Bloch's Theorem and Energy Bands

The free-electron gas treats a metal's conduction electrons as a Fermi gas in a flat box, ignoring the ions entirely. That model reproduces the electronic heat capacity and the order of magnitude of the conductivity, but it cannot explain why some solids conduct and others insulate, why a few metals carry current as though the charge carriers were positive, or why the mean free path can reach centimetres at low temperature. Every one of these failures traces to the same omission: the periodic potential of the ion cores. This module restores that potential. The single structural fact that organizes the whole subject is that the electron sees a potential with the periodicity of the lattice,

U (r + R) = U (r) for every lattice vector R,

and the eigenstates of a Hamiltonian with that symmetry have a rigidly constrained form. Bloch's theorem states that form.

The periodic potential

A crystal is a Bravais lattice of points $R = n_{1} a_{1} + n_{2} a_{2} + n_{3} a_{3}$ with a basis of ions attached to each point. An electron moving through it feels the summed Coulomb attraction of all the ion cores (and, in a mean-field sense, the averaged repulsion of the other electrons). Because the ions sit on a periodic array, the resulting one-electron potential inherits that period: translating by any $R$ maps the crystal onto itself and leaves $U$ unchanged. The one-electron Hamiltonian is

H = - \frac{ℏ ^{2}}{2 m} \nabla^{2} + U (r), U (r + R) = U (r) .

The potential is not weak in any absolute sense; near an ion it plunges to tens of electron-volts. What matters is only its periodicity. Two limiting approximations bracket every real band structure: the nearly-free-electron model, which treats the periodic modulation as a weak perturbation on plane waves, and the tight-binding model, which starts from isolated-atom orbitals and lets them hybridize. Bloch's theorem holds exactly regardless of the potential's strength, so it governs both limits.

The crystal potential U(x) along a line of ions repeats with the lattice period a; deep wells sit at the ion cores and the potential rises between them. Only the periodicity, not the depth, enters Bloch's theorem.

Bloch's theorem

The label $k$ is a wavevector; $n$ is the band index, distinguishing the discrete solutions found at fixed $k$ . A state of this form is a Bloch wave: a running plane wave $e^{i k \cdot r}$ whose amplitude is not constant but modulated by the lattice-periodic envelope $u_{n k}$ , which repeats the internal structure of the potential inside every cell.

A Bloch wave (bottom) is the product of a plane wave of wavevector k (top) and a lattice-periodic envelope u (middle) that repeats every cell; the product oscillates on the plane-wave scale while its detail recurs cell to cell.

Proof from translation operators

Define the lattice translation operator $T_{R}$ that shifts a function by a lattice vector, $T_{R} f (r) = f (r + R)$ . Because $U$ is periodic and $\nabla^{2}$ is translation-invariant, $T_{R}$ commutes with the Hamiltonian, $[T_{R}, H] = 0$ . Successive translations add, so the translations commute among themselves, $T_{R} T_{R^{'}} = T_{R + R^{'}} = T_{R^{'}} T_{R}$ .

Proof. A set of mutually commuting operators that also commute with $H$ has a common eigenbasis, so each stationary state can be chosen an eigenstate of every $T_{R}$ :

T_{R} ψ = c (R) ψ .

The multiplication law of the translations forces the eigenvalues to multiply, $c (R) c (R^{'}) = c (R + R^{'})$ . A function on the lattice obeying $c (R + R^{'}) = c (R) c (R^{'})$ is an exponential: writing $c (a_{i}) = e^{2 π i x_{i}}$ for the three primitive vectors and using the additivity gives $c (R) = e^{i k \cdot R}$ with $k = x_{1} b_{1} + x_{2} b_{2} + x_{3} b_{3}$ expressed in the reciprocal basis $b_{i}$ . Normalizability requires the $x_{i}$ real, so $∣ c (R)∣ = 1$ . Then $ψ (r + R) = T_{R} ψ (r) = e^{i k \cdot R} ψ (r)$ , which is the second form of the theorem. Setting $u (r) = e^{- i k \cdot r} ψ (r)$ and checking $u (r + R) = e^{- i k \cdot r} e^{- i k \cdot R} ψ (r + R) = e^{- i k \cdot r} ψ (r) = u (r)$ recovers the product form. $■$

The physical content is that the crystal's discrete translation symmetry, like any symmetry, labels its eigenstates by the eigenvalue of the symmetry operation. Here the eigenvalue is the phase $e^{i k \cdot R}$ picked up on translating by $R$ , and $k$ is the label. Two wavevectors differing by a reciprocal-lattice vector $G$ give identical phases, since $e^{i G \cdot R} = 1$ by definition of the reciprocal lattice. So $k$ and $k + G$ label the same translational symmetry, and every distinct label lies in one primitive cell of the reciprocal lattice — the first Brillouin zone.

Proof from the Fourier expansion

The second proof is constructive and produces the equation that the nearly-free-electron model solves. Expand the periodic potential in reciprocal -lattice plane waves, the only ones with the lattice period,

U (r) = G \sum U_{G} e^{i G \cdot r},

and expand any wavefunction obeying Born–von Kármán boundary conditions in the allowed plane waves,

ψ (r) = q \sum c_{q} e^{i q \cdot r} .

Substituting both into $H ψ = E ψ$ and matching the coefficient of each $e^{i q \cdot r}$ gives the central equation,

(\frac{ℏ ^{2} q ^{2}}{2 m} - E) c_{q} + G \sum U_{G} c_{q - G} = 0.

The potential only couples $c_{q}$ to coefficients $c_{q - G}$ whose wavevector differs by a reciprocal-lattice vector. A given $q$ is therefore linked only to the set ${q - G}$ , and never to wavevectors outside that family. Fix one representative $k$ in the first zone; the central equation is a closed linear system in the coefficients ${c_{k - G}}$ . Its solution is a wavefunction built entirely from plane waves $k - G$ ,

ψ_{k} (r) = G \sum c_{k - G} e^{i (k - G) \cdot r} = e^{i k \cdot r} u_{k} (r) G \sum c_{k - G} e^{- i G \cdot r},

and the bracketed sum is periodic because each $e^{- i G \cdot r}$ is. This reproduces Bloch's theorem and shows that at fixed $k$ the central equation has a discrete ladder of solutions — one eigenvalue $E_{n} (k)$ per band index $n$ .

The central equation couples the plane wave k only to those displaced by reciprocal-lattice vectors G; the potential component U_G links k to k minus G, and the closed chain of couplings defines one Bloch state.

Crystal momentum

The quantity $ℏ k$ is the crystal momentum. It is not the electron's true momentum. A Bloch state is not an eigenstate of the momentum operator $- i ℏ\nabla$ , because acting with $\nabla$ on $e^{i k \cdot r} u_{n k}$ also differentiates the envelope:

- i ℏ\nabla ψ_{n k} = ℏ k ψ_{n k} - i ℏ e^{i k \cdot r} \nabla u_{n k},

and the second term does not vanish. A Bloch state is a superposition of the plane-wave momenta $ℏ (k - G)$ , weighted by the $c_{k - G}$ . What $ℏ k$ does capture is the state's transformation under lattice translations, and it is that quantity which is conserved when Bloch electrons scatter off one another or off phonons. In such a process the total crystal momentum is conserved only up to a reciprocal-lattice vector,

k_{1} + k_{2} = k_{3} + k_{4} + G,

the same selection rule that governs phonon collisions: processes with $G = 0$ are normal, those with $G \neq = 0$ are Umklapp. The distinction between true momentum (conserved absolutely, from continuous translation symmetry) and crystal momentum (conserved mod $G$ , from discrete translation symmetry) is the recurring theme of transport in crystals.

The band index and $E_{n} (k)$

At each $k$ the eigenvalue problem is that of the periodic Hamiltonian restricted to Bloch waves of that wavevector: substitute $ψ = e^{i k \cdot r} u (r)$ into $H ψ = E ψ$ to get an eigenvalue equation for $u$ on a single primitive cell,

[\frac{ℏ ^{2}}{2 m} (- i \nabla + k)^{2} + U (r)] u_{n k} (r) = E_{n} (k) u_{n k} (r),

with periodic boundary conditions on the cell. A Hermitian eigenvalue problem on a finite region has a discrete, bounded-below spectrum, so for each $k$ the energies form a discrete ladder labeled $n = 1, 2, 3, \dots$ . As $k$ varies continuously through the first zone, each $E_{n} (k)$ traces out a continuous sheet — an energy band. The set of all bands is the crystal's band structure. Because $k$ and $k + G$ label the same state, each band is periodic in reciprocal space,

E_{n} (k + G) = E_{n} (k),

so the entire spectrum is determined by its values on the first Brillouin zone.

Born–von Kármán boundary conditions and the count of states

To count states one must make the crystal finite without introducing a surface. The Born–von Kármán (periodic) boundary condition wraps the crystal onto itself after $N_{i}$ cells along each primitive direction,

ψ (r + N_{i} a_{i}) = ψ (r), i = 1, 2, 3,

with $N = N_{1} N_{2} N_{3}$ the total number of primitive cells. Applying the Bloch relation $N_{i}$ times gives $e^{i N_{i} k \cdot a_{i}} = 1$ , so the allowed wavevectors are quantized,

k = i = 1 \sum 3 \frac{m _{i}}{N _{i}} b_{i}, m_{i} \in Z .

These points form a fine uniform mesh in reciprocal space. The volume per allowed $k$ is $(2 π)^{3} / V$ with $V = N v_{cell}$ the crystal volume — identical to the free-electron count, because the density of allowed wavevectors depends only on the box size, not on the potential. Counting how many fall inside one Brillouin zone (of volume $(2 π)^{3} / v_{cell}$ ) gives

\frac{( 2 π ) ^{3} / v _{cell}}{( 2 π ) ^{3} / V} = \frac{V}{v _{cell}} = N .

Each band contains exactly $N$ distinct wavevectors — one per primitive cell in the crystal. Including the two spin states, a band holds $2 N$ electrons. This integer count is the hinge of the metal–insulator distinction developed in the Fermi-surface lesson: a crystal with an even number of electrons per cell can exactly fill an integer number of bands, and a filled band carries no current.

Born–von Kármán conditions quantize k to a uniform mesh; exactly N allowed points (N the number of crystal cells) lie in the first Brillouin zone, here a square zone in two dimensions.

Zone schemes

The periodicity $E_{n} (k + G) = E_{n} (k)$ means one band structure admits three equivalent drawings. In the extended-zone scheme each band is placed in a different Brillouin zone, so the curve marches outward through successive zones and, for a weak potential, resembles the free-electron parabola $E = ℏ^{2} k^{2} /2 m$ with small gaps at the zone edges. In the reduced-zone scheme every band is translated by the appropriate $G$ back into the first zone, stacking the branches into a set of curves indexed by $n$ — the standard band-structure plot. In the repeated-zone scheme each reduced band is copied into every zone, making the full periodicity manifest and letting one follow a Fermi surface across zone boundaries without bookkeeping. The three carry identical information; the choice is a matter of convenience.

Three portrayals of the same band structure. Extended: successive bands occupy successive zones, tracing the folded free-electron parabola. Reduced: every branch folded into the first zone. Repeated: the reduced bands copied periodically in k.

The vocabulary of this lesson — Bloch waves, crystal momentum, the band index, the count of $2 N$ states per band, the zone schemes — is the coordinate system in which every later result is stated. The nearly-free-electron model solves the central equation for a weak potential and shows how the free-electron parabola breaks into bands with gaps of size $2 ∣ U_{G} ∣$ at the zone boundaries; the tight-binding model builds the same bands from the opposite limit of localized orbitals.