# Bloch's theorem

(Redirected from Bloch wave)
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 eik·r factor. The light circles represent atoms.

In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. Mathematically, they are written:[1]

Bloch function

${\displaystyle \psi (\mathbf {r} )=\mathrm {e} ^{\mathrm {i} \mathbf {k} \cdot \mathbf {r} }u(\mathbf {r} )}$

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

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

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

${\displaystyle \psi (\mathbf {r} )=\mathrm {e} ^{\mathrm {i} \mathbf {k} \cdot \mathbf {r} }u(\mathbf {r} ),}$

where u is periodic with the same periodicity as the crystal lattice. The actual quantum state of the electron is entirely determined by ${\displaystyle \psi }$, not k or u directly. This is important because k and u are not unique. Specifically, if ${\displaystyle \psi }$ 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'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).

## Bloch's theorem

Here is the statement of Bloch's theorem:

For electrons in a perfect crystal, there is a basis of wave functions with the properties:
• Each of these wave functions is an energy eigenstate
• Each of these wave functions is a Bloch state, meaning that this wave function ${\displaystyle \psi }$ can be written in the form
${\displaystyle \psi (\mathbf {r} )=\mathrm {e} ^{\mathrm {i} \mathbf {k} \cdot \mathbf {r} }u(\mathbf {r} )}$
where u has the same periodicity as the atomic structure of the crystal.
${\displaystyle u_{\mathbf {k} }(\mathbf {x} )=u_{\mathbf {k} }(\mathbf {x} +\mathbf {n} \cdot \mathbf {a} )}$

## Velocity and effective mass of Bloch electrons

If we apply the time-independent Schrödinger equation to the Bloch wave function we obtain

${\displaystyle {\hat {H_{\mathbf {k} }}}u_{\mathbf {k} }(\mathbf {r} )=\left({\frac {\hbar ^{2}}{2m}}\left(-i\nabla +\mathbf {k} \right)^{2}+U(\mathbf {r} )\right)u_{\mathbf {k} }(\mathbf {r} )=\varepsilon _{\mathbf {k} }u_{\mathbf {k} }(\mathbf {r} )}$

with boundary conditions

${\displaystyle u_{\mathbf {k} }(\mathbf {r} )=u_{\mathbf {k} }(\mathbf {r} +\mathbf {R} )}$

Given this is defined in a finite volume we expect an infinite family of eigenvalues, here ${\displaystyle {\mathbf {k} }}$ is a parameter of the Hamiltonian and therefore we arrive to a "continuous family" of eigenvalues ${\displaystyle \varepsilon _{n}(\mathbf {k} )}$ dependent on the continuous parameter ${\displaystyle {\mathbf {k} }}$ and therefore to the basic concept of an electronic band structure

This shows how the effective momentum can be seen as composed by two parts

${\displaystyle {\hat {\mathbf {p} }}_{eff}=\left(-i\hbar \nabla +\hbar \mathbf {k} \right)}$

A standard momentum ${\displaystyle -i\hbar \nabla }$ and a crystal momentum ${\displaystyle \hbar \mathbf {k} }$. More precisely the crystal momentum is not a momentum but it stands to 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

${\displaystyle {\frac {\partial \varepsilon _{n}}{\partial \mathbf {k} }}={\frac {\hbar ^{2}}{m}}\int d\mathbf {r} \psi _{n\mathbf {k} }^{*}(-i\nabla )\psi _{n\mathbf {k} }={\frac {\hbar }{m}}\langle {\hat {\mathbf {p} }}\rangle =\hbar \langle {\hat {\mathbf {v} }}\rangle }$

And for the effective mass

effective mass theorem

${\displaystyle {\frac {\partial ^{2}\varepsilon _{n}(\mathbf {k} )}{\partial k_{i}\partial k_{j}}}={\frac {\hbar ^{2}}{m}}\delta _{ij}+\left({\frac {\hbar ^{2}}{m}}\right)^{2}\sum _{n'\neq n}{\frac {\langle n\mathbf {k} |-i\nabla _{i}|n'\mathbf {k} \rangle \langle n'\mathbf {k} |-i\nabla _{j}|n\mathbf {k} \rangle +\langle n\mathbf {k} |-i\nabla _{j}|n'\mathbf {k} \rangle \langle n'\mathbf {k} |-i\nabla _{i}|n\mathbf {k} \rangle }{\varepsilon _{n}(\mathbf {k} )-\varepsilon _{n'}(\mathbf {k} )}}}$

The quantity on the right multiplied by a factor${\displaystyle {\frac {1}{\hbar ^{2}}}}$ is called effective mass tensor ${\displaystyle \mathbf {M} (\mathbf {k} )}$[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

${\displaystyle \mathbf {M} (\mathbf {k} )\mathbf {a} =\mp e(\mathbf {E} +\mathbf {v} (\mathbf {k} )\times \mathbf {B} )}$

Where ${\displaystyle \mathbf {a} }$ is an acceleration. This equation is in close analogy with the De Broglie wave type of approximation[13]

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

${\displaystyle \hbar {\dot {k}}=-e(\mathbf {E} +\mathbf {v} \times \mathbf {B} )}$

As an intuitive interpretation, both last 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]

${\displaystyle {\frac {d^{2}y}{dt^{2}}}+f(t)y=0,}$

where f(t) is a periodic potential. Specific periodic one-dimensional equations include the Kronig–Penney model and Mathieu's equation.

Mathematically Bloch's theorem is interpreted in terms of unitary characters of a lattice group, and is applied to spectral geometry.[19][20][21]

## References

1. ^ Kittel, Charles (1996). Introduction to Solid State Physics. New York: Wiley. ISBN 0-471-14286-7.
2. ^ Ashcroft & Mermin 1976, p. 134
3. ^ Ashcroft & Mermin 1976, p. 137
4. ^ Dresselhaus 2002, pp. 345-348[1]
5. ^ Representation Theory & Rick Roy 2010[2]
6. ^ Dresselhaus 2002, pp. 365-367[3]
7. ^ The vibrational spectrum and specific heat of a face centered cubic crystal, Robert B. Leighton [4]
8. ^ Group Representations and Harmonic Analysis from Euler to Langlands, Part II [5]
9. ^ Ashcroft & Mermin 1976, p. 140
10. ^ a b 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". Zeitschrift für Physik (in German). 52 (7–8): 555–600. Bibcode:1929ZPhy...52..555B. doi:10.1007/BF01339455. S2CID 120668259.
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. 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. ^ Magnus, W; Winkler, S (2004). Hill's Equation. Courier Dover. p. 11. ISBN 0-486-49565-5.
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. JSTOR 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. Bibcode:2000CMaPh.209..633K. doi:10.1007/s002200050033. S2CID 121065949.