In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, a phonon is a
collective excitation
In physics, quasiparticles and collective excitations are closely related emergent phenomena arising when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in vacuum.
For ex ...
in a periodic,
elastic
Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics.
Elastic may also refer to:
Alternative name
* Rubber band, ring-shaped band of rubber used to hold objects togeth ...
arrangement of
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, ...
s or
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...
s in
condensed matter
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 su ...
, specifically in
solid
Solid is one of the four fundamental states of matter (the others being liquid, gas, and plasma). The molecules in a solid are closely packed together and contain the least amount of kinetic energy. A solid is characterized by structural ...
s and some
liquids. A type of
quasiparticle
In physics, quasiparticles and collective excitations are closely related emergent phenomena arising when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in vacuum.
For exa ...
, a phonon is an
excited state in the
quantum mechanical
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
quantization of the
modes of vibrations for elastic structures of interacting particles. Phonons can be thought of as quantized
sound waves
In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid.
In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the ...
, similar to
photons as quantized
light waves
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 terahe ...
.
The study of phonons is an important part of condensed matter physics. They play a major role in many of the physical properties of condensed matter systems, such as
thermal conductivity
The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa.
Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
and
electrical conductivity, as well as in models of
neutron scattering
Neutron scattering, the irregular dispersal of free neutrons by matter, can refer to either the naturally occurring physical process itself or to the man-made experimental techniques that use the natural process for investigating materials. Th ...
and related effects.
The concept of phonons was introduced in 1932 by
Soviet
The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, it was nominally a federal union of fifteen nation ...
physicist
Igor Tamm
Igor Yevgenyevich Tamm ( rus, И́горь Евге́ньевич Тамм , p=ˈiɡərʲ jɪvˈɡʲenʲjɪvitɕ ˈtam , a=Ru-Igor Yevgenyevich Tamm.ogg; 8 July 1895 – 12 April 1971) was a Soviet physicist who received the 1958 Nobel Prize in ...
. The name ''phonon'' comes from the
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
word (), which translates to ''sound'' or ''voice'', because long-wavelength phonons give rise to
sound
In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid.
In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' b ...
. The name is analogous to the word ''
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
''.
Definition
A phonon is the
quantum mechanical
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
description of an elementary
vibration
Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic, such as the motion of a pendulum—or random, su ...
al motion in which a
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
of atoms or molecules uniformly oscillates at a single
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
. In
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
this designates a
normal mode of vibration. Normal modes are important because any arbitrary lattice vibration can be considered to be a
superposition of these ''elementary'' vibration modes (cf.
Fourier analysis). While normal modes are
wave-like phenomena in classical mechanics, phonons have
particle-like properties too, in a way related to the
wave–particle duality
Wave–particle duality is the concept in quantum mechanics that every particle or quantum entity may be described as either a particle or a wave. It expresses the inability of the classical concepts "particle" or "wave" to fully describe the b ...
of quantum mechanics.
Lattice dynamics
The equations in this section do not use
axioms of quantum mechanics but instead use relations for which there exists a direct
correspondence in classical mechanics.
For example: a rigid regular,
crystalline
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, macros ...
(not
amorphous) lattice is composed of ''N'' particles. These particles may be atoms or molecules. ''N'' is a large number, say of the order of 10
23, or on the order of the
Avogadro number
The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining co ...
for a typical sample of a solid. Since the lattice is rigid, the atoms must be exerting
forces on one another to keep each atom near its equilibrium position. These forces may be
Van der Waals force
In molecular physics, the van der Waals force is a distance-dependent interaction between atoms or molecules. Unlike ionic or covalent bonds, these attractions do not result from a chemical electronic bond; they are comparatively weak and th ...
s,
covalent bonds,
electrostatic attraction
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
s, and others, all of which are ultimately due to the
electric
Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
force.
Magnetic and
gravitational
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...
forces are generally negligible. The forces between each pair of atoms may be characterized by a
potential energy function ''V'' that depends on the distance of separation of the atoms. The potential energy of the entire lattice is the sum of all pairwise potential energies multiplied by a factor of 1/2 to compensate for double counting:
:
where ''r
i'' is the
position
Position often refers to:
* Position (geometry), the spatial location (rather than orientation) of an entity
* Position, a job or occupation
Position may also refer to:
Games and recreation
* Position (poker), location relative to the dealer
* ...
of the ''i''th atom, and ''V'' is the
potential energy between two atoms.
It is difficult to solve this
many-body problem
The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provid ...
explicitly in either classical or quantum mechanics. In order to simplify the task, two important
approximations are usually imposed. First, the sum is only performed over neighboring atoms. Although the electric forces in real solids extend to infinity, this approximation is still valid because the fields produced by distant atoms are effectively
screened. Secondly, the potentials ''V'' are treated as
harmonic potentials. This is permissible as long as the atoms remain close to their equilibrium positions. Formally, this is accomplished by
Taylor expanding ''V'' about its equilibrium value to quadratic order, giving ''V'' proportional to the displacement ''x''
2 and the elastic force simply proportional to ''x''. The error in ignoring higher order terms remains small if ''x'' remains close to the equilibrium position.
The resulting lattice may be visualized as a system of balls connected by springs. The following figure shows a cubic lattice, which is a good model for many types of crystalline solid. Other lattices include a linear chain, which is a very simple lattice which we will shortly use for modeling phonons. (For other common lattices, see
crystal structure
In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns ...
.)
:
The potential energy of the lattice may now be written as
:
Here, ''ω'' is the
natural frequency
Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force.
The motion pattern of a system oscillating at its natural frequency is called the normal mode (if all pa ...
of the harmonic potentials, which are assumed to be the same since the lattice is regular. ''R
i'' is the position coordinate of the ''i''th atom, which we now measure from its equilibrium position. The sum over nearest neighbors is denoted (nn).
Lattice waves
Due to the connections between atoms, the displacement of one or more atoms from their equilibrium positions gives rise to a set of vibration
wave
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
s propagating through the lattice. One such wave is shown in the figure to the right. The
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
of the wave is given by the displacements of the atoms from their equilibrium positions. The
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, t ...
''λ'' is marked.
There is a minimum possible wavelength, given by twice the equilibrium separation ''a'' between atoms. Any wavelength shorter than this can be mapped onto a wavelength longer than 2''a'', due to the periodicity of the lattice. This can be thought as one consequence of
Nyquist–Shannon sampling theorem
The Nyquist–Shannon sampling theorem is a theorem in the field of signal processing which serves as a fundamental bridge between continuous-time signals and discrete-time signals. It establishes a sufficient condition for a sample rate that per ...
, the lattice points being viewed as the "sampling points" of a continuous wave.
Not every possible lattice vibration has a well-defined wavelength and frequency. However, the
normal modes do possess well-defined wavelengths and
frequencies.
One-dimensional lattice
In order to simplify the analysis needed for a 3-dimensional lattice of atoms, it is convenient to model a 1-dimensional lattice or linear chain. This model is complex enough to display the salient features of phonons.
Classical treatment
The forces between the atoms are assumed to be linear and nearest-neighbour, and they are represented by an elastic spring. Each atom is assumed to be a point particle and the nucleus and electrons move in step (
adiabatic theorem
The adiabatic theorem is a concept in quantum mechanics. Its original form, due to Max Born and Vladimir Fock (1928), was stated as follows:
:''A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it sl ...
):
::::::::''n'' − 1 ''n'' ''n'' + 1 ← ''a'' →
···o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o···
::::::::→→→→→→
::::::::''u''
''n'' − 1''u
n'u''
''n'' + 1
where labels the th atom out of a total of , is the distance between atoms when the chain is in equilibrium, and the displacement of the th atom from its equilibrium position.
If ''C'' is the elastic constant of the spring and the mass of the atom, then the equation of motion of the th atom is
:
This is a set of coupled equations.
Since the solutions are expected to be oscillatory, new coordinates are defined by a
discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
, in order to decouple them.
Put
:
Here, corresponds and devolves to the continuous variable of scalar field theory. The are known as the ''normal coordinates'', continuum field modes .
Substitution into the equation of motion produces the following ''decoupled equations'' (this requires a significant manipulation using the orthonormality and completeness relations of the discrete Fourier transform),
:
These are the equations for decoupled
harmonic oscillators
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force ''F'' proportional to the displacement ''x'':
\vec F = -k \vec x,
where ''k'' is a positive consta ...
which have the solution
:
Each normal coordinate ''Q
k'' represents an independent vibrational mode of the lattice with wavenumber , which is known as a
normal mode.
The second equation, for , is known as the
dispersion relation
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given t ...
between the
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
and the
wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
.
In the
continuum limit
In mathematical physics and mathematics, the continuum limit or scaling limit of a lattice model refers to its behaviour in the limit as the lattice spacing goes to zero. It is often useful to use lattice models to approximate real-world processe ...
, →0, →∞, with held fixed, → , a scalar field, and
. This amounts to classical free
scalar field theory, an assembly of independent oscillators.
Quantum treatment
A one-dimensional quantum mechanical harmonic chain consists of ''N'' identical atoms. This is the simplest quantum mechanical model of a lattice that allows phonons to arise from it. The formalism for this model is readily generalizable to two and three dimensions.
In some contrast to the previous section, the positions of the masses are not denoted by ''u
i'', but, instead, by ''x''
1, ''x''
2…, as measured from their equilibrium positions (i.e. ''x
i'' = 0 if particle ''i'' is at its equilibrium position.) In two or more dimensions, the ''x
i'' are vector quantities. The
Hamiltonian
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 ...
for this system is
:
where ''m'' is the mass of each atom (assuming it is equal for all), and ''x
i'' and ''p
i'' are the position and
momentum operators, respectively, for the ''i''th atom and the sum is made over the nearest neighbors (nn). However one expects that in a lattice there could also appear waves that behave like particles. It is customary to deal with
wave
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
s in
Fourier space
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a s ...
which uses
normal modes of the
wavevector
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), ...
as variables instead coordinates of particles. The number of normal modes is same as the number of particles. However, the
Fourier space
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a s ...
is very useful given the
periodicity
Periodicity or periodic may refer to:
Mathematics
* Bott periodicity theorem, addresses Bott periodicity: a modulo-8 recurrence relation in the homotopy groups of classical groups
* Periodic function, a function whose output contains values tha ...
of the system.
A set of ''N'' "normal coordinates" ''Q
k'' may be introduced, defined as the
discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
s of the ''x
k'' and ''N'' "conjugate momenta" ''Π
k'' defined as the Fourier transforms of the ''p
k'':
:
The quantity ''k
n'' turns out to be the
wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
of the phonon, i.e. 2 divided by the
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, t ...
.
This choice retains the desired commutation relations in either real space or wavevector space
:
From the general result
:
The potential energy term is
:
where
:
The Hamiltonian may be written in wavevector space as
:
The couplings between the position variables have been transformed away; if the ''Q'' and ''Π'' were
Hermitian {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
(which they are not), the transformed Hamiltonian would describe ''N'' uncoupled harmonic oscillators.
The form of the quantization depends on the choice of boundary conditions; for simplicity, ''periodic'' boundary conditions are imposed, defining the (''N'' + 1)th atom as equivalent to the first atom. Physically, this corresponds to joining the chain at its ends. The resulting quantization is
:
The upper bound to ''n'' comes from the minimum wavelength, which is twice the lattice spacing ''a'', as discussed above.
The harmonic oscillator eigenvalues or energy levels for the mode ''ω
k'' are:
:
The levels are evenly spaced at:
:
where ''ħω'' is the
zero-point energy
Zero-point energy (ZPE) is the lowest possible energy that a quantum mechanical system may have. Unlike in classical mechanics, quantum systems constantly fluctuate in their lowest energy state as described by the Heisenberg uncertainty pri ...
of a
quantum harmonic oscillator.
An exact amount of
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
''ħω'' must be supplied to the harmonic oscillator lattice to push it to the next energy level. In comparison to the
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
case when the
electromagnetic field is quantized, the quantum of vibrational energy is called a phonon.
All quantum systems show wavelike and particlelike properties simultaneously. The particle-like properties of the phonon are best understood using the methods of
second quantization
Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as ...
and operator techniques described later.
Three-dimensional lattice
This may be generalized to a three-dimensional lattice. The wavenumber ''k'' is replaced by a three-dimensional
wavevector
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), ...
k. Furthermore, each k is now associated with three normal coordinates.
The new indices ''s'' = 1, 2, 3 label the
polarization of the phonons. In the one-dimensional model, the atoms were restricted to moving along the line, so the phonons corresponded to
longitudinal wave
Longitudinal waves are waves in which the vibration of the medium is parallel ("along") to the direction the wave travels and displacement of the medium is in the same (or opposite) direction of the wave propagation. Mechanical longitudinal waves ...
s. In three dimensions, vibration is not restricted to the direction of propagation, and can also occur in the perpendicular planes, like
transverse wave
In physics, a transverse wave is a wave whose oscillations are perpendicular to the direction of the wave's advance. This is in contrast to a longitudinal wave which travels in the direction of its oscillations. Water waves are an example of t ...
s. This gives rise to the additional normal coordinates, which, as the form of the Hamiltonian indicates, we may view as independent species of phonons.
Dispersion relation
For a one-dimensional alternating array of two types of ion or atom of mass ''m''
1, ''m''
2 repeated periodically at a distance ''a'', connected by springs of spring constant ''K'', two modes of vibration result:
[
:
where ''k'' is the wavevector of the vibration related to its wavelength by
.
The connection between frequency and wavevector, ''ω'' = ''ω''(''k''), is known as a ]dispersion relation
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given t ...
. The plus sign results in the so-called ''optical'' mode, and the minus sign to the ''acoustic'' mode. In the optical mode two adjacent different atoms move against each other, while in the acoustic mode they move together.
The speed of propagation of an acoustic phonon, which is also the speed of sound in the lattice, is given by the slope of the acoustic dispersion relation, (see group velocity.) At low values of ''k'' (i.e. long wavelengths), the dispersion relation is almost linear, and the speed of sound is approximately ''ωa'', independent of the phonon frequency. As a result, packets of phonons with different (but long) wavelengths can propagate for large distances across the lattice without breaking apart. This is the reason that sound propagates through solids without significant distortion. This behavior fails at large values of ''k'', i.e. short wavelengths, due to the microscopic details of the lattice.
For a crystal that has at least two atoms in its primitive cell, the dispersion relations exhibit two types of phonons, namely, optical and acoustic modes corresponding to the upper blue and lower red curve in the diagram, respectively. The vertical axis is the energy or frequency of phonon, while the horizontal axis is the wavevector
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), ...
. The boundaries at − and are those of 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 ...
. A crystal with ''N'' ≥ 2 different atoms in the primitive cell exhibits three acoustic modes: one longitudinal acoustic mode and two transverse acoustic modes. The number of optical modes is 3''N'' – 3. The lower figure shows the dispersion relations for several phonon modes in GaAs
Gallium arsenide (GaAs) is a III-V direct band gap semiconductor with a zinc blende crystal structure.
Gallium arsenide is used in the manufacture of devices such as microwave frequency integrated circuits, monolithic microwave integrated circ ...
as a function of wavevector k in the principal directions of its Brillouin zone.[
]
Many phonon dispersion curves have been measured by inelastic neutron scattering
Neutron scattering, the irregular dispersal of free neutrons by matter, can refer to either the naturally occurring physical process itself or to the man-made experimental techniques that use the natural process for investigating materials. Th ...
.
The physics of sound in fluids differs from the physics of sound in solids, although both are density waves: sound waves in fluids only have longitudinal components, whereas sound waves in solids have longitudinal and transverse components. This is because fluids cannot support shear stress
Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ...
es (but see viscoelastic
In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly ...
fluids, which only apply to high frequencies).
Interpretation of phonons using second quantization techniques
The above-derived Hamiltonian may look like a classical Hamiltonian function, but if it is interpreted as an operator, then it describes a quantum field theory of non-interacting boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
s.
The second quantization
Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as ...
technique, similar to the ladder operator
In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raisin ...
method used for quantum harmonic oscillators, is a means of extracting energy eigenvalues without directly solving the differential equations. Given the Hamiltonian, , as well as the conjugate position, , and conjugate momentum defined in the quantum treatment section above, we can define creation and annihilation operators
Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually d ...
:
: and
The following commutators can be easily obtained by substituting in the canonical commutation relation
In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example,
hat x,\hat p_ ...
:
:
Using this, the operators ''bk''† and ''bk'' can be inverted to redefine the conjugate position and momentum as:
: and
Directly substituting these definitions for and into the wavevector space Hamiltonian, as it is defined above, and simplifying then results in the Hamiltonian taking the form:
:
This is known as the second quantization technique, also known as the occupation number formulation, where ''nk'' = ''bk''†''bk'' is the occupation number. This can be seen to be a sum of N independent oscillator Hamiltonians, each with a unique wave vector, and compatible with the methods used for the quantum harmonic oscillator (note that ''nk'' is hermitian {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
). When a Hamiltonian can be written as a sum of commuting sub-Hamiltonians, the energy eigenstates will be given by the products of eigenstates of each of the separate sub-Hamiltonians. The corresponding energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
is then given by the sum of the individual eigenvalues of the sub-Hamiltonians.
As with the quantum harmonic oscillator, one can show that ''bk''† and ''bk'' respectively create and destroy a single field excitation, a phonon, with an energy of ''ħωk''.
Three important properties of phonons may be deduced from this technique. First, phonons are boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
s, since any number of identical excitations can be created by repeated application of the creation operator ''bk''†. Second, each phonon is a "collective mode" caused by the motion of every atom in the lattice. This may be seen from the fact that the creation and annihilation operators, defined here in momentum space, contains sums over the position and momentum operators of every atom when written in position space (See position and momentum space
In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension.
Position space (also real space or coordinate space) is the set of all ''position vectors'' r in space, and h ...
). Finally, using the ''position–position correlation function'', it can be shown that phonons act as waves of lattice displacement.
This technique is readily generalized to three dimensions, where the Hamiltonian takes the form:
:
Which can be interpreted as the sum of 3N independent oscillator Hamiltonians, one for each wave vector and polarization.
Acoustic and optical phonons
Solids with more than one atom in the smallest unit cell
In geometry, biology, mineralogy and solid state physics, a unit cell is a repeating unit formed by the vectors spanning the points of a lattice. Despite its suggestive name, the unit cell (unlike a unit vector, for example) does not necessaril ...
exhibit two types of phonons: acoustic phonons and optical phonons.
Acoustic phonons are coherent movements of atoms of the lattice out of their equilibrium positions. If the displacement is in the direction of propagation, then in some areas the atoms will be closer, in others farther apart, as in a sound wave in air (hence the name acoustic). Displacement perpendicular to the propagation direction is comparable to waves on a string. If the wavelength of acoustic phonons goes to infinity, this corresponds to a simple displacement of the whole crystal, and this costs zero deformation energy. Acoustic phonons exhibit a linear relationship between frequency and phonon wave-vector for long wavelengths. The frequencies of acoustic phonons tend to zero with longer wavelength. Longitudinal and transverse acoustic phonons are often abbreviated as LA and TA phonons, respectively.
Optical phonons are out-of-phase movements of the atoms in the lattice, one atom moving to the left, and its neighbor to the right. This occurs if the lattice basis consists of two or more atoms. They are called ''optical'' because in ionic crystals, such as sodium chloride, fluctuations in displacement create an electrical polarization that couples to the electromagnetic field. Hence, they can be excited by infrared radiation
Infrared (IR), sometimes called infrared light, is electromagnetic radiation (EMR) with wavelengths longer than those of visible light. It is therefore invisible to the human eye. IR is generally understood to encompass wavelengths from around ...
, the electric field of the light will move every positive sodium ion in the direction of the field, and every negative chloride ion in the other direction, causing the crystal to vibrate.
Optical phonons have a non-zero frequency at the 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 ...
center and show no dispersion near that long wavelength limit. This is because they correspond to a mode of vibration where positive and negative ions at adjacent lattice sites swing against each other, creating a time-varying electrical dipole moment
The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI unit for electric dipole moment is the coulomb-meter (C⋅m). The ...
. Optical phonons that interact in this way with light are called ''infrared active''. Optical phonons that are ''Raman active'' can also interact indirectly with light, through Raman scattering
Raman scattering or the Raman effect () is the inelastic scattering of photons by matter, meaning that there is both an exchange of energy and a change in the light's direction. Typically this effect involves vibrational energy being gained by a ...
. Optical phonons are often abbreviated as LO and TO phonons, for the longitudinal and transverse modes respectively; the splitting between LO and TO frequencies is often described accurately by the Lyddane–Sachs–Teller relation
In condensed matter physics, the Lyddane–Sachs–Teller relation (or LST relation) determines the ratio of the natural frequency of longitudinal optic lattice vibrations ( phonons) (\omega_\text) of an ionic crystal to the natural frequency of t ...
.
When measuring optical phonon energy experimentally, optical phonon frequencies are sometimes given in spectroscopic wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
notation, where the symbol ''ω'' represents ordinary frequency (not angular frequency), and is expressed in units of cm−1. The value is obtained by dividing the frequency by the speed of light in vacuum
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit f ...
. In other words, the wave-number in cm−1 units corresponds to the inverse of the wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, t ...
of a photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
in vacuum that has the same frequency as the measured phonon.
Crystal momentum
By analogy to photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
s and matter waves
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 wav ...
, phonons have been treated with wavevector ''k'' as though it has a momentum ''ħk''; however, this is not strictly correct, because ''ħk'' is not actually a physical momentum; it is called the ''crystal momentum'' or ''pseudomomentum''. This is because ''k'' is only determined up to addition of constant vectors (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 fu ...
s and integer multiples thereof). For example, in the one-dimensional model, the normal coordinates ''Q'' and ''Π'' are defined so that
:
where
:
for any integer ''n''. A phonon with wavenumber ''k'' is thus equivalent to an infinite family of phonons with wavenumbers ''k'' ± , ''k'' ± , and so forth. Physically, the reciprocal lattice vectors act as additional chunks of momentum which the lattice can impart to the phonon. Bloch electron
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. The theorem is named after the physicist Felix Bloch, who d ...
s obey a similar set of restrictions.
It is usually convenient to consider phonon wavevectors ''k'' which have the smallest magnitude , ''k'', in their "family". The set of all such wavevectors defines 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 ...
''. Additional Brillouin zones may be defined as copies of the first zone, shifted by some reciprocal lattice vector.
Thermodynamics
The thermodynamic
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
properties of a solid are directly related to its phonon structure. The entire set of all possible phonons that are described by the phonon dispersion relations combine in what is known as the phonon density of states
In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as D(E) = N(E)/V , where N(E)\delta E is the number of states i ...
which determines the heat capacity
Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K).
Heat capacity ...
of a crystal. By the nature of this distribution, the heat capacity is dominated by the high-frequency part of the distribution, while thermal conductivity is primarily the result of the low-frequency region.
At absolute zero temperature, a crystal lattice lies in its ground state, and contains no phonons. A lattice at a nonzero temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer.
Thermometers are calibrated in various Conversion of units of temperature, temp ...
has an energy that is not constant, but fluctuates random
In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual ra ...
ly about some mean value
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set.
For a data set, the ''arithm ...
. These energy fluctuations are caused by random lattice vibrations, which can be viewed as a gas of phonons. Because these phonons are generated by the temperature of the lattice, they are sometimes designated thermal phonons.
Thermal phonons can be created and destroyed by random energy fluctuations. In the language of statistical mechanics this means that the chemical potential for adding a phonon is zero. This behavior is an extension of the harmonic potential into the anharmonic regime. The behavior of thermal phonons is similar to the photon gas
In physics, a photon gas is a gas-like collection of photons, which has many of the same properties of a conventional gas like hydrogen or neon – including pressure, temperature, and entropy. The most common example of a photon gas in equilibr ...
produced by an electromagnetic cavity
An electromagnetic cavity is a cavity that acts as a container for electromagnetic fields such as photons, in effect containing their wave function inside. The size of the cavity determines the maximum photon wavelength that can be trapped. Additi ...
, wherein photons may be emitted or absorbed by the cavity walls. This similarity is not coincidental, for it turns out that the electromagnetic field behaves like a set of harmonic oscillators, giving rise to Black-body radiation. Both gases obey the Bose–Einstein statistics
In quantum statistics, Bose–Einstein statistics (B–E statistics) describes one of two possible ways in which a collection of non-interacting, indistinguishable particles may occupy a set of available discrete energy states at thermodynamic ...
: in thermal equilibrium and within the harmonic regime, the probability of finding phonons or photons in a given state with a given angular frequency is:
:
where ''ω''''k'',''s'' is the frequency of the phonons (or photons) in the state, ''k''B is the Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
, and ''T'' is the temperature.
Phonon tunneling
Phonons have been shown to exhibit Quantum tunneling
In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizati ...
behavior (or ''phonon tunneling'') where, across gaps up to a nanometer wide, heat can flow via phonons that "tunnel" between two materials. This type of heat transfer works between distances too large for conduction to occur but too small for radiation to occur and therefore cannot be explained by classical heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
models.
Operator formalism
The phonon Hamiltonian is given by
:
In terms of the creation and annihilation operators
Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually d ...
, these are given by
:
Here, in expressing the Hamiltonian
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 ...
in operator formalism, we have not taken into account the ''ħωq'' term as, given a continuum or infinite lattice, the ''ħωq'' terms will add up yielding an infinite term. Hence, it is " renormalized" by setting the factor of ''ħωq'' to 0, arguing that the difference in energy is what we measure and not the absolute value of it. Hence, the ''ħωq'' factor is absent in the operator formalized expression for the Hamiltonian
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 ...
.
The ground state, also called the "vacuum state
In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used as ...
", is the state composed of no phonons. Hence, the energy of the ground state is 0. When a system is in the state , we say there are ''nα'' phonons of type ''α'', where ''nα'' is the occupation number of the phonons. The energy of a single phonon of type ''α'' is given by ''ħωq'' and the total energy of a general phonon system is given by ''n''1''ħω''1 + ''n''2''ħω''2 +…. As there are no cross terms (e.g. ''n''1''ħω''2), the phonons are said to be non-interacting. The action of the creation and annihilation operators is given by:
:
and,
:
The creation operator, ''aα''† creates a phonon of type ''α'' while ''aα'' annihilates one. Hence, they are respectively the creation and annihilation operators for phonons. Analogous to the quantum harmonic oscillator case, we can define particle number operator
In quantum mechanics, for systems where the total number of particles may not be preserved, the number operator is the observable that counts the number of particles.
The number operator acts on Fock space. Let
:, \Psi\rangle_\nu=, \phi_1,\phi_2 ...
as
:
The number operator commutes with a string of products of the creation and annihilation operators if and only if the number of creation operators is equal to number of annihilation operators.
It can be shown that phonons are symmetric under exchange (i.e. = ), so therefore they are considered bosons
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer ...
.
Nonlinearity
As well as photons, phonons can interact via parametric down conversion and form squeezed coherent state
In physics, a squeezed coherent state is a quantum state that is usually described by two non-commuting observables having continuous spectra of eigenvalues. Examples are position x and momentum p of a particle, and the (dimension-less) elect ...
s.
Predicted properties
Recent research has shown that phonons and roton
In theoretical physics, a roton is an elementary excitation, or quasiparticle, seen in superfluid helium-4 and Bose–Einstein condensates with long-range dipolar interactions or spin-orbit coupling. The dispersion relation of elementary excita ...
s may have a non-negligible mass and be affected by gravity just as standard particles are. In particular, phonons are predicted to have a kind of negative mass
In theoretical physics, negative mass is a type of exotic matter whose mass is of opposite sign to the mass of normal matter, e.g. −1 kg. Such matter would violate one or more energy conditions and show some strange properties such as the ...
and negative gravity. This can be explained by how phonons are known to travel faster in denser materials. Because the part of a material pointing towards a gravitational source is closer to the object, it becomes denser on that end. From this, it is predicted that phonons would deflect away as it detects the difference in densities, exhibiting the qualities of a negative gravitational field. Although the effect would be too small to measure, it is possible that future equipment could lead to successful results.
Phonons have also been predicted to play a key role in superconductivity in materials and the prediction of superconductive compounds.[Enamul Haque and M. Anwar Hossain. (2018)]
First-principles prediction of phonon-mediated superconductivity in XBC (X= Mg, Ca, Sr, Ba)
Arviv.org, Retrieved November 27, 2018
In 2019, researchers were able to isolate individual phonons without destroying them for the first time.
See also
* Boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
* Brillouin scattering Brillouin scattering (also known as Brillouin light scattering or BLS), named after Léon Brillouin, refers to the interaction of light with the material waves in a medium (e.g. electrostriction and magnetostriction). It is mediated by the refractiv ...
* Fracton
* Linear elasticity
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mec ...
* Mechanical wave
In physics, a mechanical wave is a wave that is an oscillation of matter, and therefore transfers energy through a medium. While waves can move over long distances, the movement of the medium of transmission—the material—is limited. Therefor ...
* Phonon scattering Phonons can scatter through several mechanisms as they travel through the material. These scattering mechanisms are: Umklapp phonon-phonon scattering, phonon-impurity scattering, phonon-electron scattering, and phonon-boundary scattering. Each sca ...
* Carrier scattering
* 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 manipul ...
* Rayleigh wave
Rayleigh waves are a type of surface acoustic wave that travel along the surface of solids. They can be produced in materials in many ways, such as by a localized impact or by piezo-electric transduction, and are frequently used in non-destructi ...
* Relativistic heat conduction
Relativistic heat conduction refers to the modelling of heat conduction (and similar diffusion processes) in a way compatible with special relativity. In special (and general) relativity, the usual heat equation for non-relativistic heat conduct ...
* Rigid unit modes
* SASER
Sound amplification by stimulated emission of radiation (SASER) refers to a device that emits acoustic radiation. It focuses sound waves in a way that they can serve as accurate and high-speed carriers of information in many kinds of applications ...
* Second sound
Second sound is a quantum mechanical phenomenon in which heat transfer occurs by wave-like motion, rather than by the more usual mechanism of diffusion. Its presence leads to a very high thermal conductivity. It is known as "second sound" because t ...
* Surface acoustic wave
A surface acoustic wave (SAW) is an acoustic wave traveling along the surface of a material exhibiting elasticity, with an amplitude that typically decays exponentially with depth into the material, such that they are confined to a depth of about ...
* Surface phonon
In solid state physics, a surface phonon is the quantum of a lattice vibration mode associated with a solid surface. Similar to the ordinary lattice vibrations in a bulk solid (whose quanta are simply called phonons), the nature of surface vibra ...
* Thermal conductivity
The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa.
Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
* Vibration
Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic, such as the motion of a pendulum—or random, su ...
References
External links
*
Explained: Phonons
MIT News, 2010.
* Phonons in a One Dimensional Microfluidic Crysta
an
with movies i
{{Authority control
Quasiparticles
Bosons
1932 introductions
Subatomic particles with spin 0