Bloch wave

Bloch wave equipotential in silicon lattice

A Bloch wave or Bloch state, named after Swiss physicist Felix Bloch, is the wavefunction of a particle (usually, an electron) placed in a periodic potential. 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) $\, u_{n \mathbf{k}}(r)$ 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

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.

More generally, a Bloch-wave description applies 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.

The plane wave vector (Bloch wave vector) k, which when multiplied by the reduced Planck's constant is the particle's crystal momentum, is unique only up to a reciprocal lattice vector, so one only needs to consider the wave vectors inside the first Brillouin zone. For a given wave vector and potential, there are a number of solutions, indexed by n, to Schrödinger's equation for a Bloch electron. These solutions, called bands, are separated in energy by a finite spacing at each k; if there is a separation that extends over all wave vectors, it is called a (complete) band gap. The band structure is the collection of energy eigenstates within the first Brillouin zone. All the properties of electrons in a periodic potential can be calculated from this band structure and the associated wave functions, at least within the independent electron approximation.

A corollary of this result is that the Bloch wave vector k is a conserved quantity in a crystalline system (modulo addition of reciprocal lattice vectors), and hence the group velocity of the wave is conserved. This means that electrons can propagate without scattering through a crystalline material, almost like free particles, and that electrical resistance in a crystalline conductor only results from imperfections and finite size which break the periodicity and induce interaction with phonons.

Proof

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 $\mathbf{a}_1,\mathbf{a}_2,\mathbf{a}_3$. 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 $\mathbf{b}_1,\mathbf{b}_2,\mathbf{b}_3$ (with units of inverse length), with the property that $\mathbf{a}_i \cdot \mathbf{b}_i = 1$, but $\mathbf{a}_i \cdot \mathbf{b}_j = 0$ when $i\neq j$. (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 $n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3$ (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 $u(\mathbf{r})e^{i\mathbf{k}\cdot \mathbf{r}}$, where $u(\mathbf{r})$ 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 $n_i$ are integers. As a special case of this, $\psi(\mathbf{r}+\mathbf{a}_i) = e^{i\theta_i} \psi(\mathbf{r})$ for i=1,2,3, where $\theta_i$ are three numbers which vary for different Bloch waves but do not depend on r. Define $\mathbf{k} = \theta_1 \mathbf{b}_1 + \theta_2 \mathbf{b}_2 + \theta_3 \mathbf{b}_3$, 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

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

This proves that u has the periodicity of the lattice. Since $\psi(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}} u(\mathbf{r})$, 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 $\theta_n$ 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.