Bloch's theorem

File:BlochWave in Silicon.png

Isosurface of the square modulus of a Bloch state in a silicon lattice

File:Bloch function.svg

Solid line: A schematic of the real part of a typical Bloch state in one dimension. The dotted line is from the factor e^ik·r. The light circles represent atoms.

In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential can be expressed as plane waves modulated by periodic functions. The theorem is named after the Swiss physicist Felix Bloch, who discovered the theorem in 1929. Mathematically, they are written Template:Equation box 1 where $\mathbf{𝐫}$ is position, $\psi$ is the wave function, $u$ is a periodic function with the same periodicity as the crystal, the wave vector $\mathbf{𝐤}$ is the crystal momentum vector, $e$ is Euler's number, and $i$ is the imaginary unit.

Functions of this form are known as Bloch functions or Bloch states, and serve as a suitable basis for the wave functions or states of electrons in crystalline solids.

The description of electrons in terms of Bloch functions, termed Bloch electrons (or less often Bloch Waves), underlies the concept of electronic band structures.

These eigenstates are written with subscripts as ${\psi }_{n\mathbf{𝐤}}$, where $n$ is a discrete index, called the band index, which is present because there are many different wave functions with the same $\mathbf{𝐤}$ (each has a different periodic component $u$). Within a band (i.e., for fixed $n$), ${\psi }_{n\mathbf{𝐤}}$ varies continuously with $\mathbf{𝐤}$, as does its energy. Also, ${\psi }_{n\mathbf{𝐤}}$ is unique only up to a constant reciprocal lattice vector $\mathbf{𝐊}$, or, ${\psi }_{n\mathbf{𝐤}}={\psi }_{n(\mathbf{𝐤}\mathbf{+}\mathbf{𝐊})}$. Therefore, the wave vector $\mathbf{𝐤}$ can be restricted to the first Brillouin zone of the reciprocal lattice without loss of generality.

Applications and consequences

Applicability

The most common example of Bloch's theorem is describing electrons in a crystal, especially in characterizing the crystal's electronic properties, such as electronic band structure. However, a Bloch-wave description applies more generally to any wave-like phenomenon in a periodic medium. For example, a periodic dielectric structure 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.

Wave vector

File:BlochWaves1D.svg

A Bloch wave function (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 state broken up in two different ways, involving the wave vector k₁ (left) or k₂ (right). The difference (k₁ − k₂) 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 $\psi$, not k or u directly. This is important because k and u are not unique. Specifically, if $\psi$ 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 values of k 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 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).

Statement

A second and equivalent way to state the theorem is the following

Proof

Using lattice periodicity

Bloch's theorem, being a statement about lattice periodicity, all the symmetries in this proof are encoded as translation symmetries of the wave function itself. Template:Math proof

Using operators

In this proof all the symmetries are encoded as commutation properties of the translation operators Template:Math proof

Using group theory

Apart from the group theory technicalities this proof is interesting because it becomes clear how to generalize the Bloch theorem for groups that are not only translations. This is typically done for space groups which are a combination of a translation and a point group and it is used for computing the band structure, spectrum and specific heats of crystals given a specific crystal group symmetry like FCC or BCC and eventually an extra basis.Template:Rp In this proof it is also possible to notice how it is key that the extra point group is driven by a symmetry in the effective potential but it shall commute with the Hamiltonian. Template:Math proof

In the generalized version of the Bloch theorem, the Fourier transform, i.e. the wave function expansion, gets generalized from a discrete Fourier transform which is applicable only for cyclic groups, and therefore translations, into a character expansion of the wave function where the characters are given from the specific finite point group.

Also here is possible to see how the characters (as the invariants of the irreducible representations) can be treated as the fundamental building blocks instead of the irreducible representations themselves.

Velocity and effective mass

If we apply the time-independent Schrödinger equation to the Bloch wave function we obtain $${\hat{H}}_{\mathbf{𝐤}}{u}_{\mathbf{𝐤}}(\mathbf{𝐫})=[\frac{{\hslash }^2}{2m}{(-i\mathrm{\nabla }+\mathbf{𝐤})}^2+U(\mathbf{𝐫})]u_{\mathbf{𝐤}}(\mathbf{𝐫})={\epsilon }_{\mathbf{𝐤}}{u}_{\mathbf{𝐤}}(\mathbf{𝐫})$$ with boundary conditions $$u_{\mathbf{𝐤}}(\mathbf{𝐫})=u_{\mathbf{𝐤}}(\mathbf{𝐫}+\mathbf{𝐑})$$ Given this is defined in a finite volume we expect an infinite family of eigenvalues; here $\mathbf{𝐤}$ is a parameter of the Hamiltonian and therefore we arrive at a "continuous family" of eigenvalues ${\epsilon }_n(\mathbf{𝐤})$ dependent on the continuous parameter $\mathbf{𝐤}$ and thus at the basic concept of an electronic band structure.

This shows how the effective momentum can be seen as composed of two parts, $${\widehat{\mathbf{𝐩}}}_{\text{eff}}=-i\hslash \mathrm{\nabla }+\hslash \mathbf{𝐤},$$ a standard momentum $-i\hslash \mathrm{\nabla }$ and a crystal momentum $\hslash \mathbf{𝐤}$. More precisely the crystal momentum is not a momentum but it stands for the momentum in the same way as the electromagnetic momentum in the minimal coupling, and as part of a canonical transformation of the momentum.

For the effective velocity we can derive Template:Equation box 1 Template:Math proof

For the effective mass Template:Equation box 1 Template:Math proof The quantity on the right multiplied by a factor$\frac{1}{{\hslash }^2}$ is called effective mass tensor $\mathbf{𝐌}(\mathbf{𝐤})$ and we can use it to write a semi-classical equation for a charge carrier in a band Template:Equation box 1 where $\mathbf{𝐚}$ is an acceleration. This equation is analogous to the de Broglie wave type of approximation Template:Equation box 1 As an intuitive interpretation, both of the previous two equations resemble formally and are in a semi-classical analogy with Newton's second law for an electron in an external Lorentz force.

Mathematical caveat

Mathematically, a rigorous theorem such as Bloch's theorem cannot exist in Quantum Mechanics: The spectral values of a band structure in a solid crystal or lattice system belong to the continuous spectrum, for which no finite norm eigenstates in the Hilbert space exist, i.e, no eigenstates with finite energy or finite probability can exist – cf. decomposition of spectrum –, because eigenvalues belong to the point spectrum by definition. Therefore, all physicists' calculations in Bloch's theorem with eigenstate decompositions in a Hilbert space are in some sense purely formal: The decomposition series do not converge in Hilbert space, and no proper spatially periodic function can be a finite norm state in the full Hilbert space.

Decompositions of periodic continuous functions – similarly to Bloch – can possibly be performed in spaces of bounded or bounded continuous functions, but not in spaces of functions square integrable over full x-space, which would be the required Hilbert space setting for Quantum Mechanics.

In Mathematical Physics, as a substitute, different rigorous decompositions can be obtained which also provide the band structure, by exploiting lattice symmetry based on a Hilbert space direct integral decomposition. By that method, the Hamiltonian operator is decomposed into a parameter dependent family of so-called reduced Hamiltonian operators on a corresponding family of Hilbert spaces and with corresponding domains of definitions (e.g. characterized by different boundary conditions). Each of these Hamiltonians has (in general) a discrete point spectrum with finite eigenstates of finite multiplicity, corresponding to the physicist's eigenvalue computations. Superposing these states with the direct integral would throw the states out of the original Hilbert space (and - possibly - provide only generalized eigenstates in a larger space, e.g. in the top space of a Gelfand triple), but the spectra of these Hamiltonians combine into the continuous band spectrum of the original Hamiltonian.

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). The general form of a one-dimensional periodic potential equation is Hill's equation: $$\frac{d^{2}y}{d{t}^2}+f(t)y=0,$$ where f(t) is a periodic potential. Specific periodic one-dimensional equations include the Kronig–Penney model and Mathieu's equation.

Mathematically, various theorems similar to Bloch's theorem are for instance interpreted in terms of unitary characters of a lattice group, and applied to spectral geometry.

Floquet theory is usually not done in a Hilbert space of functions square integrable with respect to the periodic independent variable, but in Banach spaces of continuous or differentiable functions, or in Frechet or nuclear spaces. So the methods used there do not directly apply to the Hilbert space setting required in Quantum Mechanics and require proper adaptation, such as using a Hilbert space direct integral.

See also

• Bloch oscillations
• Bloch wave – MoM method
• Electronic band structure
• Nearly free electron model
• Periodic boundary conditions
• Symmetries in quantum mechanics
• Tight-binding model
• Wannier function