Bloch's theorem

thumb|upright=1.2|[[Isosurface of the square modulus of a Bloch state in a silicon lattice]]

thumb|upright=1.7|Solid line: A schematic of the real part of a typical Bloch state in one dimension. The dotted line is from the factor . 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

u(\mathbf{r})</math>

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where <math>\mathbf{r}</math> is position, <math>\psi</math> is the wave function, <math>u</math> is a periodic function with the same periodicity as the crystal, the wave vector <math>\mathbf{k}</math> is the crystal momentum vector, <math>e</math> is Euler's number, and <math>i</math> 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 <math>\psi_{n\mathbf{k</math>, where <math>n</math> is a discrete index, called the band index, which is present because there are many different wave functions with the same <math>\mathbf{k}</math> (each has a different periodic component <math>u</math>). Within a band (i.e., for fixed <math>n</math>), <math>\psi_{n\mathbf{k</math> varies continuously with <math>\mathbf{k}</math>, as does its energy. Also, <math>\psi_{n\mathbf{k</math> is unique only up to a constant reciprocal lattice vector <math>\mathbf{K}</math>, or, <math>\psi_{n\mathbf{k=\psi_{n(\mathbf{k+K})}</math>. Therefore, the wave vector <math>\mathbf{k}</math> 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

thumb|upright=1.75|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 (left) or (right). The difference () 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

<math display="block">\psi ( \mathbf{r} ) = e^{ i \mathbf{k} \cdot \mathbf{r} } u ( \mathbf{r} ) ,</math>

where is periodic with the same periodicity as the crystal lattice. The actual quantum state of the electron is entirely determined by <math>\psi</math>, not or directly. This is important because and are not unique. Specifically, if <math>\psi</math> 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 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).

Statement

u(\mathbf{r}),</math> where <math>u(\mathbf{r})</math> has the same periodicity as the atomic structure of the crystal, such that <math display="block">u_{\mathbf{k(\mathbf{x}) = u_{\mathbf{k(\mathbf{x} + \mathbf{n} \cdot \mathbf{a}).</math>

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

(\mathbf{x}+\mathbf{a}) = e^{i\mathbf{k}\cdot\mathbf{a\psi_{\mathbf{k(\mathbf{x}).</math>

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.

e^{-i\mathbf{k}\cdot \mathbf{a}_j} \big) \big( e^{2\pi i \theta_j} \psi(\mathbf{r}) \big) \\

&= e^{-i\mathbf{k} \cdot \mathbf{r e^{-2\pi i \theta_j} e^{2\pi i \theta_j} \psi(\mathbf{r}) \\

&= u(\mathbf{r}).

\end{align}</math>

This proves that has the periodicity of the lattice. Since <math>\psi(\mathbf{r}) = e^{i \mathbf{k}\cdot\mathbf{r u(\mathbf{r}),</math> that proves that the state is a Bloch state.

Finally, we are ready for the main proof of Bloch's theorem which is as follows.

As above, let <math> \hat{T}_{n_1,n_2,n_3} </math> 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 <math> \hat{T}_{n_1,n_2,n_3} \!</math> 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

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

_{\mathbf{n\psi(\mathbf{r})&= \psi(\mathbf{r}+\mathbf{T}_{\mathbf{n) \\

&= \psi(\mathbf{r}+n_1\mathbf{a}_1+n_2\mathbf{a}_2+n_3\mathbf{a}_3) \\

&= \psi(\mathbf{r}+\mathbf{A}\mathbf{n})

\end{align}</math>

with

\mathbf{A} = \begin{bmatrix} \mathbf{a}_1 & \mathbf{a}_2 & \mathbf{a}_3 \end{bmatrix}, \quad \mathbf{n} = \begin{pmatrix} n_1 \\ n_2 \\ n_3 \end{pmatrix}

</math>

We use the hypothesis of a mean periodic potential

<math display="block">U(\mathbf{x}+\mathbf{T}_{\mathbf{n)= U(\mathbf{x})</math>

and the independent electron approximation with a Hamiltonian

<math display="block">\hat{H}=\frac{\hat{\mathbf{p^2}{2m}+U(\mathbf{x})</math>

Given the Hamiltonian is invariant for translations it shall commute with the translation operator

<math display="block">[\hat{H},\hat{\mathbf{T_{\mathbf{n] = 0</math>

and the two operators shall have a common set of eigenfunctions.

Therefore, we start to look at the eigen-functions of the translation operator:

<math display="block">\hat{\mathbf{T_{\mathbf{n\psi(\mathbf{x})=\lambda_{\mathbf{n\psi(\mathbf{x})</math>

Given <math>\hat{\mathbf{T_{\mathbf{n</math> is an additive operator

\hat{\mathbf{T_{\mathbf{n}_1} \hat{\mathbf{T_{\mathbf{n}_2}\psi(\mathbf{x}) =

\psi(\mathbf{x} + \mathbf{A} \mathbf{n}_1 + \mathbf{A} \mathbf{n}_2) = \hat{\mathbf{T_{\mathbf{n}_1 + \mathbf{n}_2} \psi(\mathbf{x})

</math>

If we substitute here the eigenvalue equation and dividing both sides for <math>\psi(\mathbf{x})</math> we have

\lambda_{\mathbf{n}_1} \lambda_{\mathbf{n}_2} =

\lambda_{\mathbf{n}_1 + \mathbf{n}_2}

</math>

This is true for

<math display="block">\lambda_{\mathbf{n = e^{s \mathbf{n} \cdot \mathbf{a} } </math>

where <math>s \in \Complex </math> if we use the normalization condition over a single primitive cell of volume V

1 = \int_V |\psi(\mathbf{x})|^2 d \mathbf{x} =

\int_V \left|\hat\mathbf{T}_\mathbf{n} \psi(\mathbf{x})\right|^2 d \mathbf{x} =

|\lambda_{\mathbf{n|^2 \int_V |\psi(\mathbf{x})|^2 d \mathbf{x}

</math>

and therefore

<math display="block">1 = |\lambda_{\mathbf{n|^2</math> and <math display="block">s = i k </math> where <math>k \in \mathbb{R}</math>. Finally,

\mathbf{\hat{T}_n}\psi(\mathbf{x})=

\psi(\mathbf{x} + \mathbf{n} \cdot \mathbf{a} ) =

e^{i k \mathbf{n} \cdot \mathbf{a} }\psi(\mathbf{x})

,</math>

which is true for a Bloch wave i.e. for <math>\psi_{\mathbf{k(\mathbf{x}) = e^{i \mathbf{k} \cdot \mathbf{x} } u_{\mathbf{k(\mathbf{x})</math> with <math>u_{\mathbf{k(\mathbf{x}) = u_{\mathbf{k(\mathbf{x} + \mathbf{A}\mathbf{n})</math>

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 it is key that the extra point group is driven by a symmetry in the effective potential but it shall commute with the Hamiltonian.

\hat{\boldsymbol{\tau_1 \hat{\boldsymbol{\tau_2 \hat{\boldsymbol{\tau_3

</math> where <math display="block">\hat{\boldsymbol{\tau_i = n_i \hat{\mathbf{a_i</math>

The commutativity of the <math>\hat{\boldsymbol{\tau_i</math> operators gives three commuting cyclic subgroups (given they can be generated by only one element) which are infinite, 1-dimensional and abelian. All irreducible representations of abelian groups are one dimensional.

Given they are one dimensional the matrix representation and the character are the same. The character is the representation over the complex numbers of the group or also the trace of the representation which in this case is a one dimensional matrix.

All these subgroups, given they are cyclic, they have characters which are appropriate roots of unity. In fact they have one generator <math>\gamma</math> which shall obey to <math>\gamma^n = 1</math>, and therefore the character <math>\chi(\gamma)^n = 1</math>. Note that this is straightforward in the finite cyclic group case but in the countable infinite case of the infinite cyclic group (i.e. the translation group here) there is a limit for <math>n \to \infty</math> where the character remains finite.

Given the character is a root of unity, for each subgroup the character can be then written as

<math display="block">\chi_{k_1}(\hat{\boldsymbol{\tau_1 (n_1,a_1)) = e^{i k_1 n_1 a_1}</math>

If we introduce the Born–von Karman boundary condition on the potential:

<math display="block">V \left(\mathbf {r} +\sum_i N_{i} \mathbf {a}_{i}\right) = V (\mathbf {r} +\mathbf{L}) = V (\mathbf {r} )</math>

where L is a macroscopic periodicity in the direction <math>\mathbf{a}</math> that can also be seen as a multiple of <math>a_i</math> where <math display="inline">\mathbf{L} = \sum_i N_{i}\mathbf {a}_{i}</math>

This substituting in the time independent Schrödinger equation with a simple effective Hamiltonian

<math display="block">\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r})</math>

induces a periodicity with the wave function:

<math display="block">\psi \left(\mathbf {r} + \sum_i N_{i}\mathbf {a}_{i}\right) = \psi (\mathbf {r} )</math>

And for each dimension a translation operator with a period L

<math display="block">\hat{P}_{\varepsilon|\tau_i + L_i} = \hat{P}_{\varepsilon|\tau_i}</math>

From here we can see that also the character shall be invariant by a translation of <math>L_i</math>:

and from the last equation we get for each dimension a periodic condition:

where <math>m_1 \in \mathbb{Z}</math> is an integer and <math>k_1=\frac {2 \pi m_1}{L_1}</math>

The wave vector <math>k_1</math> identify the irreducible representation in the same manner as <math>m_1</math>, and <math>L_1</math> is a macroscopic periodic length of the crystal in direction <math>a_1</math>. In this context, the wave vector serves as a quantum number for the translation operator.

We can generalize this for 3 dimensions

<math>\chi_{k_1}(n_1,a_1)\chi_{k_2}(n_2,a_2)\chi_{k_3}(n_3,a_3) = e^{i\mathbf{k} \cdot \boldsymbol{\tau</math>

and the generic formula for the wave function becomes:

<math display="block">\hat{P}_R\psi_j = \sum_{\alpha} \psi_{\alpha} \chi_{\alpha j}(R)</math>

i.e. specializing it for a translation

<math display="block">\hat{P}_{\varepsilon|\boldsymbol{\tau \psi(\mathbf{r}) =\psi(\mathbf{r}) e^{i \mathbf{k} \cdot \boldsymbol{\tau = \psi(\mathbf{r} + \boldsymbol{\tau})</math>

and we have proven Bloch’s theorem.

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

<math display="block">\hat{H}_\mathbf{k} u_\mathbf{k}(\mathbf{r}) =

\left[ \frac{\hbar^2}{2m} \left( -i \nabla + \mathbf{k} \right)^2 + U(\mathbf{r}) \right] u_\mathbf{k}(\mathbf{r}) =

\varepsilon_\mathbf{k} u_\mathbf{k}(\mathbf{r})

</math>

with boundary conditions

<math display="block">u_\mathbf{k}(\mathbf{r}) = u_\mathbf{k}(\mathbf{r} + \mathbf{R})</math>

Given this is defined in a finite volume we expect an infinite family of eigenvalues; here <math>{\mathbf{k</math> is a parameter of the Hamiltonian and therefore we arrive at a "continuous family" of eigenvalues <math>\varepsilon_n(\mathbf{k})</math> dependent on the continuous parameter <math>{\mathbf{k</math> 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,

<math display="block">\hat{\mathbf{p_\text{eff} = -i \hbar \nabla + \hbar \mathbf{k} ,</math>

a standard momentum <math>-i \hbar \nabla</math> and a crystal momentum <math>\hbar \mathbf{k}</math>. 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

= \frac {\hbar^2}{m} \int

d\mathbf{r}\, \psi^{*}_{n\mathbf{k (-i \nabla)\psi_{n\mathbf{k = \frac {\hbar}{m}\langle\hat{\mathbf{p\rangle = \hbar \langle\hat{\mathbf{v\rangle</math>

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</math> and <math>\frac{\partial^2 \varepsilon_n(\mathbf{k})}{\partial k_i \partial k_j}</math>

given they are the coefficients of the following expansion in where is considered small with respect to

\varepsilon_n(\mathbf{k} + \mathbf{q}) = \varepsilon_n(\mathbf{k}) +

\sum_i \frac{\partial \varepsilon_n}{\partial k_i} q_i +

\frac{1}{2} \sum_{ij} \frac{\partial^2 \varepsilon_n}{\partial k_i \partial k_j} q_i q_j +

O(q^3)

</math>

Given <math>\varepsilon_n(\mathbf{k}+\mathbf{q})</math> are eigenvalues of <math>\hat{H}_{\mathbf{k}+\mathbf{q</math>

We can consider the following perturbation problem in q:

\hat{H}_{\mathbf{k}+\mathbf{q =

\hat{H}_\mathbf{k} +

\frac{\hbar^2}{m} \mathbf{q} \cdot ( -i\nabla + \mathbf{k} ) +

\frac{\hbar^2}{2m} q^2

</math>

Perturbation theory of the second order states that

E_n =E^0_n + \int d\mathbf{r}\, \psi^{*}_n \hat{V} \psi_n +

\sum_{n' \neq n} \frac{|\int d\mathbf{r} \,\psi^{*}_n \hat{V} \psi_n|^2}{E^0_n - E^0_{n' + ...

</math>

To compute to linear order in

\sum_i \frac{\partial \varepsilon_n}{\partial k_i} q_i =

\sum_i \int d\mathbf{r}\, u_{n\mathbf{k^{*} \frac{\hbar^2}{m} ( -i\nabla + \mathbf{k} )_i q_i u_{n\mathbf{k

</math>

where the integrations are over a primitive cell or the entire crystal, given if the integral

<math display="block">\int d\mathbf{r}\, u_{n\mathbf{k^{*} u_{n\mathbf{k</math>

is normalized across the cell or the crystal.

We can simplify over to obtain

\frac{\partial \varepsilon_n}{\partial \mathbf{k =

\frac{\hbar^2}{m} \int d\mathbf{r} \, u_{n\mathbf{k^{*}( -i\nabla + \mathbf{k} ) u_{n\mathbf{k

</math>

and we can reinsert the complete wave functions

\frac{\partial \varepsilon_n}{\partial \mathbf{k =

\frac{\hbar^2}{m} \int d\mathbf{r} \, \psi_{n\mathbf{k^{*}( -i\nabla) \psi_{n\mathbf{k

</math>

For the effective mass