5 Bands outro

This is a conclusive wrap-up on the emergence and calculation energy bands in periodic systems. Fundamental assumptions and approaches will be quickly reviewed, but interested students are advised to proceed to more specialized textbook - a good starting point is Grosso and Pastori Parravicini (2014, ch. 5) - for further details. Various tools to build intuition on band dispersions and on their cross-sections are provided.

5.1 Key approximations and methods

The discussion so far rested on a few tacit assumptions, worth recalling before moving on:

Rigid lattice approximation

Ions are treated as fixed at their equilibrium positions, so that \(V(\mathbf{r})\) is time-independent. The justification is fundamentally based on the large ion–electron mass ratio, which in turns allow the adiabatic approach of Born–Oppenheimer approximation. Because the two dynamics live on such different time scales, the ion configuration will eventually enter the electronic problem typically as a fixed parameter, with its time variation as a perturbation. This is discussed in Chapter 12.

What is left out. Lattice vibrations (phonons) and the electron–phonon coupling, which can be included as a perturbation and controls scattering times and resistivity. Many other phenomena will be linked to the fact that ions are not really fixed, for instance thermal expansion and fusion, as well as superconductivity. Finally mechanical strain is also an interesting ingredient in materials, acting as a knob to engineer band structures and electronic states.

One-electron approximation

A crystal is a many-body system, solved here using a single-particle approach. This is most of the times a reasonable approximation, which can be justified at several levels:

even when not explicitly stated, \(V(\mathbf{r})\) is obviously generated by all the charges in the crystal, including the electrons themselves. In the simplest possible mean-field approach, \(V(\mathbf{r})\) is generated by the total charge density \(\rho(\mathbf{r})\), this is called Hartree approximation. Enforcing the antisymmetry required by fermionic statistics leads to the Hartree–Fock (HF) approximation, which ends up again in a single-particle problem, now with a new exchange term accounting for Pauli exclusion principle.
Genuine many-body correlations are still missing in HF. Modern band calculation methods typically rely on the density-functional theory (DFT), which recasts the full many-body problem into an auxiliary single-particle one, where interactions are carried by a functional of the electron density. Notably, this is in principle an exact reformulation, if the correct functional were known (it’s not known, but a variety of good approximations exist).
Ultimately, Landau’s Fermi-liquid theory explains why the single-electron picture survives at all, which is far from obvious: in short, energy and momentum conservation restrict the available scattering channels so severely that charged excitations close to \(E_F\) behave as long-lived quasiparticles, in one-to-one correspondence with the states of the corresponding non-interacting system.

What is left out. Sometimes many-body effect acquire a major role and qualitatively different ground states can emerge. Superconductivity is the textbook case where interactions simply cannot be dropped. Moving to even more exotic limits, fractional quantum Hall liquids and Luttinger liquids go further, and show that in a many-body system it may not even make sense to speak of individual electrons carrying individual charge and spin.

Non-relativistic approximation

Bands are obtained from the Schrödinger, not the Dirac, equation. Relativistic corrections, in particular spin–orbit coupling, have been ignored here. Clearly, in any serious band calculation, especially in the presence of heavy (large \(Z\)) elements. This is for instance respondible for the spli-off valence band in cubic semiconductors such as Si and Ge.

What is left out. Spin has always been considered a simple two-fold degeneracy, an independent degree of freedom. Sometimes this is too brutal and orbital and spin degree of freedom can be strongly intertwined, as is the case, for instance, of the Rashba and Dresselhaus effects. In addition, also our discussion on groups ignored spin: this leads to double groups, interested students are redirected to more specialistic books such as Yu and Cardona (2010) for further details.

We recall here the main families of methods, which mainly differ in the basis chosen to describe the crystal states, or in the way the problem is reformulated altogether.

Tight binding / LCAO

This method has already been discussed in detail in Chapter 4, and will be skipped here. The idea is to start from localized atomic orbitals and build band states using Bloch sums.

Plane-wave expansions

Here, one starts from a delocalized plane wave basis, basically building on the approach already introduced in 1D with the NFE model of Section 3.2.2, with the only difference that a full diagonalization is performed and the formalism is extended to 3D. While some metals (e.g. Al) can be described by models strikingly close to a free electron system, a few subtle steps should be analyzed to clarify how this is even possible in real crystals: this is addressed in broad terms in Section 5.2.

Further approaches

A number of alternative methods exist, many of which have only an historical value today, refer to Grosso and Pastori Parravicini (2014) for an exhaustive review. One interesting concept is the one of cellular methods, which follows a logic similar to the one we adpted in Section 3.2.1 with the transfer matrix formalism:

Cell description: electron states are described in the single cell, at a given trial energy \(E\);
Boundary: \(\mathbf{k}\)-dependent phase matching conditions prescribed for Bloch states;
Solution: combining the previous two, a band dispersion \(E(\mathbf{k})\) can be extracted.

This typically leads to a transcendental relations, such as the final

\[F(qa) = \cos(ka)\,\qquad E=\frac{\hbar^2q^2}{2m}\]

of Equation 3.14. Cellular methods were introduced in 1933 by Wigner and Seitz and today basically have just a historical value. The extension to 3D case brings significant complexity and approximations: the original method was based on an \(E\)-dependent spherical basis which, combined to the cell polyhedron and limited available computational tools, allowed a phase matching only on a finite set of points at the cell edge. Green-function (KKR) methods restate the same idea in scattering language, which is even closer to our transfer matrix exercise: each cell is characterized by how it scatters at energy \(E\), and bands appear where the multiply scattered waves interfere constructively under the Bloch condition. While the cellular method proper is basically obsolete, some of its descendants are still used, for instance in the description of disordered alloys. These go beyond the scope of these notes.

The single-particle approach is probably the hardest to cope with, thus the next optional sections provide a few further clues. These are not intended to be, by any mean, exhaustive: the interested student is strongly suggested to shift to specialized textbooks and courses for further details. An possible starting point is again Grosso and Pastori Parravicini (2014).

Hartree and Hartree–Fock approximations

Both approximations are based on a variational approach: the initial all-electron Hamiltonian is exact

\[\mathcal{H}_{all-e} = \sum_{i}\left[-\frac{\hbar^2}{2m}\nabla_i^2 + V_\text{ion}(\mathbf{r}_i)\right] + \frac{1}{2}\sum_{i\neq j}\frac{e^2}{4\pi\varepsilon_0|\mathbf{r}_i-\mathbf{r}_j|}, \tag{5.1}\]

as well as disastrously unmanageable: the wavefunction of the \(N\) electrons \(\Psi(\mathbf{r}_1,\cdots,\mathbf{r}_N)\) lives in \(3N\) dimensions. In a variational approach, instead of figuring out eigenvalues, we look for the best ground state approximation as the one minimizing

\[E[\Psi] = \frac{\langle\Psi|\mathcal{H}|\Psi\rangle}{\langle\Psi|\Psi\rangle} \;\geq\; E_0,\]

but we do so using a specific family of \(\Psi\) functions: this choice will be the core of the approximation.

Hartree approximation. The simplest option is to ignore correlations between particles altogether, and just restrict the all-electron wavefunction to a simple product

\[\Psi_H(\mathbf{r}_1\ldots\mathbf{r}_N) = \varphi_1(\mathbf{r}_1)\,\varphi_2(\mathbf{r}_2)\cdots\varphi_N(\mathbf{r}_N).\]

The energy is a functional of the \(N\) orbitals and making it stationary with respect to \(\varphi_i^*\), with the orthonormality \(\langle\varphi_i|\varphi_j\rangle=\delta_{ij}\) enforced by Lagrange multipliers \(\varepsilon_i\), gives

\[\left[-\frac{\hbar^2}{2m}\nabla^2 + V_\text{ion}(\mathbf{r}) + V_H(\mathbf{r})\right]\varphi_i = \varepsilon_i\varphi_i,\]

i.e. a single-particle equation per orbital, each electron moving in the mean field

\[V_H(\mathbf{r}) = \int d^3r'\,\frac{\sum_{j\neq i}e^2|\varphi_j(\mathbf{r}')|^2}{4\pi\varepsilon_0|\mathbf{r}-\mathbf{r}'|} \approx \int d^3r'\,\frac{e^2\,n(\mathbf{r}')}{4\pi\varepsilon_0|\mathbf{r}-\mathbf{r}'|}, \qquad n(\mathbf{r}) = \sum_j |\varphi_j(\mathbf{r})|^2,\]

where \(n\) is the particle density, not a charge density: the two charges of the interacting pair are already accounted for by the \(e^2\). The approximation consists in extending the sum to \(j=i\), so that each electron also feels its own charge — a self-interaction error, which the exchange term of Hartree–Fock will cancel exactly. Note here that since \(n\) is built from the very orbitals one is solving for, the equations are non-linear and must be iterated until self-consistency is reached.

Hartree–Fock approximation. The minimalistic correction to Hartree consists in making the wavefunction antisymmetric, so that it at least satisfies statistical correlations. We now need a Slater determinant of \(N\) spin-orbitals,

\[\Psi_{HF} = \frac{1}{\sqrt{N!}}\,\det\big[\varphi_i(\mathbf{x}_j)\big],\]

where \(\mathbf{x} = (\mathbf{r},\sigma)\) describes both position and spin. Here we can expand \(\langle\Psi_{HF}|\mathcal{H}|\Psi_{HF}\rangle\) as

\[E[\{\varphi\}] = \sum_i \langle\varphi_i|\,T+V_\text{ion}|\varphi_i\rangle + \frac{1}{2}\sum_{i,j}\Big(J_{ij} - \delta_{\sigma_i\sigma_j}K_{ij}\Big), \tag{5.2}\]

with the direct (Coulomb) and exchange integrals

\[\begin{aligned} J_{ij} &= \!\iint\! d^3r\,d^3r'\,\frac{e^2\,|\varphi_i(\mathbf{r})|^2|\varphi_j(\mathbf{r}')|^2}{4\pi\varepsilon_0|\mathbf{r}-\mathbf{r}'|},\\ K_{ij} &= \!\iint\! d^3r\,d^3r'\,\frac{e^2\,\varphi_i^*(\mathbf{r})\varphi_j^*(\mathbf{r}')\varphi_j(\mathbf{r})\varphi_i(\mathbf{r}')}{4\pi\varepsilon_0|\mathbf{r}-\mathbf{r}'|}. \end{aligned}\]

The exchange term is the new entry caused by the antisymmetry: for equal spins it lowers the energy, since two electrons with the same spin cannot reside in the same place/state, thus the reduced electrostatic cost. Its diagonal part settles the debt left open by Hartree: \(J_{ii} - K_{ii} = 0\), so the self-interaction cancels exactly. Minimising Equation 5.2 under orthonormality gives the Hartree–Fock equations,

\[\begin{aligned} \varepsilon_i\,\varphi_i(\mathbf{r}) &=\left[-\frac{\hbar^2}{2m}\nabla^2 + V_\text{ion}(\mathbf{r}) + V_H(\mathbf{r})\right]\varphi_i(\mathbf{r})\\ &- \sum_{j}^{\,\sigma_j=\sigma_i} \int\!d^3r'\,\frac{e^2\varphi_j^*(\mathbf{r}')\varphi_i(\mathbf{r}')}{4\pi\varepsilon_0|\mathbf{r}-\mathbf{r}'|}\,\,\varphi_j(\mathbf{r}). \end{aligned} \tag{5.3}\]

Again a non-linear equation, again self-consistently, now also non-local. What is still missing is correlation, and the omission is not innocent: in the homogeneous electron gas bare exchange makes \(d\varepsilon/dk\) diverge at \(k_F\), so that the density of states vanishes at \(E_F\), and in solids the gaps come out overestimated by several eV — the opposite error of local density functionals.

5.2 About plane-wave expansion

The end of this section reviews the empirical pseudopotential method, which — de facto — simply extends to 3D and to full diagonalization our simple NFE model built on a plane wave basis. Only a very few Fourier components are used to describe the periodic potential in fairly complex crystals such as Si, Ge or GaAs. Indeed, also many metals - for instance Al - are presented as being minor deviations from a free electron dispersion. In the following we want to sketch the argument on why and in what sense can this actually even make sense.

5.2.1 Step 1 - Plane waves are problematic

Electrons in Al or Si are far from being free and far from being easy to expand in plane waves: they are subject to the divergent \(1/r\) potential of the ion cores; some of them reside in deep orbitals that are tightly bound to the nucleus; some more reside in much more delocalized bands responsible for chemical bonds or metallic conduction. This situation is summarized in the qualitative sketch of Figure 5.1, depicting a surely not desirable starting point for plane wave expansion: length scales of core and valence orbitals can easily span two orders of magnitude, plus a \(1/r\) potential is surely not smooth nor it contains only a few Fourier components. As a consequence, an enormous plane wave basis is apparently necessary to address the problem; this was defined by C. Herring as a numerical catastrophe.

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Figure 5.1: Comparison between energy and length scales of various orbitals in Si. Top: two Si atoms at the bond distance in the silicon crystal, with the spheres containing 90% of the \(1s\), \(2s\) and \(3s\) density (Clementi and Raimondi (1963)). Bottom: from the point of view of an electron in a 3s orbital, the bare ion potential \(-Ze^2/r\) is largely screened by the electrons in the first two shells; in addition, valence electrons have to be in a state which is orthogonal to such core states. The key question here is: can all this be described in terms of an effective potential only acting on the valence electron?

How catastrophic is Herring’s catastrophe?

We consider the problem originally discussed by Herring, i.e. silicon. The two length scales at stake are drawn one against the other in Figure 5.1. The \(3s/3p\) valence states of Si extend over the bond length (\(\approx 2.35\,\)Å), but the same states near each nucleus must oscillate rapidly to stay orthogonal to the deep \(1s\), \(2s\), \(2p\) core states, with a characteristic size \(a_0/Z\) (\(\approx 0.04\,\)Å for \(Z=14\)). These two length scales differ about two orders of magnitude, thus clearly many more than \(100^3=10^6\), millions Fourier components are required to properly describe the Bloch states, meaning the matrix one should diagonalize has a size of the order of \(10^6\times 10^6\).

This is a very hard limit: any cutoff makes the mapping of the core electrons unreliable and, as a consequence, the full calculation unreliable; very short wavelengths are also unavoidable if one wants do describe structures at the scale of the Si-Si bond, which is obviously very important.

5.2.2 Step 2 - Orthogonalization

The problem of the divergent number of Fourier components is particularly disappointing: exceptional computational resources are needed to describe core electrons… but core electrons are the least interesting part of the crystal, and we just need them because the more important valence electrons have to be orthogonal to the core. Moreover, core-levels are so deep in energy and they experience so little hopping and mixing with anything else… that they must be almost exact Bloch sums of the (known) core orbitals.

Building on this knowledge, it is possible to expand the valence electron wavefunction using plane waves that are preliminarily made orthogonal to core bands, these are called Orthogonalized Plane Waves (OPW). Since the request to describe core electrons drops, no catastrophically large Fourier expansion is anymore needed. The resulting eigenvalue equation includes a repulsive potential term that basically makes it more energetically expensive for valence electron to problem the core and its divergent potential.

Formal OPW approach

Core states are essentially blind to crystal formation: core orbitals on neighboring atoms do not overlap, hopping integrals vanish, and the corresponding bands are flat. As such they can be calculated for a single Si atom, using conventional techniques. The crystalline core states are then simply Bloch sums (Section 4.1), thus they are known eigenstates

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

with \(E_c\) corresponding to the atomic core level.

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

\[|\chi_{\mathbf{k}+\mathbf{g}}\rangle = (1 - \mathbb{P}_{\mathbf k})\,|w_{\mathbf{k}+\mathbf{g}}\rangle. \tag{5.5}\]

where the core projector

\[\mathbb{P}_{\mathbf k} = \sum_c |\psi_{c\mathbf{k}}\rangle\langle\psi_{c\mathbf{k}}|, \tag{5.6}\]

was introduced. Each \(\chi(\mathbf{r})\) is smooth and plane-wave-like far from the nuclei, but near each nucleus the projection makes it rapidly oscillate as required by orthogonality to the core: this is exactly the feature that bare plane waves struggle to reproduce.

Secular equation. We now expand a valence state on the OPWs

\[|\psi\rangle = \sum_\mathbf{g} c_\mathbf{g} |\chi_{\mathbf{k}+\mathbf{g}}\rangle = (1-\mathbb{P}_\mathbf{k})|\tilde\psi\rangle,\]

where \(|\tilde\psi\rangle = \sum_\mathbf{g} a_\mathbf{g}|w_{\mathbf{k}+\mathbf{g}}\rangle\) is a smooth function. Inserting this expansion into \(\mathcal{H}|\psi\rangle = E|\psi\rangle\) and using \(\mathcal{H}\mathbb{P}_\mathbf{k} = \sum_c E_c |\psi_{c\mathbf{k}}\rangle\langle\psi_{c\mathbf{k}}|\) we obtain

\[\Bigl[\mathcal{H} + \sum_c (E - E_c)\,|\psi_{c\mathbf{k}}\rangle\langle\psi_{c\mathbf{k}}|\Bigr]\,|\tilde\psi\rangle = E\,|\tilde\psi\rangle. \tag{5.7}\]

Here obviously \(E > E_c\) thus the new projector term is repulsive and localized close to the nucleus: it raises the energy of any particle close to the core and partially cancels the deep attractive potential hiding inside \(\mathcal{H}\). Note this is an interesting diagonalization problem where the matrix itself is energy-dependent: it thus requires peculiar solution strategies, going beyond the scope of these notes.

5.2.3 Step 3 - Effective potential

We now move to the bold question: can we reproduce the valence electron behavior using an effective potential in place of the repulsive term in the OPW formalism, while completely removing core electrons from the equation? This idea goes under the general name pseudopotential (PP) method. It should be noted that outside the core such a potential is fundamentally known: valence electrons experience the electrostatic potential of a screened ion. In the region inside the core the situation is far less obvious, but this only accounts for a minor portion of the crystal volume, thus the overall target is less hopeless than it seems. Over time, many methods have been created to “complete” the screened potential in the core region, all of them leading to a removal of the \(1/r\) divergence.

Pseudopotential approach

Let us start from evaluation the relative weight of the cores, in terms of volumes. The 10-electron \(1s^22s^22p^6\) core of Si has no well-defined size but \(90\%\) of the charge can be estimated to fall within \(\approx 0.45\,\)Å from the Si nucleus (Clementi and Raimondi (1963) and Bunge et al. (1993)). This leads to a core volume \(\approx0.4\,\)Å\(^3\). This should be compared with the volume per atom in the Si crystal (8 atoms per \(5.43^3\,\)Å\(^3\) unit cell) \(\approx20\,\)Å\(^3\). Building on these facts, the key idea is rather bold: forget the energy-dependent and non-local term \(\sum_c (E - E_c)\,|\psi_{c\mathbf{k}}\rangle\langle\psi_{c\mathbf{k}}|\) of the OPW formalism and directly model a weak, smooth effective potential \(V_\text{ps}(\mathbf{r})\) such that the pseudo-wavefunction \(|\tilde{\psi}\rangle\) solving

\[\left[-\frac{\hbar^2}{2m}\nabla^2 + V_\text{ps}(\mathbf{r})\right]|\tilde{\psi}\rangle = E\,|\tilde{\psi}\rangle \tag{5.8}\]

reproduces the properties of valence electrons in the true problem. In the process, core electrons are removed from the equation, with an unavoidable impact on \(\tilde{\psi}\) (for instance, we expect many less nodes). The key properties we are interested in are first of all the orbital energies, but also the resulting charge density. We recall here two classic realisations for \(V_\text{ps}(\mathbf{r})\)

Flat core. The crudest model consists in assuming the cancellation is complete inside a core radius \(r_c\), while the \(Z_c\) electrons in the core are screened outside \(r_c\). A more general options includes also a constant

\[ V_\text{ps}(r) = \begin{cases} A & r < r_c \\ -\dfrac{(Z-Z_c)e^2}{4\pi\varepsilon_0 r} & r > r_c, \end{cases} \tag{5.9}\]

with fitted parameters \(A\) and \(r_c\) per element (plus electrostatic screening). Crude as it is, it captures simple metals remarkably well: the aluminum form factors used in Figure 5.6 come exactly from this construction.

Norm-conserving pseudopotentials. More modern approach require: (i) the pseudo eigenvalue should equal the all-electron valence eigenvalue; (ii) the pseudo-wavefunction coincides exactly with the all-electron one beyond a chosen cutoff radius \(r_c\). The second requirement is crucial to preserve charge distribution in the crystal, which is of course important for electrostatic reasons. The procedure is a bit different: one parameterises is the pseudo-wavefunction, which is smooth and nodeless inside \(r_c\) and identical to the all-electron one outside \(r_c\), and the potential is then obtained by inverting the radial Schrödinger equation. Given the angular-momentum dependence of the wavefunctions, the pseudopotential will also depend on the angular momentum. These are also the pseudopotentials underlying modern DFT codes.

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Figure 5.2: Left: Ashcroft’s empty core, with one single fit parameter \(r_c\). Right: a non-local pseudopotential, one channel per angular momentum: the cancellation is strong for \(l=0\), weaker for \(l=1\) and even more for \(l=2\). See Grosso and Pastori Parravicini (2014) for further details.

5.3 Empirical pseudopotentials

As in the case of empirical TB, empirical pseudopotentials aim at producing a simple analytical model with a handful of parameters obtained by data fitting. In this sense, the whole argument we made on plane waves only serves to justify the somewhat surprising fact that only a few Fourier components are sufficient to reproduce real bands. As argued in the following, in symmetric crystal the number of parameters can be truely small.

We do not need to reformulate the plane-wave eigenvalue problem, since it is identical to the one already presented in Section 3.2.2. The key point is what is the relevant Fourier expansion of the potential \(V_\text{ps}(\mathbf{r})\): in highly symmetric FCC crystal their number can be rather small. The Fourier components of an FCC-periodic function correspond to reciprocal vectors \(\mathbf{g}\) belonging to the BCC dual lattice. The harmonics are classified in terms of the normalized square modulus \(|\mathbf{g}a/2\pi|^2\):

The first \(8\) harmonics at \(\mathbf{g}=(\pm 1,\pm 1, \pm 1)\cdot(2\pi/a)\), marked as amplitude \(V(3)\);
The next \(6\) harmonics a \(\mathbf{g}=(\pm 2,0,0)\cdot(2\pi/a)\) and cyclic permutations, marked as \(V(4)\);
Then we have \(12\) at \(\mathbf{g}=(\pm 2,\pm 2,0)\cdot(2\pi/a)\) and cyclic permutations, marked as \(V(8)\);
Then we have \(24\) at \(\mathbf{g}=(\pm 3,\pm 1,\pm 1)\cdot(2\pi/a)\) and cyclic permutations, marked as \(V(11)\).

In diamond crystals, the transform of the two sublattices at \(\mathbf{d}_1\) and \(\mathbf{d}_2\) give rise to interference and lead to a cancellation of \(V(4)\) (optionally check Definition 2.5 and the worked examples that follow it): only three real coefficients \(V(3)\), \(V(8)\) and \(V(11)\) remain, covering the first \(50\) (!) harmonic terms of \(V_\text{ps}(\mathbf{r})\). Moving to zinc-blende crystals, the situation is marginally more complicated since the two sublattices are unequal and six parameters are necessary: three real coefficient describing the average potential \(V_S(3)\), \(V_S(8)\) and \(V_S(11)\), with \(S\) standing for symmetric; three more describing the difference between the sublattices \(V_A(3)\), \(V_A(4)\) and \(V_A(11)\), with \(A\) standing for anti-symmetric and \(V_A(8)\) again canceled by destructive interference.

The resulting diagonalization for Si, Ge, GaAs and InAs is reported in Figure 5.3.

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Figure 5.3: Empirical pseudopotential band structures. 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 gray and the absolute gap \([E_v, E_c]\) bounded by dashed lines. Left: select free (empty lattice), Si, Ge, GaAs or InAs to activate the correct Fourier components. Note how Si and Ge are indirect (CBM on \(\Delta\) and at \(L\), respectively) while GaAs and InAs are direct at \(\Gamma\).

In the band dispersions, occupied bands are highlighted in blue: as discussed in the next chapters, band filling will be crucial for the conductive or insulating nature of a crystal and can be determined by comparing the number of available electrons with the available bands. For instance, in silicon every primitive cell contains two Si atoms, each one with electronic configuration \(1s^22s^22p^63s^23p^2\) and thus \(4\) valence electrons. The resulting \(8\) valence electrons per primitive cell fill \(4\) bands, meaning there is an finite energy gap between the last occupied electron state and the first empty one; the case of Ge is completely similar. In the case of GaAs and InAs, the atoms in the two sublattices have \(5\) and \(3\) valence electrons, but the total is still \(8\) electrons per primitive cell and \(4\) bands are filled: gapped again.

5.4 Fermi energy and surface

In previous pseudopotential examples, the crystal were gapped, but in many other cases one or more band is partially filled. Any crystal with at least one partially filled band will be called a metal for reasons that will become obvious in the next chapter.

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 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 \(E_F=\hbar^2k_F^2/2m\). When a single band is partially filled, a quite similar situation can emerge in crystals as well, but generally speaking Fermi surfaces can give rise to counterintuitive situations:

a metal can have multiple Fermi surfaces, emerging from one or more partially filled bands;
a Fermi surfaces can well be open and extend over multiple Brillouin zones
a closed surface Fermi surface might enclose empty states and not occupied ones.

All these options can be explored in the optional sections in the final part of the chapter.

In conclusion…

Take-home message

Key approximations: what are the key approximations on which band theory is based? Qualitative understanding of the approximations, of their origin and of what
Pseudopotential: why is a direct application of plane-wave expansion hard on a real crystal? How does this lead to the pseudopotential approach? Empirical pseudopotential description of cubic semiconductors
Fermi surfaces: basic definition.

As in the previous chapter, the material that follows is not compulsory but you are strongly advised to go through it as a way to verify your understanding of the course subject, and to build an intuition on band dispersions, Brillouin zone cross-sections and Fermi surfaces.

Test your understanding - 2D toy model

In the following we explore a simple model on a square lattice: we start from folding in 2D and see how a fairly arbitrary potential impacts the dispersions. The band structure is traced by diagonalising at each \(\mathbf{k}\) along \(\Gamma\)–\(M\)–\(X\)–\(\Gamma\). The natural starting point is \(V_0 = 0\) where the “band structure” is nothing but a folded free-electron paraboloid. Folding in 2D is significantly less obvious than in 1D and can be explored in the interactive Figure 5.4.

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Figure 5.4: Zone folding of the free-electron parabola on the square lattice. This is a trivial extension of the 1D case, but significant complication come from geometry.

Figure 5.5 allows turning on a generic gaussian confinement potential of amplitude \(V_0\) and tunable symmetry. The following exploration path is suggested:

Start from \(V_0 = 0\), bands correspond to the folded free-electron spectrum of Figure 5.4.
Turn on \(V_0\) with the default rotationally symmetric potential and appreciate how the potential lifts most degeneracies and opens gaps at the zone boundaries.
Note how the bottom band is very close to the TB results of the single-orbital squarium, both in terms of the shape of the band dispersion and of wavefunction configuration as a function of \(\mathbf{k}\).
Note how the second and third band remain degenerate at the \(M\) point \((1,1)\cdot\pi/a\): the two eigenfunctions are clearly mapped one onto the other by \(\pi/2\) rotation, thus degeneracy come from an irreducible representation of the rotational symmetry of the crystal.
Test this by stretching the gaussian potential in the \(y\) direction: degeneracy is lifted.
Note how the second and third band still cross along the MX direction: this is clearly due to a different issue, bands have a well-defined \(xy\) parity, thus they are not mixed by the hamiltonian
Test this by breaking the rectangular symmetry of the potential: residual degeneracy is also lifted.

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Figure 5.5: Band dispersions of a single electron in a square crystal cell: check out how the potential \(V(x,y)=V_0\exp\left[-\mathbf{r}\cdot(\sigma^{-2})\mathbf{r}/2\right]\) in the single cell lifts degeneracies of the folded parabola.

Test your understanding — folding a Fermi sphere

It is instructive to consider what happens when a Fermi sphere grows beyond the boundary of the Brillouin zone, in the case of a free-electron system. Here we assume an FCC crystal: as long as there is one electron per cell \(N_e/N=1\) (\(N_e\) electrons, \(N\) cells), all the electrons can fit in the first band, filling it by half (recall a band holds \(2N\) electrons, the \(2\) from spin). When \(N_e/N\) grows beyond 1, the sphere grows beyond the first Brillouin zone: ideally at \(N_e/N\) all electrons can fit in the first band, but given band overlaps in a folded parabola in 3D, the result will be partial filling of multiple bands, and multiple Fermi surface.

Figure 5.6 allows tracking this evolution up to \(N_e/N\) beyond 3. The selection on top allows moving away from the free-electron limit and implement the pseudopotentials corresponding to the Al and Ca crystals, which are both FCC lattices with a trivial one-atom basis. While the resulting Fermi surfaces can appear as fairly complicated, they are very close to a simple folded sphere.

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Figure 5.6: Nearly-free-electron Fermi surfaces of an fcc lattice with the Ashcroft 1963 aluminum pseudopotential (\(V_{111} \simeq 0.244\) eV, \(V_{200} \simeq 0.765\) eV). Left: Brillouin zone and Fermi surfaces; the orange side of each shell is occupied, the light-blue side empty. Right: band-energy ranges with occupied (orange) and empty (blue) ranges, the Fermi surface can be dragged. The corresponding \(N_e/N\) value is indicated in the bottom-right corner of the figure. Based on the number of valence electrons, the value for Al and Ca are \(3\) and \(2\), respectively.

Test your understanding — cross-sections and Fermi surfaces

Figure 5.7 reports a small gallery of band dispersions and Fermi surfaces for selected real metals. In particular:

alkali metals Li, Na, K, Rb, Cs (BCC, monovalent): a nearly perfect Fermi sphere, well inside the zone;
calcium (FCC, divalent): as shown in Figure 5.6, the sphere spills just past the boundary;
aluminum (FCC, trivalent): the sphere overflows into a second-band hole surface and a third-band “monster” of small pockets;
noble metals Cu, Ag, Au (FCC): the filled \(d\) complex sits a few eV below \(E_F\); the sphere bulges towards the hexagonal faces and opens necks at \(L\) — an open surface;
palladium (FCC transition metal): \(E_F\) falls inside the partially filled \(d\) complex, fragmenting the surface into multiple sheets.

Notes on the calculations. The alkalis, Ca and Al use a weak local pseudopotential (two form factors each, as in Section 5.3; the Animalu–Heine values of Figure 5.6 for Ca); the noble metals and Pd a minimal \(s\)+\(d\) tight-binding model (Section 4.2) tuned to the known \(d\)-complex position and \(L\)-point levels — semi-quantitative but topologically faithful.

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Figure 5.7: 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; colored dots mark the crossings \(E_n(\mathbf k)=E_F\). For the FCC metals the final leg runs straight from \(\Gamma\) through \(K\) to the \(X\) point of the neighboring zone (dashed alignment line; the gray square is the facing square face of that zone): by periodicity the bands of the last column close exactly on those of the \(X\) column. Right: bands along the path, referred to \(E_F\) (dashed); the sheet-crossing bands carry the color of their sheet. At the cursor’s \(k\) every band carries a dot: filled if the band is occupied there (\(E<E_F\)), open if empty.