# Bloch wave

Bloch wave equipotential in silicon lattice
Solid line: A schematic of a typical Bloch wave in one dimension. (The actual wave is complex; this is the real part.) The dotted line is from the eik·r factor. The light circles represent atoms.

A Bloch wave or Bloch state, named after Swiss physicist Felix Bloch, is a type of wavefunction for a particle in a periodically-repeating environment, most commonly an electron in a crystal. Bloch's theorem states that the energy eigenfunction for such a system may be written as the product of a plane wave envelope function and a periodic function (periodic Bloch function) un k(r) that has the same periodicity as the potential, giving:

$\psi_{n\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}u_{n\mathbf{k}}(\mathbf{r}).$

The corresponding energy eigenvalues are ϵn(k) = ϵn(k + K), periodic with periodicity K of a reciprocal lattice vector. The energies associated with the index n vary continuously with wave vector k and form an energy band identified by band index n. The eigenvalues for given n are periodic in k; all distinct values of ϵn(k) occur for k-values within the first Brillouin zone of the reciprocal lattice.

## Applications and consequences

### Applicability

The most common example of Bloch's theorem is describing electrons in a crystal. However, a Bloch-wave description applies more generally to any wave-like phenomenon in a periodic medium. For example, a periodic dielectric in electromagnetism leads to photonic crystals, and a periodic acoustic medium leads to phononic crystals. It is generally treated in the various forms of the dynamical theory of diffraction.

### Bloch states versus energy eigenstates

Every Bloch state ψ is an energy eigenstate. The converse, however, is not true: There are energy eigenstates that are not Bloch states. This occurs when two or more Bloch states have the same energy but different values of k. A linear combination of these states, e.g. (ψ12)/√2, would be an energy eigenstate that is not a Bloch state.

Moreover, the set of all Bloch states is a basis for all normalizable electron states in an infinite crystal. Therefore, all the properties of electrons in a periodic potential can be calculated from knowing the energy and wave function for each Bloch state, at least within the independent electron approximation.

### Meaning and non-uniqueness of the k-vector

A Bloch wave (bottom) can be broken up into the product of a periodic function (top) and a plane-wave (center). The left side and right side represent the same Bloch wave broken up in two different ways, involving the wave vector k1 (left) or k2 (right). The difference (k1k2) is a reciprocal lattice vector. In all plots, blue is real part and red is imaginary part.

Suppose an electron is in a Bloch state

$\psi ( \mathbf{r} ) = e^{ i \mathbf{k} \cdot \mathbf{r} } u ( \mathbf{r} ) ,$

where u is periodic with the same periodicity as the crystal lattice. The actual quantum state of the electron is entirely determined by ψ, not k or u directly. This is important because k and u are not unique. Specifically, if ψ can be written as above using k, it can also be written using (k + K), where K is any reciprocal lattice vector (see figure at right). Therefore, wave vectors that differ by a reciprocal lattice vector are equivalent, in the sense that they characterize the same set of Bloch states.

The first Brillouin zone is a restricted set of k-vectors with the property that no two of them are equivalent, yet every possible k is equivalent to one (and only one) vector in the first Brillouin zone. Therefore, if we restrict k to the first Brillouin zone, then every Bloch state has a unique k. Therefore the first Brillouin zone is often used to depict all of the Bloch states without redundancy, for example in a band structure, and it is used for the same reason in many calculations.

When k is multiplied by the reduced Planck's constant, it equals the electron's crystal momentum. Related to this, the group velocity of an electron can be calculated based on how the energy of a Bloch state varies with k; for more details see crystal momentum.

### Detailed example

For a detailed example in which the consequences of Bloch's theorem are worked out in a specific situation, see the article: Particle in a one-dimensional lattice (periodic potential).

## Proof of Bloch's theorem

Next, we prove Bloch's theorem:

For electrons in a perfect crystal, there is a basis of wavefunctions called "Bloch states" (or "Bloch waves"), with the properties:
• Every Bloch state is an energy eigenstate
• Every Bloch state $\psi$ can be written as
$\psi(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}} u(\mathbf{r})$
where u has the same periodicity as the atomic structure of the crystal.

### Preliminaries: Crystal symmetries, lattice, and reciprocal lattice

The defining property of a crystal is translational symmetry, which means that if the crystal is shifted an appropriate amount, it winds up with all its atoms in the same places. (A finite-size crystal cannot have perfect translational symmetry, but it is a useful approximation.)

A three-dimensional crystal has three primitive lattice vectors a1, a2, a3. If the crystal is shifted by any of these three vectors, or a combination of them of the form

$n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3$

where ni are three integers, then the atoms end up in the same set of locations as they started.

Another helpful ingredient in the proof is the reciprocal lattice vectors. These are three vectors b1, b2, b3 (with units of inverse length), with the property that ai · bi = 2π, but ai · bj = 0 when ij. (For the formula for bi, see reciprocal lattice vector.)

### Proof

Let $\hat{T}_{n_1,n_2,n_3} \!$ denote a translation operator that shifts every wave function by the amount n1a1 + n2a2 + n3a3 (as above, ni are integers). Because the crystal is symmetric, this operator commutes with the Hamiltonian operator. Moreover every such translation operator commutes with every other. Therefore, there is a simultaneous eigenbasis of the Hamiltonian operator and every possible $\hat{T}_{n_1,n_2,n_3} \!$ operator. The states in this basis are called Bloch waves. The goal below is to show that they have the expected properties.

Each Bloch wave is (by construction) an eigenstate of the Hamiltonian, and also an eigenstate of all the translation operators. The former proves that the Bloch waves are energy eigenstates. As the next paragraph will show, the latter proves that the Bloch waves have the form u(r)eik · r, where u(r) is a periodic function (with the same periodicity as the crystal).

Let ψ(r) be a Bloch wave. It is, by construction, an eigenstate of all the translation operators $\hat{T}_{n_1,n_2,n_3}$, where ni are integers. As a special case of this,

$\psi(\mathbf{r}+\mathbf{a}_i) = C_i \psi(\mathbf{r})$

for i = 1, 2, 3, where Ci are three numbers (the eigenvalues) which vary for different Bloch waves but do not depend on r. It is helpful to write the numbers Ci in a different form, by choosing three numbers θ1, θ2, θ3 with e2πiθi = Ci:

$\psi(\mathbf{r}+\mathbf{a}_i) = e^{2 \pi i \theta_i} \psi(\mathbf{r})$

Again, the θi are three numbers which vary for different Bloch waves but do not depend on r. Define k = θ1b1 + θ2b2 + θ3b3, where bi are the reciprocal lattice vectors (see above). Finally, define

$u(\mathbf{r}) = e^{-i\mathbf{k}\cdot\mathbf{r}} \psi(\mathbf{r})\,.$

Then

$u(\mathbf{r} + \mathbf{a}_i) = e^{-i\mathbf{k} \cdot (\mathbf{r} + \mathbf{a}_i)} \psi(\mathbf{r}+\mathbf{a}_i) = \big( e^{-i\mathbf{k} \cdot \mathbf{r}} e^{-i\mathbf{k}\cdot \mathbf{a}_i} \big) \big( e^{2\pi i \theta_i} \psi(\mathbf{r}) \big) = e^{-i\mathbf{k} \cdot \mathbf{r}} e^{-2\pi i \theta_i} e^{2\pi i \theta_i} \psi(\mathbf{r}) = u(\mathbf{r})$.

This proves that u has the periodicity of the lattice. Since ψ(r) = ei k · ru(r), that proves that the Bloch wave has the expected form.

## History and related equations

The concept of the Bloch state was developed by Felix Bloch in 1928, to describe the conduction of electrons in crystalline solids. The same underlying mathematics, however, was also discovered independently several times: by George William Hill (1877), Gaston Floquet (1883), and Alexander Lyapunov (1892). As a result, a variety of nomenclatures are common: applied to ordinary differential equations, it is called Floquet theory (or occasionally the Lyapunov–Floquet theorem). Various one-dimensional periodic potential equations have special names, for example, Hill's equation:[1]

$\frac {d^2y}{dx^2}+\left(\theta_0+2\sum_{n=1}^\infty \theta_n \cos(2nx) \right ) y=0,$

where the θn are constants. Hill's equation is very general, as the θ-related terms may be viewed as a Fourier series expansion of a periodic potential. Other much studied periodic one-dimensional equations are the Kronig–Penney model and Mathieu's equation.

## References

1. ^ Magnus, W; Winkler, S (2004). Hill's Equation. Courier Dover. p. 11. ISBN 0-486-49565-5.