Bloch's theorem

Bloch's theorem
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 physicist Felix Bloch, who discovered the theorem in 1929. Mathematically, they are written
where is position, is the wave function, is a periodic function with the same periodicity as the crystal, the wave vector is the crystal momentum vector, is Euler's number, and 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.
Named after Swiss physicist Felix Bloch, 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 , where is a discrete index, called the band index, which is present because there are many different wave functions with the same (each has a different periodic component ). Within a band (i.e., for fixed ), varies continuously with , as does its energy. Also, is unique only up to a constant reciprocal lattice vector , or, . Therefore, the wave vector 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
Suppose an electron is in a Bloch state
where is periodic with the same periodicity as the crystal lattice. The actual quantum state of the electron is entirely determined by , not or directly. This is important because and are not unique. Specifically, if can be written as above using , it can also be written using , where 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 with the property that no two of them are equivalent, yet every possible is equivalent to one (and only one) vector in the first Brillouin zone. Therefore, if we restrict to the first Brillouin zone, then every Bloch state has a unique . 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 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 ; 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).
Theorem
Bloch's theorem is as follows:
For electrons in a perfect crystal, there is a basis of wave functions with the following two properties:
each of these wave functions is an energy eigenstate,
each of these wave functions is a Bloch state, meaning that this wave function can be written in the form where has the same periodicity as the atomic structure of the crystal, such that
Proof
Using lattice periodicity
Source:
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 . If the crystal is shifted by any of these three vectors, or a combination of them of the form
where 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 (with units of inverse length), with the property that , but when . (For the formula for , see reciprocal lattice vector.)
Lemma about translation operators
Let denote a translation operator that shifts every wave function by the amount (as above, are integers). The following fact is helpful for the proof of Bloch's theorem:
Finally, we are ready for the main proof of Bloch's theorem which is as follows.
As above, let denote a translation operator that shifts every wave function by the amount , where are integers. Because the crystal has translational symmetry, 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 operator. This basis is what we are looking for. The wave functions in this basis are energy eigenstates (because they are eigenstates of the Hamiltonian), and they are also Bloch states (because they are eigenstates of the translation operators; see Lemma above).
Using operators
Source:
We define the translation operator
with
We use the hypothesis of a mean periodic potential
and the independent electron approximation with an Hamiltonian
Given the Hamiltonian is invariant for translations it shall commute with the translation operator
and the two operators shall have a common set of eigenfunctions.
Therefore we start to look at the eigen-functions of the translation operator:
Given is an additive operator
If we substitute here the eigenvalue equation and dividing both sides for we have
This is true for
where
if we use the normalization condition over a single primitive cell of volume V
and therefore
and where . Finally,
Which is true for a Bloch wave i.e. for with
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.
In this proof it is also possible to notice how is key that the extra point group is driven by a symmetry in the effective potential but it shall commute with the Hamiltonian.
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
with boundary conditions
Given this is defined in a finite volume we expect an infinite family of eigenvalues; here is a parameter of the Hamiltonian and therefore we arrive at a "continuous family" of eigenvalues dependent on the continuous parameter 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,
a standard momentum and a crystal momentum . 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
For the effective mass
The quantity on the right multiplied by a factor is called effective mass tensor and we can use it to write a semi-classical equation for a charge carrier in a band
where is an acceleration. This equation is analogous to the de Broglie wave type of approximation
As an intuitive interpretation, both of the previous two equations resemble formally and are in a semi-classical analogy with the newton equation in an external Lorentz force.
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:
where is a periodic potential. Specific periodic one-dimensional equations include the Kronig–Penney model and Mathieu's equation.
Mathematically Bloch's theorem is interpreted in terms of unitary characters of a lattice group, and is applied to spectral geometry.
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
References
Further reading
Eponymous theorems of physics
Quantum mechanics
Condensed matter physics