu, Ag, Au and Pd with a minimal $s$+$d$ tight-binding model (Section 4.2) whose few parameters are tuned — in the true semi-empirical spirit — to reproduce the known position and width of the $d$ complex and the $L$-point levels; the result is semi-quantitative but topologically faithful. Pick an element in the periodic table, then drag the grey cursor in the band panel: the purple marker travels along the same $\Gamma$-path drawn inside the Brillouin zone on the left, and every time a band crosses $E_F$ (circled dots on the right) the marker pierces the corresponding Fermi sheet (matching dots on the path in 3D).

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Figure 5.6: Band structures and Fermi surfaces of simple metals. Top: periodic table — the highlighted elements are selectable. Left: first Brillouin zone with the Fermi-surface sheets, the $k$-path of the band plot (dark tube, spheres on the high-symmetry points) and the purple $k$ marker; coloured dots mark the crossings $E_n(\mathbf k)=E_F$. Right: bands along the path, referred to $E_F$ (dashed); the sheet-crossing bands carry the colour of their sheet. Drag the grey bar to move along the path; drag the 3D view to rotate.

Test your understanding: almost a folded sphere

Figure 5.5 illustrates how a free-electron spherical Fermi surface evolves when folded according to an FCC Bravais lattice, as a function of the number of electrons per unit cell $N_e/N$, where $N_e$ is the total number of electrons and $N$ the total number of cells. Here, we should recall that a band dispersion in the FBZ can host $2N$ electrons, where $2$ comes from spin degeneracy. A few limits can be easily reached thanks to a snapping in the interactive figure:

At one electron per cell, $N_e=N$, the first band is half filled and we automatically have a metal; in this limit the Fermi surface is just a simple spherical surface, completely inside the FBZ.
When we reach two electrons per cell, $N_e=2N$, the Fermi sphere does not fit inside the FBZ anymore: folding starts to appear close to the $L$ points and basically leads to the formation of a second Fermi surface.
While seeing the configuration of the previous point as two separate Fermi surfaces might seem artificial, it will be much less so after selecting “Ca” (Calcium, an FCC metal with two valence electrons per cell), which turns on a perturbation reproducing the bands in Ca: this leads to an energy anticrossing and to a much clearer separation between the different surfaces.

Stop for a moment now and consider that this is just the extension to a 3D FCC of the classic simple 1D folded parabola, plus a minor anticrossing at $k=\pm\pi/a$. Anything making the current picture more complicated comes only from geometry in 3D space; conceptually, we are depicting exactly the same situation.

If we continue to three electrons per cell, $N_e=3N$, we reach a situation which closely describes the electronic configuration of aluminium (an FCC metal with three valence electrons per cell). Now the sphere inflates even further: the first band gets completely full with no Fermi surface; the “lenses” at the $L$ points grow, merging into a very non-spherical Fermi surface; the third band starts to be filled, giving rise to an even less obvious Fermi surface, falling under the class of so-called “monsters”, due to their complicated geometry.

A few concluding remarks:

Typically one thinks about a Fermi surface as a closed surface, encircling a compact, simply connected region of the FBZ. This is not always the case and all the replicas in the repeated zone scheme might well “touch”, leading to a single surface with no closed boundary. This is called an open Fermi surface, with “necks” connecting the replicas. The transition between an open and closed energy surface is connected to the crossing of a van Hove singularity, as can also be seen in the cubium example of Section 4.4.
In the case of closed Fermi surfaces, one can distinguish between an electron surface enclosing occupied states (typically surrounding a band minimum, appearing as blue in Figure 5.5) and a hole surface enclosing empty states (typically surrounding a band maximum, appearing as orange in Figure 5.5).

In conclusion…

Take-home message

Plane waves converge slowly near nuclei; orthogonalising them to the core states (OPW) reveals that the effective valence potential is weak — the pseudopotential.
A weak pseudopotential justifies a posteriori the nearly-free electron model: simple metals like aluminium are free electrons folded into the zone, and the Fermi surface follows directly from the folded bands.
The empirical pseudopotential method turns this into a few-parameter recipe that reproduces the bands of Si, Ge and GaAs quantitatively.

\(S \bmod 4\)	\(\|\cos(\pi S/4)\|\)	Structure factor	Examples (\(\mathbf{g}^2\) in \((2\pi/a)^2\))
0	\(1\)	\(2v\)	0, 8, 16, 24
1	\(1/\sqrt{2}\)	\(\sqrt{2}\,v\)	3, 11, 19
2	\(0\)	\(0\) (extinct)	4, 12, 20
3	\(1/\sqrt{2}\)	\(\sqrt{2}\,v\)	3, 11, 19

Material	\(a\) (Å)	\(V_S^3\)	\(V_S^8\)	\(V_S^{11}\)	\(V_A^3\)	\(V_A^4\)	\(V_A^{11}\)
Si	5.43	\(-0.21\)	\(+0.04\)	\(+0.08\)	\(0\)	\(0\)	\(0\)
Ge	5.66	\(-0.23\)	\(0\)	\(+0.06\)	\(0\)	\(0\)	\(0\)
GaAs	5.64	\(-0.23\)	\(+0.01\)	\(+0.06\)	\(+0.07\)	\(+0.05\)	\(+0.01\)
InAs	6.04	\(-0.22\)	\(0\)	\(+0.05\)	\(+0.08\)	\(+0.05\)	\(+0.03\)