Lattices, Reciprocal Space, and Bloch Electrons

A crystal repeats a motif in space. This repetition organizes its electronic states into Bloch waves and makes reciprocal space a natural language for band energies, diffraction and transport. The chapter develops that language before introducing two complementary band constructions: a weak periodic potential and hopping between localized orbitals.

Helpful background. Vectors, Fourier phases and quantum eigenstates are enough to begin. The condensed-matter roadmap and mathematical background review these ideas. Conventions collects the normalization and sign choices used in calculations.

From repeated structure to Bloch states

Start with crystals and lattices. The lattice specifies translations; the motif specifies what is repeated. A primitive cell contains one copy of that motif, although a conventional cell may contain several.

The reciprocal lattice contains vectors $\mathbf G$ whose phase $e^{i\mathbf G\cdot\mathbf R}$ is one for every lattice translation $\mathbf R$. Wavevectors differing by $\mathbf G$ consequently have the same translation phases. A Brillouin zone selects one representative of each such class. Its faces are identified boundaries in momentum space, not physical walls surrounding the electrons.

Bloch’s theorem then organizes a periodic one-particle eigenproblem into sectors labeled by $\mathbf k$. The theorem does not require a weak potential. It reorganizes the stated periodic operator exactly; an approximation enters when the operator itself is an effective model or its basis is truncated. For interacting particles, simultaneous translation gives a total crystal-momentum label, but does not by itself produce independent single-electron bands.

An open finite sample generally lacks exact bulk translation symmetry. A slab can retain translations parallel to its surface, and a periodic finite sample has a discrete momentum grid. Keeping these cases separate prevents a bulk band plot from being mistaken for the complete spectrum of a finite device.

Two ways to construct a band

Starting physical picture	Calculation	Useful question
Almost free electrons coupled by weak Fourier components of a periodic potential	Nearly free electrons	How does Bragg scattering split a free-electron degeneracy?
Localized orbitals coupled by hopping amplitudes	Tight-binding models	How do orbital energies and neighboring bonds determine the dispersion?

These descriptions are complementary. A complete plane-wave basis and a complete localized basis can describe the same operator. Practical calculations keep finite subsets, so their errors depend on the physics being retained. Tight binding is not a required preliminary step for every density-of-states calculation, and nearly-free-electron theory is not a proof of Bloch’s theorem.

Two further pages develop information beyond the eigenvalues. Wannier functions connect a chosen band subspace to localized states. Symmetry of Bloch states explains symmetry labels, protected degeneracies and the restrictions that symmetries impose on a Bloch Hamiltonian.

What can a dispersion tell us?

The density of states counts available states near an energy. It requires an integral over the full Brillouin zone with the correct spin, band and volume factors. A line through a few symmetry points does not supply that integral.

The Fermi surface is the relevant zero-temperature energy level set of a metal once its filling or chemical potential is fixed. In an interacting system, the relation to quasiparticle poles and spectral weight needs additional care. A useful band velocity is

$$\mathbf v_n(\mathbf k)=\frac{1}{\hbar}\nabla_{\mathbf k}\varepsilon_n(\mathbf k).$$

Its direction is normal to a smooth constant-energy surface. It need not be parallel to $\mathbf k$: crystal momentum is not generally mechanical momentum.

A dispersion alone does not determine an optical intensity, scattering rate or conductivity. Those also require the appropriate matrix elements, occupations, interactions or relaxation processes. Continue to band theory and electronic structure for material band interpretation, or lattice vibrations when the moving degrees of freedom are ions.

Example: what two honeycomb bands mean

A honeycomb network has a triangular Bravais lattice and two motif sites per primitive cell. If one spinless orbital is retained on each site, its Bloch Hamiltonian is a $2\times2$ matrix and has two bands. Adding spin without spin-dependent terms doubles the state count and makes each band spin degenerate.

Nearest-neighbor hopping can produce crossings at the inequivalent zone corners. This establishes a property of that orbital model. To place the Fermi level at the crossing, the filling must be specified. To interpret a perturbation as opening a topological gap, its symmetry and the wavefunctions across the Brillouin zone must also be studied. Merely redrawing the same lattice with a larger cell folds bands into a smaller zone; it does not open a physical gap.

Exercises

1. Count states before filling them. A periodic honeycomb sample has $N_c$ primitive cells and one spinful orbital on each of its two sites. How many allowed momenta, spatial bands and single-electron states does the model have? How many electrons give half filling?

Solution

There are $N_c$ momenta and two spatial bands. Each has two spin states at each momentum, giving $4N_c$ single-electron states. Half filling contains $2N_c$ electrons, or one electron per site. Spin-dependent terms can remove spin degeneracy without changing the number of retained states.

2. A gap on a plotted path. Two bands never touch along a displayed high-symmetry path. What further information is needed to establish a band-insulating ground state at a specified integer filling?

Solution

One must check the entire Brillouin zone and compare the highest occupied energy with the lowest unoccupied energy, which may occur at different momenta. A direct separation at each plotted point does not exclude an off-path crossing or an indirect overlap. The state count and filling must place the chemical potential in the resulting global gap, within the assumptions of the band description.

References

• Ashcroft, N. W., and N. D. Mermin. Solid State Physics. Holt, Rinehart and Winston, 1976.
• Martin, R. M. Electronic Structure: Basic Theory and Practical Methods. Cambridge University Press, 2004.