Physics:Bloch's theorem

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Short description: Fundamental theorem in condensed matter physics
Isosurface of the square modulus of a Bloch state in a silicon lattice
Solid line: A schematic of the real part of a typical Bloch state in one dimension. The dotted line is from the factor eik·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 physicist Felix Bloch, who discovered the theorem in 1929.[1] Mathematically, they are written[2]

Bloch function

[math]\displaystyle{ \psi(\mathbf{r}) = e^{i \mathbf{k}\cdot\mathbf{r}} u(\mathbf{r}) }[/math]

where [math]\displaystyle{ \mathbf{r} }[/math] is position, [math]\displaystyle{ \psi }[/math] is the wave function, [math]\displaystyle{ u }[/math] is a periodic function with the same periodicity as the crystal, the wave vector [math]\displaystyle{ \mathbf{k} }[/math] is the crystal momentum vector, [math]\displaystyle{ e }[/math] is Euler's number, and [math]\displaystyle{ 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.

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 [math]\displaystyle{ \psi_{n\mathbf{k}} }[/math], where [math]\displaystyle{ n }[/math] is a discrete index, called the band index, which is present because there are many different wave functions with the same [math]\displaystyle{ \mathbf{k} }[/math] (each has a different periodic component [math]\displaystyle{ u }[/math]). Within a band (i.e., for fixed [math]\displaystyle{ n }[/math]), [math]\displaystyle{ \psi_{n\mathbf{k}} }[/math] varies continuously with [math]\displaystyle{ \mathbf{k} }[/math], as does its energy. Also, [math]\displaystyle{ \psi_{n\mathbf{k}} }[/math] is unique only up to a constant reciprocal lattice vector [math]\displaystyle{ \mathbf{K} }[/math], or, [math]\displaystyle{ \psi_{n\mathbf{k}}=\psi_{n(\mathbf{k+K})} }[/math]. Therefore, the wave vector [math]\displaystyle{ \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

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 (k1k2) 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]\displaystyle{ \psi ( \mathbf{r} ) = e^{ i \mathbf{k} \cdot \mathbf{r} } u ( \mathbf{r} ) , }[/math] where u is periodic with the same periodicity as the crystal lattice. The actual quantum state of the electron is entirely determined by [math]\displaystyle{ \psi }[/math], not k or u directly. This is important because k and u are not unique. Specifically, if [math]\displaystyle{ \psi }[/math] 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'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 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).

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 [math]\displaystyle{ \psi }[/math] can be written in the form [math]\displaystyle{ \psi(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}} u(\mathbf{r}), }[/math] where u(r) has the same periodicity as the atomic structure of the crystal, such that [math]\displaystyle{ u_{\mathbf{k}}(\mathbf{x}) = u_{\mathbf{k}}(\mathbf{x} + \mathbf{n} \cdot \mathbf{a}). }[/math]

Proof

Using lattice periodicity

Source:[3]

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 a1, a2, a3. If the crystal is shifted by any of these three vectors, or a combination of them of the form [math]\displaystyle{ n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3, }[/math] 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 b1, b2, b3 (with units of inverse length), with the property that ai · bi = 2π, but ai · bj = 0 when ij. (For the formula for bi, see reciprocal lattice vector.)

Lemma about translation operators

Let [math]\displaystyle{ \hat{T}_{n_1,n_2,n_3} }[/math] denote a translation operator that shifts every wave function by the amount n1a1 + n2a2 + n3a3 (as above, nj are integers). The following fact is helpful for the proof of Bloch's theorem:

Lemma — If a wave function ψ is an eigenstate of all of the translation operators (simultaneously), then ψ is a Bloch state.

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

As above, let [math]\displaystyle{ \hat{T}_{n_1,n_2,n_3} }[/math] denote a translation operator that shifts every wave function by the amount n1a1 + n2a2 + n3a3, where ni 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]\displaystyle{ \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

Source:[4]

We define the translation operator [math]\displaystyle{ \begin{align} \hat{\mathbf{T}}_{\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]\displaystyle{ \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]\displaystyle{ U(\mathbf{x}+\mathbf{T}_{\mathbf{n}})= U(\mathbf{x}) }[/math] and the independent electron approximation with an Hamiltonian [math]\displaystyle{ \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]\displaystyle{ [\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]\displaystyle{ \hat{\mathbf{T}}_{\mathbf{n}}\psi(\mathbf{x})=\lambda_{\mathbf{n}}\psi(\mathbf{x}) }[/math] Given [math]\displaystyle{ \hat{\mathbf{T}}_{\mathbf{n}} }[/math] is an additive operator [math]\displaystyle{ \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]\displaystyle{ \psi(\mathbf{x}) }[/math] we have [math]\displaystyle{ \lambda_{\mathbf{n}_1} \lambda_{\mathbf{n}_2} = \lambda_{\mathbf{n}_1 + \mathbf{n}_2} }[/math]

This is true for [math]\displaystyle{ \lambda_{\mathbf{n}} = e^{s \mathbf{n} \cdot \mathbf{a} } }[/math] where [math]\displaystyle{ s \in \Complex }[/math]

if we use the normalization condition over a single primitive cell of volume V [math]\displaystyle{ 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]\displaystyle{ 1 = |\lambda_{\mathbf{n}}|^2 }[/math] and [math]\displaystyle{ s = i k }[/math] where [math]\displaystyle{ k \in \mathbb{R} }[/math]. Finally, [math]\displaystyle{ \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]\displaystyle{ \psi_{\mathbf{k}}(\mathbf{x}) = e^{i \mathbf{k} \cdot \mathbf{x} } u_{\mathbf{k}}(\mathbf{x}) }[/math] with [math]\displaystyle{ 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.[6]:{{{1}}}[7]

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.[8]

Velocity and effective mass

If we apply the time-independent Schrödinger equation to the Bloch wave function we obtain [math]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ {\mathbf{k}} }[/math] is a parameter of the Hamiltonian and therefore we arrive at a "continuous family" of eigenvalues [math]\displaystyle{ \varepsilon_n(\mathbf{k}) }[/math] dependent on the continuous parameter [math]\displaystyle{ {\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]\displaystyle{ \hat{\mathbf{p}}_\text{eff} = -i \hbar \nabla + \hbar \mathbf{k} , }[/math] a standard momentum [math]\displaystyle{ -i \hbar \nabla }[/math] and a crystal momentum [math]\displaystyle{ \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

mean velocity of a Bloch electron

[math]\displaystyle{ \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}} = \frac {\hbar}{m}\langle\hat{\mathbf{p}}\rangle = \hbar \langle\hat{\mathbf{v}}\rangle }[/math]

For the effective mass

effective mass theorem

[math]\displaystyle{ \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]\displaystyle{ \frac{1}{\hbar^2} }[/math] is called effective mass tensor [math]\displaystyle{ \mathbf{M}(\mathbf{k}) }[/math][11] and we can use it to write a semi-classical equation for a charge carrier in a band[12]

Second order semi-classical equation of motion for a charge carrier in a band

[math]\displaystyle{ \mathbf{M}(\mathbf{k}) \mathbf{a} = \mp e \left(\mathbf {E} + \mathbf{v}(\mathbf{k}) \times \mathbf{B}\right) }[/math]

where [math]\displaystyle{ \mathbf{a} }[/math] is an acceleration. This equation is analogous to the de Broglie wave type of approximation[13]

First order semi-classical equation of motion for electron in a band

[math]\displaystyle{ \hbar \dot{k} = - e \left(\mathbf {E} + \mathbf{v} \times \mathbf{B}\right) }[/math]

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[14] 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),[15] Gaston Floquet (1883),[16] and Alexander Lyapunov (1892).[17] 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:[18] [math]\displaystyle{ \frac {d^2y}{dt^2}+f(t) y=0, }[/math] where f(t) 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.[19][20][21]

See also


References

  1. Bloch, F. (1929). Über die quantenmechanik der elektronen in kristallgittern. Zeitschrift für physik, 52(7), 555-600.
  2. Kittel, Charles (1996). Introduction to Solid State Physics. New York: Wiley. ISBN 0-471-14286-7. 
  3. Ashcroft & Mermin 1976, p. 134
  4. Ashcroft & Mermin 1976, p. 137
  5. Roy, Ricky (May 2, 2010). "Representation Theory". University of Puget Sound. http://buzzard.pugetsound.edu/courses/2010spring/projects/roy-representation-theory-ups-434-2010.pdf. 
  6. Cite error: Invalid <ref> tag; no text was provided for refs named Dresselhaus2002
  7. The vibrational spectrum and specific heat of a face centered cubic crystal, Robert B. Leighton [1]
  8. Group Representations and Harmonic Analysis from Euler to Langlands, Part II [2]
  9. Ashcroft & Mermin 1976, p. 140
  10. 10.0 10.1 Ashcroft & Mermin 1976, p. 765 Appendix E
  11. Ashcroft & Mermin 1976, p. 228
  12. Ashcroft & Mermin 1976, p. 229
  13. Ashcroft & Mermin 1976, p. 227
  14. Felix Bloch (1928). "Über die Quantenmechanik der Elektronen in Kristallgittern" (in de). Zeitschrift für Physik 52 (7–8): 555–600. doi:10.1007/BF01339455. Bibcode1929ZPhy...52..555B. 
  15. George William Hill (1886). "On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon". Acta Math. 8: 1–36. doi:10.1007/BF02417081. https://zenodo.org/record/1691491.  This work was initially published and distributed privately in 1877.
  16. Gaston Floquet (1883). "Sur les équations différentielles linéaires à coefficients périodiques". Annales Scientifiques de l'École Normale Supérieure 12: 47–88. doi:10.24033/asens.220. 
  17. Alexander Mihailovich Lyapunov (1992). The General Problem of the Stability of Motion. London: Taylor and Francis.  Translated by A. T. Fuller from Edouard Davaux's French translation (1907) of the original Russian dissertation (1892).
  18. Hill's Equation. Courier Dover. 2004. p. 11. ISBN 0-486-49565-5. https://books.google.com/books?id=ML5wm-T4RVQC&q=%22hill's+equation%22. 
  19. Kuchment, P.(1982), Floquet theory for partial differential equations, RUSS MATH SURV., 37, 1–60
  20. Katsuda, A.; Sunada, T (1987). "Homology and closed geodesics in a compact Riemann surface". Amer. J. Math. 110 (1): 145–156. doi:10.2307/2374542. 
  21. Kotani M; Sunada T. (2000). "Albanese maps and an off diagonal long time asymptotic for the heat kernel". Comm. Math. Phys. 209 (3): 633–670. doi:10.1007/s002200050033. Bibcode2000CMaPh.209..633K. 

Further reading