Bloch wave

A Bloch wave (also called Bloch state or Bloch function or Bloch wavefunction), named after Swiss physicist Felix Bloch, is a wave function which can be written as a plane wave modulated by a periodic function. By definition, a Bloch wave can be described mathematically with a wave function that can be written in the form:[1]

Bloch Wave

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

Isosurface of the square modulus of a Bloch wave in silicon lattice

Solid line: A schematic of a typical Bloch wave in one dimension. (The actual wave is complex; this is the real part.) The dotted line is from the e^ik·r factor. The light circles represent atoms.

where ${\displaystyle \mathbf {r} }$ is position, ${\displaystyle \psi }$ is the Bloch wave, ${\displaystyle u}$ 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.

Bloch waves are important in solid-state physics, where they are often used to describe an electron in a crystal. This application is motivated by Bloch's theorem, which states that the energy eigenstates for an electron in a crystal can be written as Bloch waves (more precisely, it states that the electron wave functions in a crystal have a basis consisting entirely of Bloch wave energy eigenstates). This fact underlies the concept of electronic band structures.

These Bloch wave energy 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 Bloch waves 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, a Bloch wave, ${\displaystyle \psi _{n\mathbf {k} }}$, 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, Bloch waves 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 (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 wave broken up in two different ways, involving the wave vector k₁ (left) or k₂ (right). The difference (k₁−k₂) 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 wavefunctions with the properties:

• Each of these wavefunctions is an energy eigenstate
• Each of these wavefunctions is a Bloch wave, meaning that this wavefunction ${\displaystyle \psi }$ 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.

${\displaystyle u_{\mathbf {k} }(\mathbf {x} )=u_{\mathbf {k} }(\mathbf {x} +\mathbf {n} \cdot \mathbf {a} )}$

Proof of theorem

Proof with lattice periodicity[note 1]

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

${\displaystyle n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3}}$

where n_i 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 a_i · b_i = 2π, but a_i · b_j = 0 when i ≠ j. (For the formula for b_i, see reciprocal lattice vector.)

Lemma about translation operators

Let ${\displaystyle {\hat {T}}_{n_{1},n_{2},n_{3}}\!}$ denote a translation operator that shifts every wave function by the amount n₁a₁ + n₂a₂ + n₃a₃ (as above, n_j are integers). The following fact is helpful for the proof of Bloch's theorem:

Lemma: If a wavefunction ${\displaystyle \psi }$ is an eigenstate of all of the translation operators (simultaneously), then ${\displaystyle \psi }$ is a Bloch wave.

Proof: Assume that we have a wavefunction ${\displaystyle \psi }$ which is an eigenstate of all the translation operators. As a special case of this,

${\displaystyle \psi (\mathbf {r} +\mathbf {a} _{j})=C_{j}\psi (\mathbf {r} )}$

for j = 1, 2, 3, where C_j are three numbers (the eigenvalues) which do not depend on r. It is helpful to write the numbers C_j in a different form, by choosing three numbers θ₁, θ₂, θ₃ with e^2πiθ_j = C_j:

${\displaystyle \psi (\mathbf {r} +\mathbf {a} _{j})=\mathrm {e} ^{2\pi \mathrm {i} \theta _{j}}\psi (\mathbf {r} )}$

Again, the θ_j are three numbers which do not depend on r. Define k = θ₁b₁ + θ₂b₂ + θ₃b₃, where b_j are the reciprocal lattice vectors (see above). Finally, define

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

Then

${\displaystyle u(\mathbf {r} +\mathbf {a} _{j})=\mathrm {e} ^{-\mathrm {i} \mathbf {k} \cdot (\mathbf {r} +\mathbf {a} _{j})}\psi (\mathbf {r} +\mathbf {a} _{j})={\big (}\mathrm {e} ^{-\mathrm {i} \mathbf {k} \cdot \mathbf {r} }\mathrm {e} ^{-\mathrm {i} \mathbf {k} \cdot \mathbf {a} _{j}}{\big )}{\big (}\mathrm {e} ^{2\pi \mathrm {i} \theta _{j}}\psi (\mathbf {r} ){\big )}=\mathrm {e} ^{-\mathrm {i} \mathbf {k} \cdot \mathbf {r} }\mathrm {e} ^{-2\pi \mathrm {i} \theta _{j}}\mathrm {e} ^{2\pi \mathrm {i} \theta _{j}}\psi (\mathbf {r} )=u(\mathbf {r} )}$.

This proves that u has the periodicity of the lattice. Since ${\displaystyle \psi (\mathbf {r} )=\mathrm {e} ^{\mathrm {i} \mathbf {k} \cdot \mathbf {r} }u(\mathbf {r} )}$, that proves that the state is a Bloch wave.

Proof

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

As above, let ${\displaystyle {\hat {T}}_{n_{1},n_{2},n_{3}}\!}$ denote a translation operator that shifts every wave function by the amount n₁a₁ + n₂a₂ + n₃a₃, where n_i 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 ${\displaystyle {\hat {T}}_{n_{1},n_{2},n_{3}}\!}$ operator. This basis is what we are looking for. The wavefunctions in this basis are energy eigenstates (because they are eigenstates of the Hamiltonian), and they are also Bloch waves (because they are eigenstates of the translation operators; see Lemma above).

Another proof

Proof with operators [note 2]

We define the translation operator

${\displaystyle {\hat {\mathbf {T} }}_{\mathbf {n} }\psi (\mathbf {r} )=\psi (\mathbf {r} +\mathbf {T} _{\mathbf {n} })=\psi (\mathbf {r} +n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3})=\psi (\mathbf {r} +\mathbf {n} \cdot \mathbf {a} )}$

We use the hyphotesis of a mean periodic potential

${\displaystyle U(\mathbf {x} +\mathbf {T} _{\mathbf {n} })=U(\mathbf {x} )}$

and the independent electron approximation with an hamiltonian

${\displaystyle {\hat {H}}={\frac {{\hat {\mathbf {p} }}^{2}}{2m}}+U(\mathbf {x} )}$

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

${\displaystyle [{\hat {H}},{\hat {\mathbf {T} }}_{\mathbf {n} }]=0}$

and the two operators shall have a common set of eigenfunctions. Therefore we start to look at the eigen-functions of the translation operator:

${\displaystyle {\hat {\mathbf {T} }}_{\mathbf {n} }\psi (\mathbf {x} )=\lambda _{\mathbf {n} }\psi (\mathbf {x} )}$

Given ${\displaystyle {\hat {\mathbf {T} }}_{\mathbf {n} }}$ is an additive operator

${\displaystyle {\hat {\mathbf {T} }}_{\mathbf {n_{1}} }{\hat {\mathbf {T} }}_{\mathbf {n_{2}} }\psi (\mathbf {x} )=\psi (\mathbf {x} +\mathbf {n_{1}} \cdot \mathbf {a} +\mathbf {n_{2}} \cdot \mathbf {a} )={\hat {\mathbf {T} }}_{\mathbf {n_{1}} +\mathbf {n_{2}} }\psi (\mathbf {x} )}$

If we substitute here the eigenvalue equation and diving both sides for ${\displaystyle \psi (\mathbf {x} )}$ we have

${\displaystyle \lambda _{\mathbf {n_{1}} }\lambda _{\mathbf {n_{2}} }=\lambda _{\mathbf {n_{1}} +\mathbf {n_{2}} }}$

This is true for

${\displaystyle \lambda _{\mathbf {n} }=e^{s\mathbf {n} \cdot \mathbf {a} }}$

where ${\displaystyle s\in \mathbb {C} }$

if we use the normalization condition over a single primitive cell of volume V

${\displaystyle 1=\int _{V}|\psi (\mathbf {x} )|^{2}d\mathbf {x} =\int _{V}|{\hat {\mathbf {T_{n}} }}\psi (\mathbf {x} )|^{2}d\mathbf {x} =|\lambda _{\mathbf {n} }|^{2}\int _{V}|\psi (\mathbf {x} )|^{2}d\mathbf {x} }$

and therefore

${\displaystyle 1=|\lambda _{\mathbf {n} }|^{2}}$ and ${\displaystyle s=ik}$ where ${\displaystyle k\in \mathbb {R} }$

Finally

${\displaystyle {\hat {\mathbf {T_{n}} }}\psi (\mathbf {x} )=\psi (\mathbf {x} +\mathbf {n} \cdot \mathbf {a} )=e^{ik\mathbf {n} \cdot \mathbf {a} }\psi (\mathbf {x} )}$

Which is true for a block wave i.e for ${\displaystyle \psi _{\mathbf {k} }(\mathbf {x} )=e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )}$ with ${\displaystyle u_{\mathbf {k} }(\mathbf {x} )=u_{\mathbf {k} }(\mathbf {x} +\mathbf {n} \cdot \mathbf {a} )}$

Group theory proof

Proof with Character Theory [note 3]

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

${\displaystyle {\vec {\mathbf {\tau } }}=\sum _{i=1}^{3}n_{i}{\vec {\mathbf {a} }}_{i}}$

We can think of these as commuting operators

${\displaystyle {\hat {\vec {\mathbf {\tau } }}}={\hat {\vec {\mathbf {\tau } }}}_{1}{\hat {\vec {\mathbf {\tau } }}}_{2}{\hat {\vec {\mathbf {\tau } }}}_{3}}$ where ${\displaystyle {\hat {\vec {\mathbf {\tau } }}}_{i}=n_{i}{\hat {\vec {\mathbf {a} }}}_{i}}$

The commutativity of the ${\displaystyle {\hat {\vec {\mathbf {\tau } }}}_{i}}$ 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 [2]

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 ${\displaystyle \gamma }$ which shall obey to ${\displaystyle \gamma ^{n}=1}$, and therefore the character ${\displaystyle \chi (\gamma )^{n}=1}$. 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 ${\displaystyle n\to \infty }$ where the character remains finite.

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

${\displaystyle \chi _{k_{1}}({\hat {\vec {\tau }}}_{1}(n_{1},a_{1}))=e^{ik_{1}n_{1}a_{1}}}$

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

${\displaystyle V(\mathbf {r} +\sum N_{i}\mathbf {a} _{i})=V(\mathbf {r} +\mathbf {L} )=V(\mathbf {r} )}$

Where L is a macroscopic periodicity in the direction ${\displaystyle {\vec {a}}}$ that can also be seen as a multiple of ${\displaystyle a_{i}}$ where ${\displaystyle \mathbf {L} =\sum N_{i}\mathbf {a} _{i}}$

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

${\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V({\vec {r}})}$

induces a periodicity with the wave function:

${\displaystyle \psi (\mathbf {r} +\sum N_{i}\mathbf {a} _{i})=\psi (\mathbf {r} )}$

And for each dimension a translation operator with a period L

${\displaystyle {\hat {P}}_{\varepsilon |\tau _{i}+L_{i}}={\hat {P}}_{\varepsilon |\tau _{i}}}$

From here we can see that also the character shall be invariant by a translation of ${\displaystyle L_{i}}$:

${\displaystyle e^{ik_{1}n_{1}a_{1}}=e^{ik_{1}(n_{1}a_{1}+L_{1})}}$

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

${\displaystyle k_{1}n_{1}a_{1}=k_{1}(n_{1}a_{1}+L_{1})-2\pi m_{1}}$

where ${\displaystyle m_{1}\in \mathbb {Z} }$ is an integer and ${\displaystyle k_{1}={\frac {2\pi m_{1}}{L_{1}}}}$

The wave vector ${\displaystyle k_{1}}$ identify the irreducible representation in the same manner as ${\displaystyle m_{1}}$, and ${\displaystyle L_{1}}$ is a macroscopic periodic length of the crystal in direction ${\displaystyle a_{1}}$. In this context, the wave vector serves as a quantum number for the translation operator.

We can generalize this for 3 dimensions ${\displaystyle \chi _{k_{1}}(n_{1},a_{1})\chi _{k_{2}}(n_{2},a_{2})\chi _{k_{3}}(n_{3},a_{3})=e^{i{\vec {k}}\cdot {\vec {\tau }}}}$

and the generic formula for the wave function becomes:

${\displaystyle {\hat {P}}_{R}\psi _{j}=\sum _{\alpha }\psi _{\alpha }\chi _{\alpha j}(R)}$

i.e. specializing it for a translation

${\displaystyle {\hat {P}}_{\varepsilon |{\vec {\tau }}}\psi ({\vec {\mathbf {r} }})=\psi ({\vec {\mathbf {r} }})e^{i{\vec {k}}\cdot {\vec {\tau }}}=\psi ({\vec {\mathbf {r} }}+{\vec {\mathbf {\tau } }})}$

and we have proven Bloch’s theorem.

A part 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[note 4][3].

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[4].

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

Proof [note 5]

${\displaystyle [{\frac {-\hbar ^{2}}{2m}}\nabla ^{2}+U(\mathbf {x} )]\left(e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )\right)=E_{\mathbf {k} }\left(e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )\right)}$

We remain with

${\displaystyle {\frac {-\hbar ^{2}}{2m}}\nabla \cdot \left(i\mathbf {k} e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )+e^{i\mathbf {k} \cdot \mathbf {x} }\nabla u_{\mathbf {k} }(\mathbf {x} )\right)+U(\mathbf {x} )e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )=E_{\mathbf {k} }e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )}$

${\displaystyle {\frac {-\hbar ^{2}}{2m}}\left(i\mathbf {k} \cdot \left(i\mathbf {k} e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )+e^{i\mathbf {k} \cdot \mathbf {x} }\nabla u_{\mathbf {k} }(\mathbf {x} )\right)+i\mathbf {k} e^{i\mathbf {k} \cdot \mathbf {x} }\nabla u_{\mathbf {k} }(\mathbf {x} )+e^{i\mathbf {k} \cdot \mathbf {x} }\nabla ^{2}u_{\mathbf {k} }(\mathbf {x} )\right)+U(\mathbf {x} )e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )=E_{\mathbf {k} }e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )}$

${\displaystyle {\frac {\hbar ^{2}}{2m}}\left(\mathbf {k} ^{2}e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )-2i\mathbf {k} e^{i\mathbf {k} \cdot \mathbf {x} }\nabla u_{\mathbf {k} }(\mathbf {x} )-e^{i\mathbf {k} \cdot \mathbf {x} }\nabla ^{2}u_{\mathbf {k} }(\mathbf {x} )\right)+U(\mathbf {x} )e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )=E_{\mathbf {k} }e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )}$

${\displaystyle {\frac {\hbar ^{2}}{2m}}\left(-i\nabla +\mathbf {k} \right)^{2}u_{\mathbf {k} }(\mathbf {x} )+U(\mathbf {x} )u_{\mathbf {k} }(\mathbf {x} )=E_{\mathbf {k} }u_{\mathbf {k} }(\mathbf {x} )}$

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

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

Proof [note 6]

We evaluate the derivatives ${\displaystyle {\frac {\partial \varepsilon _{n}}{\partial \mathbf {k} }}}$ and ${\displaystyle {\frac {\partial ^{2}\varepsilon _{n}(\mathbf {k} )}{\partial k_{i}\partial k_{j}}}}$ given they are the coefficients of the following expansion in q where q is considered small with respect to k

${\displaystyle \varepsilon _{n}(\mathbf {k} +\mathbf {q} )=\varepsilon _{n}(\mathbf {k} )+\sum _{i}{\frac {\partial \varepsilon _{n}}{\partial k_{i}}}q_{i}+{\frac {1}{2}}\sum _{ij}{\frac {\partial ^{2}\varepsilon _{n}}{\partial k_{i}\partial k_{j}}}+O(q^{3})}$

Given ${\displaystyle \varepsilon _{n}(\mathbf {k} +\mathbf {q} )}$ are eigenvalues of ${\displaystyle {\hat {H}}_{\mathbf {k} +\mathbf {q} }}$ We can consider the following perturbation problem in q:

${\displaystyle {\hat {H}}_{\mathbf {k} +\mathbf {q} }={\hat {H}}_{\mathbf {k} }+{\frac {\hbar ^{2}}{m}}\mathbf {q} \cdot (-i\nabla +\mathbf {k} )+{\frac {\hbar ^{2}}{2m}}q^{2}}$

Perturbation theory of the second order tells that:

${\displaystyle E_{n}=E_{n}^{0}+\int d\mathbf {r} \psi _{n}^{*}{\hat {V}}\psi _{n}+\sum _{n'\neq n}{\frac {|\int d\mathbf {r} \psi _{n}^{*}{\hat {V}}\psi _{n}|^{2}}{E_{n}^{0}-E_{n'}^{0}}}+...}$

To compute to linear order in q

${\displaystyle \sum _{i}{\frac {\partial \varepsilon _{n}}{\partial k_{i}}}q_{i}=\sum _{i}\int d\mathbf {r} u_{n\mathbf {k} }^{*}{\frac {\hbar ^{2}}{m}}(-i\nabla +\mathbf {k} )_{i}q_{i}u_{n\mathbf {k} }}$

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

${\displaystyle \int d\mathbf {r} u_{n\mathbf {k} }^{*}u_{n\mathbf {k} }}$

is normalized across the cell or the crystal.

We can simplify over q and remain with

${\displaystyle {\frac {\partial \varepsilon _{n}}{\partial \mathbf {k} }}={\frac {\hbar ^{2}}{m}}\int d\mathbf {r} u_{n\mathbf {k} }^{*}(-i\nabla +\mathbf {k} )u_{n\mathbf {k} }}$

And we can reinsert the complete wave functions

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

And for the effective mass

effective mass theorem

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

Proof [note 6]

The second order term

${\displaystyle {\frac {1}{2}}\sum _{ij}{\frac {\partial ^{2}\varepsilon _{n}}{\partial k_{i}\partial k_{j}}}q_{i}q_{j}={\frac {\hbar ^{2}}{2m}}q^{2}+\sum _{n'\neq n}{\frac {|\int d\mathbf {r} u_{n\mathbf {k} }^{*}{\frac {\hbar ^{2}}{m}}\mathbf {q} \cdot (-i\nabla +\mathbf {k} )u_{n'\mathbf {k} }|^{2}}{\varepsilon _{n\mathbf {k} }-\varepsilon _{n'\mathbf {k} }}}}$

Again with ${\displaystyle \psi _{n\mathbf {k} }=|n\mathbf {k} \rangle =e^{i\mathbf {k} \mathbf {x} }u_{n\mathbf {k} }}$

${\displaystyle {\frac {1}{2}}\sum _{ij}{\frac {\partial ^{2}\varepsilon _{n}}{\partial k_{i}\partial k_{j}}}q_{i}q_{j}={\frac {\hbar ^{2}}{2m}}q^{2}+\sum _{n'\neq n}{\frac {|\langle n\mathbf {k} |{\frac {\hbar ^{2}}{m}}\mathbf {q} \cdot (-i\nabla )|n'\mathbf {k} \rangle |^{2}}{\varepsilon _{n\mathbf {k} }-\varepsilon _{n'\mathbf {k} }}}}$

And getting rid of ${\displaystyle q_{i}}$ and ${\displaystyle q_{j}}$ we have the 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 ${\displaystyle \mathbf {M} (\mathbf {k} )}$[note 7] and we can use it to write a semi-classical equation for a charge carrier in a band[note 8]

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

${\displaystyle \mathbf {M} (\mathbf {k} )\mathbf {a} =\mp e(\mathbf {E} +{\frac {1}{c}}\mathbf {v} (\mathbf {k} )\times \mathbf {H} )}$

In close analogy with the De Broglie wave type of approximation[note 9]

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

${\displaystyle \hbar {\dot {k}}=-e(\mathbf {E} +{\frac {1}{c}}\mathbf {v} \times \mathbf {H} )}$

History and related equations

The concept of the Bloch state was developed by Felix Bloch in 1928,[5] 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),[6] Gaston Floquet (1883),[7] and Alexander Lyapunov (1892).[8] 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:[9]

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

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.[10][11][12]

See also

• 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

References

1. Kittel, Charles (1996). Introduction to Solid State Physics. New York: Wiley. ISBN 0-471-14286-7.
2. Representation Theory & Rick Roy 2010 harvnb error: no target: CITEREFRepresentation_TheoryRick_Roy2010 (help)
3. The vibrational spectrum and specific heat of a face centered cubic crystal, Robert B. Leighton
4. Group Representations and Harmonic Analysis from Euler to Langlands, Part II
5. 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.
6. 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.
7. 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.
8. 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).
9. Magnus, W; Winkler, S (2004). Hill's Equation. Courier Dover. p. 11. ISBN 0-486-49565-5.
10. Kuchment, P.(1982), Floquet theory for partial differential equations, RUSS MATH SURV., 37,1-60
11. 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.
12. 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.

Citations

1. Ashcroft & Mermin 1976, pp. 134
2. Ashcroft & Mermin 1976, pp. 137
3. Dresselhaus 2002, pp. 345-348 harvnb error: no target: CITEREFDresselhaus2002 (help)
4. Dresselhaus 2002, pp. 365-367 harvnb error: no target: CITEREFDresselhaus2002 (help)
5. Ashcroft & Mermin 1976, pp. 140
6. Ashcroft & Mermin 1976, pp. 765 Appendix E
7. Ashcroft & Mermin 1976, pp. 228
8. Ashcroft & Mermin 1976, pp. 229
9. Ashcroft & Mermin 1976, pp. 227

General

• Ashcroft, Neil; Mermin, N. David (1976). Solid State Physics. New York: Holt, Rinehart and Winston. ISBN 978-0-03-083993-1.CS1 maint: ref=harv (link)

Further reading

• Neil W. Ashcroft & N. David Mermin (1976). Solid State Physics. Orlando: Harcourt. ISBN 0-03-049346-3.
• 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.
• Nick Thompson, "Bloch's Theorem via Representation Theory".