![Bloch_function](https://upload.wikimedia.org/wikipedia/commons/2/28/Bloch_function.svg)
In
condensed matter physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the sub ...
, Bloch's theorem states that solutions to the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
in a periodic potential take the form of a
plane wave
In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space.
For any position \vec x in space and any time t, th ...
modulated by a
periodic function
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to desc ...
. The theorem is named after the physicist
Felix Bloch
Felix Bloch (23 October 1905 – 10 September 1983) was a Swiss-American physicist and Nobel physics laureate who worked mainly in the U.S. He and Edward Mills Purcell were awarded the 1952 Nobel Prize for Physics for "their development of ne ...
, who discovered the theorem in 1929. Mathematically, they are written
where
is position,
is the
wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
,
is a
periodic function
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to desc ...
with the same periodicity as the crystal, the
wave vector
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
is the
crystal momentum vector,
is
Euler's number
The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of a logarithm, base of the natural logarithms. It is the Limit of a sequence, limit ...
, and
is the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
.
Functions of this form are known as Bloch functions or Bloch states, and serve as a suitable
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
for the
wave functions
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements m ...
or
states of electrons in
crystalline solids
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macrosc ...
.
Named after Swiss
physicist
A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe.
Physicists generally are interested in the root or ultimate caus ...
Felix Bloch
Felix Bloch (23 October 1905 – 10 September 1983) was a Swiss-American physicist and Nobel physics laureate who worked mainly in the U.S. He and Edward Mills Purcell were awarded the 1952 Nobel Prize for Physics for "their development of ne ...
, the description of electrons in terms of Bloch functions, termed Bloch electrons (or less often ''Bloch Waves''), underlies the concept of
electronic band structure
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or '' ...
s.
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
In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial fu ...
vector
, or,
. Therefore, the wave vector
can be restricted to the first
Brillouin zone
In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice i ...
of the reciprocal lattice
without loss of generality
''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicate ...
.
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
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or '' ...
. However, a Bloch-wave description applies more generally to any wave-like phenomenon in a periodic medium. For example, a periodic
dielectric
In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mate ...
structure in
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
leads to
photonic crystal
A photonic crystal is an optical nanostructure in which the refractive index changes periodically. This affects the propagation of light in the same way that the structure of Crystal structure, natural crystals gives rise to X-ray crystallograp ...
s, and a periodic acoustic medium leads to
phononic crystal
An acoustic metamaterial, sonic crystal, or phononic crystal, is a material designed to control, direct, and manipulate sound waves or phonons in gases, liquids, and solids (crystal lattices). Sound wave control is accomplished through manipulating ...
s. It is generally treated in the various forms of the
dynamical theory of diffraction
The dynamical theory of diffraction describes the interaction of waves with a regular lattice. The wave fields traditionally described are X-rays, neutrons or electrons and the regular lattice are atomic crystal structures or nanometer-scale multi ...
.
Wave vector
![BlochWaves1D](https://upload.wikimedia.org/wikipedia/commons/5/56/BlochWaves1D.svg)
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
In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial f ...
(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
In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice i ...
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
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or '' ...
, and it is used for the same reason in many calculations.
When k is multiplied by the
reduced Planck's constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
, it equals the electron's
crystal momentum
In solid-state physics crystal momentum or quasimomentum is a momentum-like vector associated with electrons in a crystal lattice. It is defined by the associated wave vectors \mathbf of this lattice, according to
:_ \equiv \hbar
(where \hba ...
. Related to this, the
group velocity
The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space.
For example, if a stone is thrown into the middl ...
of an electron can be calculated based on how the energy of a Bloch state varies with k; for more details see
crystal momentum
In solid-state physics crystal momentum or quasimomentum is a momentum-like vector associated with electrons in a crystal lattice. It is defined by the associated wave vectors \mathbf of this lattice, according to
:_ \equiv \hbar
(where \hba ...
.
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)
In quantum mechanics, the particle in a one-dimensional lattice is a problem that occurs in the model of a periodic crystal lattice. The potential is caused by ions in the periodic structure of the crystal creating an electromagnetic field so el ...
.
Bloch's theorem
Bloch's theorem is as follows:
Proof of theorem
Proof with lattice periodicity
= 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'' a
1, a
2, a
3. If the crystal is shifted by any of these three vectors, or a combination of them of the form
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 vector
In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial f ...
s''. These are three vectors b
1, b
2, b
3 (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
In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial f ...
.)
= Lemma about translation operators
=
Let
denote a
translation operator that shifts every wave function by the amount (as above, ''n
j'' are integers). The following fact is helpful for the proof of Bloch's theorem:
= Proof
=
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 , where ''n
i'' are integers. Because the crystal has translational symmetry, this operator commutes with the
Hamiltonian operator
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
. 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
We define the translation operator
We use the hypothesis of a mean periodic potential
and the
independent electron approximation
In condensed matter physics, the independent electron approximation is a simplification used in complex systems, consisting of many electrons, that approximates the electron-electron interaction in crystals as null. It is a requirement for both the ...
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 proof
Velocity and effective mass of Bloch electrons
If we apply the time-independent
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
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
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or '' ...
.
This shows how the effective momentum can be seen as composed of two parts,
a standard momentum
and a
crystal momentum
In solid-state physics crystal momentum or quasimomentum is a momentum-like vector associated with electrons in a crystal lattice. It is defined by the associated wave vectors \mathbf of this lattice, according to
:_ \equiv \hbar
(where \hba ...
. More precisely the
crystal momentum
In solid-state physics crystal momentum or quasimomentum is a momentum-like vector associated with electrons in a crystal lattice. It is defined by the associated wave vectors \mathbf of this lattice, according to
:_ \equiv \hbar
(where \hba ...
is not a momentum but it stands for the momentum in the same way as the electromagnetic momentum in the
minimal coupling In analytical mechanics and quantum field theory, minimal coupling refers to a coupling between field theory (physics), fields which involves only the electric charge, charge distribution and not higher multipole moments of the charge distribution. ...
, and as part of a
canonical transformation
In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations. This is sometimes known as form invariance. It need not preserve the form of the Hamiltonian itself. Canoni ...
of the momentum.
For the effective velocity we can derive
For the effective mass
The quantity on the right multiplied by a factor
is called
effective mass tensor
and we can use it to write a semi-classical equation for a
charge carrier
In physics, a charge carrier is a particle or quasiparticle that is free to move, carrying an electric charge, especially the particles that carry electric charges in electrical conductors. Examples are electrons, ions and holes. The term is used ...
in a band
where
is an
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
. This equation is analogous to the
De Broglie wave
Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave ...
type of approximation
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
In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
.
History and related equations
The concept of the Bloch state was developed by
Felix Bloch
Felix Bloch (23 October 1905 – 10 September 1983) was a Swiss-American physicist and Nobel physics laureate who worked mainly in the U.S. He and Edward Mills Purcell were awarded the 1952 Nobel Prize for Physics for "their development of ne ...
in 1928 to describe the conduction of electrons in crystalline solids. The same underlying mathematics, however, was also discovered independently several times: by
George William Hill
George William Hill (March 3, 1838 – April 16, 1914) was an American astronomer and mathematician. Working independently and largely in isolation from the wider scientific community, he made major contributions to celestial mechanics and t ...
(1877),
Gaston Floquet
Achille Marie Gaston Floquet (15 December 1847, Épinal – 7 October 1920, Nancy, France, Nancy) was a French mathematician, best known for his work in mathematical analysis, especially in theory of differential equations.
See also
*Floquet the ...
(1883), and
Alexander Lyapunov
Aleksandr Mikhailovich Lyapunov (russian: Алекса́ндр Миха́йлович Ляпуно́в, ; – 3 November 1918) was a Russian mathematician, mechanician and physicist. His surname is variously romanized as Ljapunov, Liapunov, Liap ...
(1892). As a result, a variety of nomenclatures are common: applied to
ordinary differential equations
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
, it is called
Floquet theory
Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form
:\dot = A(t) x,
with \displaystyle A(t) a Piecewise#Continuity, piecewise contin ...
(or occasionally the ''Lyapunov–Floquet theorem''). The general form of a one-dimensional periodic potential equation is
Hill's equation:
[
]
where ''f''(''t'') is a periodic potential. Specific periodic one-dimensional equations include the
Kronig–Penney model
In quantum mechanics, the particle in a one-dimensional lattice is a problem that occurs in the model of a periodic crystal lattice. The potential is caused by ions in the periodic structure of the crystal creating an electromagnetic field so elec ...
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 Spectral geometry is a field in mathematics which concerns relationships between geometric structures of manifolds and spectra of canonically defined differential operators. The case of the Laplace–Beltrami operator on a closed Riemannian m ...
.
A relevant new theory
Based on Bloch's theorem, the conventional theory of electronic states in crystals could not correctly explain genuine crystals' boundary and size effects. However, about half a century ago, the mathematical theory of periodic differential equations had some significant progress.
Based on those new mathematical understandings, a recent new theory of electronic states in low dimensional systems
[
]
[
]
[
]
aims to understand such effects. The new theory found that the size and boundary effects of electronic states in each specific dimension in the low-dimensional system are separated in some simple but essential cases.
That is, the energies and properties of some electronic states (including but not limited to the surface states) depend only on the system boundary in that dimension. In contrast, the numbers, energies, and properties
of other electronic states (they are stationary Bloch states, usually many times more) depend only on the system size in that dimension,
see "A more general model: particle in a box with a period potential in
Particle in a box
In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypo ...
."
There is a significant difference between the band structures of Bloch waves in one-dimensional and multi-dimensional space. The Schrödinger differential equation for a one-dimensional periodic potential
is an ordinary differential equation that cannot have more than two linearly independent solutions; this leads to each permitted band and each band gap existing alternatively as the energy increases.
Correspondingly, a theorem in the theory of ordinary periodic differential equations
limits that a boundary-dependent state is either in a band gap or at a band edge. On the other hand, the Schrödinger differential equation for a multi-dimensional periodic potential is a partial differential equation
with no limitation to the number of independent solutions. As a result, the permitted bands in a multi-dimensional crystal are often overlapped. The number of band gaps in a multi-dimensional crystal is always finite. Furthermore, there are no band gaps if the potential is minimal. Correspondingly, a theorem in the theory of partial periodic differential equations
limits that the energy of a boundary-dependent state in a multi-dimensional crystal must be higher or equal to the upper band edge of the relevant permitted band without giving an upper limit. Therefore, a boundary-dependent state decaying in a specific direction can have energy in the range of a permitted band of the bulk. Theoretically, such cases are rather general in multi-dimensional crystals.
The very existence of the boundary-dependent states or sub-bands leads to the properties of electronic states in a simple low-dimensional system being substantially different from the properties of electronic states based on Bloch's theorem as in conventional solid-state physics. And also significantly different from what is widely believed in the solid-state physics community regarding the properties of electronic states in a low-dimensional system or finite crystal, such as ideas based on effective mass concepts. Since ''the energy of each boundary-dependent state is always higher than the energies of its relevant Bloch stationary states''
(see
Particle in a box
In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypo ...
),
the energy gap between occupied and vacant states in an ideal low-dimensional system of a cubic semiconductor is smaller than the band gap of the bulk semiconductor. An essential difference between a bulk metal and a bulk semiconductor would not be so clear when the size of the crystal becomes small enough, so the effects of the boundary-dependent electronic states become more significant. A low-dimensional system of a cubic semiconductor crystal could even have the electrical conductivity properties of metal.
As a one-electron and non-spin theory, this new theory is more general than the conventional theory of electronic states in crystals based on Bloch's theorem and the well-known "
Particle in a box
In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypo ...
" model in quantum mechanics:
The new theory contains the physics cores that the each of the two classical theories has separately: That is, the
former's potential periodicity and the latter's boundary and finite size.
See also
*
Bloch oscillations
Bloch oscillation is a phenomenon from solid state physics. It describes the oscillation of a particle (e.g. an electron) confined in a periodic potential when a constant force is acting on it.
It was first pointed out by Felix Bloch and Clarence ...
*
Bloch wave – MoM method
*
Electronic band structure
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or '' ...
*
Nearly free electron model
In solid-state physics, the nearly free electron model (or NFE model) or quasi-free electron model is a quantum mechanical model of physical properties of electrons that can move almost freely through the crystal lattice of a solid. The model i ...
*
Periodic boundary conditions
Periodic boundary conditions (PBCs) are a set of boundary conditions which are often chosen for approximating a large (infinite) system by using a small part called a ''unit cell''. PBCs are often used in computer simulations and mathematical mode ...
*
Symmetries in quantum mechanics
*
Tight-binding model
In solid-state physics, the tight-binding model (or TB model) is an approach to the calculation of electronic band structure using an approximate set of wave functions based upon Quantum superposition, superposition of wave functions for isolated ...
*
Wannier function
The Wannier functions are a complete set of orthogonal functions used in solid-state physics. They were introduced by Gregory Wannier in 1937. Wannier functions are the localized molecular orbitals of crystalline systems.
The Wannier functions for ...
References
Further reading
*
*
*
*
*
*
*
{{Refend
Quantum states
Condensed matter physics