Bloch's theorem

Summary

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]

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.
Bloch function

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 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

 
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

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

Bloch's theoremEdit

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 u has the same periodicity as the atomic structure of the crystal.
 

Proof of theoremEdit

Proof with lattice periodicity[2]Edit

Preliminaries: Crystal symmetries, lattice, and reciprocal latticeEdit

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

 
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 operatorsEdit

Let   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   is an eigenstate of all of the translation operators (simultaneously), then   is a Bloch state.

Proof of Lemma

Assume that we have a wave function   which is an eigenstate of all the translation operators. As a special case of this,

 
for j = 1, 2, 3, where Cj are three numbers (the eigenvalues) which do not depend on r. It is helpful to write the numbers Cj in a different form, by choosing three numbers θ1, θ2, θ3 with e2πiθj = Cj:
 
Again, the θj are three numbers which do not depend on r. Define k = θ1b1 + θ2b2 + θ3b3, where bj are the reciprocal lattice vectors (see above). Finally, define
 
Then
 
This proves that u has the periodicity of the lattice. Since  , that proves that the state is a Bloch state.
ProofEdit

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 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   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).

Proof with operators[3]Edit

We define the translation operator

 
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  

Group theory proofEdit

Proof with Character Theory[4]

All Translations are unitary and Abelian. Translations can be written in terms of unit vectors

 
We can think of these as commuting operators
 
where
 

The commutativity of the   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.[5]

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   which shall obey to  , and therefore the character  . 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   where the character remains finite.

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

 

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

 
where L is a macroscopic periodicity in the direction   that can also be seen as a multiple of   where  

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

 
induces a periodicity with the wave function:
 

And for each dimension a translation operator with a period L

 

From here we can see that also the character shall be invariant by a translation of  :

 
and from the last equation we get for each dimension a periodic condition:
 
where   is an integer and  

The wave vector   identify the irreducible representation in the same manner as  , and   is a macroscopic periodic length of the crystal in direction  . In this context, the wave vector serves as a quantum number for the translation operator.

We can generalize this for 3 dimensions   and the generic formula for the wave function becomes:

 
i.e. specializing it for a translation
 
and we have proven Bloch’s theorem.

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][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 of Bloch electronsEdit

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.
Proof[9]

 

We remain with

 
 
 
 

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

mean velocity of a bloch electron

 

Proof[10]

We evaluate the derivatives   and   given they are the coefficients of the following expansion in q where q is considered small with respect to k

 
Given   are eigenvalues of   We can consider the following perturbation problem in q:
 
Perturbation theory of the second order states that
 
To compute to linear order in q
 
where the integrations are over a primitive cell or the entire crystal, given if the integral
 
is normalized across the cell or the crystal.

We can simplify over q to obtain

 
and we can reinsert the complete wave functions
 

For the effective mass

effective mass theorem

 

Proof[10]

The second order term

 
Again with  
 
Eliminating   and   we have the theorem
 

The quantity on the right multiplied by a factor  is called effective mass tensor  [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

 

where   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

 

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 equationsEdit

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]

 

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 alsoEdit

ReferencesEdit

  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.

Further readingEdit

  • Ashcroft, Neil; Mermin, N. David (1976). Solid State Physics. New York: Holt, Rinehart and Winston. ISBN 978-0-03-083993-1.
  • Dresselhaus, M. S. (2002). "Applications of Group Theory to the Physics of Solids" (PDF). MIT. Archived (PDF) from the original on 1 November 2019. Retrieved 12 September 2020.
  • Dresselhaus, M. S. (2010). Group theory: application to the physics of condensed matter. Springer-Verlag. ISBN 978-3-642-06945-1. OCLC 692760083.
  • H. Föll. "Periodic Potentials and Bloch's Theorem – lectures in "Semiconductors I"". The University of Kiel.
  • M.S.P. Eastham (1973). The Spectral Theory of Periodic Differential Equations. Texts in Mathematics. Edinburgh: Scottish Academic Press.
  • J. Gazalet; S. Dupont; J.C. Kastelik; Q. Rolland & B. Djafari-Rouhani (2013). "A tutorial survey on waves propagating in periodic media: Electronic, photonic and phononic crystals. Perception of the Bloch theorem in both real and Fourier domains". Wave Motion. 50 (3): 619–654. doi:10.1016/j.wavemoti.2012.12.010.