Bloch's theorem
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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.<ref>Bloch, F. (1929). Über die quantenmechanik der elektronen in kristallgittern. Zeitschrift für physik, 52(7), 555-600.</ref> Mathematically, they are written<ref>Template:Cite book</ref> Template:Equation box 1 u(\mathbf{r})</math> |cellpadding |border |border colour = rgb(80,200,120) |background colour = rgb(80,200,120,10%)}} 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 consequencesEdit
ApplicabilityEdit
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 vectorEdit
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 Template:Math 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 Template:Math or Template:Math directly. This is important because Template:Math and Template:Math are not unique. Specifically, if <math>\psi</math> can be written as above using Template:Math, it can also be written using Template:Math, where Template:Math 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 Template:Math with the property that no two of them are equivalent, yet every possible Template:Math is equivalent to one (and only one) vector in the first Brillouin zone. Therefore, if we restrict Template:Math to the first Brillouin zone, then every Bloch state has a unique Template:Math. 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 Template:Math 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 Template:Math; for more details see crystal momentum.
Detailed exampleEdit
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).
StatementEdit
Template:Math theorem 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<ref name="ziman:1">Template:Cite book</ref>
Template:Math theorem(\mathbf{x}+\mathbf{a}) = e^{i\mathbf{k}\cdot\mathbf{a}}\psi_{\mathbf{k}}(\mathbf{x}).</math> }}
ProofEdit
Using lattice periodicityEdit
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 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 Template:Mvar 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 Template:Math, where Template:Mvar 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 operatorsEdit
In this proof all the symmetries are encoded as commutation properties of the translation operators Template:Math proof_{\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 <math display="block"> \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 an 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 <math display="block"> \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 <math display="block"> \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 <math display="block"> 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, <math display="block"> \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 theoryEdit
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.<ref name="Dresselhaus2002"/>Template:Rp<ref>The vibrational spectrum and specific heat of a face centered cubic crystal, Robert B. Leighton [1]</ref> 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 = \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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
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>: <math display="block">e^{i k_1 n_1 a_1} = e^{i k_1 ( n_1 a_1 + L_1)}</math> and from the last equation we get for each dimension a periodic condition: <math display="block"> k_1 n_1 a_1 = k_1 ( n_1 a_1 + L_1) - 2 \pi m_1</math> 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.<ref>Group Representations and Harmonic Analysis from Euler to Langlands, Part II [2]</ref>
Velocity and effective massEdit
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 Template:Equation box 1 = \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> |cellpadding |border |border colour = rgb(80,200,120) |background colour = rgb(80,200,120,10%)}} Template:Math proof</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 Template:Math where Template:Math is considered small with respect to Template:Math <math display="block"> \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: <math display="block"> \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 <math display="block"> 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 Template:Math <math display="block"> \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 Template:Math to obtain <math display="block"> \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 <math display="block"> \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 Template:Equation box 1 Template:Math proof^{*} \frac{\hbar^2}{m} \mathbf{q} \cdot (-i\nabla + \mathbf{k}) u_{n'\mathbf{k}} |^2} {\varepsilon_{n\mathbf{k}} - \varepsilon_{n'\mathbf{k}}} </math> Again with <math> \psi_{n\mathbf{k}} =| n\mathbf{k}\rangle = e^{i\mathbf{k}\mathbf{x}} u_{n\mathbf{k}}</math> <math display="block"> \frac{1}{2} \sum_{ij} \frac{\partial^2 \varepsilon_n}{\partial k_i \partial k_j} q_i q_j = \frac {\hbar^2}{2m} q^2 + \sum_{n' \neq n} \frac{| \langle n\mathbf{k} | \frac{\hbar^2}{m} \mathbf{q} \cdot (-i\nabla) | n'\mathbf{k}\rangle |^2} {\varepsilon_{n\mathbf{k}} - \varepsilon_{n'\mathbf{k}}} </math> Eliminating <math>q_i</math> and <math>q_j</math> we have the theorem <math display="block"> \frac{\partial^2 \varepsilon_n(\mathbf{k})}{\partial k_i \partial k_j} = \frac {\hbar^2}{m} \delta_{ij} + \left( \frac {\hbar^2}{m} \right)^2 \sum_{n' \neq n} \frac{ \langle n\mathbf{k} | -i \nabla_i | n'\mathbf{k} \rangle \langle n'\mathbf{k} | -i \nabla_j | n\mathbf{k} \rangle + \langle n\mathbf{k} | -i \nabla_j | n'\mathbf{k} \rangle \langle n'\mathbf{k} | -i \nabla_i | n\mathbf{k} \rangle }{ \varepsilon_n(\mathbf{k}) - \varepsilon_{n'}(\mathbf{k}) } </math> }} The quantity on the right multiplied by a factor<math>\frac{1}{\hbar^2}</math> is called effective mass tensor <math>\mathbf{M}(\mathbf{k})</math><ref name=":5">Template:Harvnb</ref> and we can use it to write a semi-classical equation for a charge carrier in a band<ref name=":6">Template:Harvnb</ref> Template:Equation box 1 where <math>\mathbf{a}</math> is an acceleration. This equation is analogous to the de Broglie wave type of approximation<ref name=":7">Template:Harvnb</ref> 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.
Edit
The concept of the Bloch state was developed by Felix Bloch in 1928<ref>Template:Cite journal</ref> 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),<ref>Template:Cite journal This work was initially published and distributed privately in 1877.</ref> Gaston Floquet (1883),<ref>Template:Cite journal</ref> and Alexander Lyapunov (1892).<ref>Template:Cite book Translated by A. T. Fuller from Edouard Davaux's French translation (1907) of the original Russian dissertation (1892).</ref> 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:<ref name=Magnus_Winkler> Template:Cite book </ref> <math display="block">\frac {d^2y}{dt^2}+f(t) y=0, </math> where Template:Math 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.<ref>Kuchment, P.(1982), Floquet theory for partial differential equations, RUSS MATH SURV., 37, 1–60</ref><ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>
See alsoEdit
- 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
ReferencesEdit
Further readingEdit
- Template:Cite book
- Template:Cite book
- {{#invoke:citation/CS1|citation
|CitationClass=web }}
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