Phonon
In
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
The concept of phonons was introduced in 1932 by
Definition
A phonon is the quantum mechanical description of an elementary vibrational motion in which a lattice of atoms or molecules uniformly oscillates at a single frequency.[3] In classical mechanics 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 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,
where ri is the position of the ith atom, and V is the potential energy between two atoms.
It is difficult to solve this
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.)
The potential energy of the lattice may now be written as
Here, ω is the natural frequency of the harmonic potentials, which are assumed to be the same since the lattice is regular. Ri is the position coordinate of the ith atom, which we now measure from its equilibrium position. The sum over nearest neighbors is denoted (nn).
It is important to mention that the mathematical treatment given here is highly simplified in order to make it accessible to non-experts. The simplification has been achieved by making two basic assumptions in the expression for the total potential energy of the crystal. These assumptions are that (i) the total potential energy can be written as a sum of pairwise interactions, and (ii) each atom interacts with only its nearest neighbors. These are used only sparingly in modern lattice dynamics.[5] A more general approach is to express the potential energy in terms of force constants.[5] See, for example, the Wiki article on multiscale Green's functions.
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 waves propagating through the lattice. One such wave is shown in the figure to the right. The amplitude of the wave is given by the displacements of the atoms from their equilibrium positions. The wavelength 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 2a, due to the periodicity of the lattice. This can be thought as one consequence of Nyquist–Shannon sampling theorem, 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):
- n − 1 n n + 1 ← a →
···o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o···
- →→ → →→→
- un − 1 un un + 1
where n labels the nth atom out of a total of N, a is the distance between atoms when the chain is in equilibrium, and un the displacement of the nth atom from its equilibrium position.
If C is the elastic constant of the spring and m the mass of the atom, then the equation of motion of the nth 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 order to decouple them.[6]
Put
Here, na corresponds and devolves to the continuous variable x of scalar field theory. The Qk are known as the normal coordinates, continuum field modes φk.
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),[7]
These are the equations for decoupled
Each normal coordinate Qk represents an independent vibrational mode of the lattice with wavenumber k, which is known as a normal mode.
The second equation, for ωk, is known as the dispersion relation between the angular frequency and the wavenumber.
In the continuum limit, a→0, N→∞, with Na held fixed, un → φ(x), 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 ui, but, instead, by x1, x2..., as measured from their equilibrium positions (i.e. xi = 0 if particle i is at its equilibrium position.) In two or more dimensions, the xi are vector quantities. The Hamiltonian for this system is
where m is the mass of each atom (assuming it is equal for all), and xi and pi are the position and
A set of N "normal coordinates" Qk may be introduced, defined as the discrete Fourier transforms of the xk and N "conjugate momenta" Πk defined as the Fourier transforms of the pk:
The quantity kn turns out to be the wavenumber of the phonon, i.e. 2π divided by the wavelength.
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
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 ωk are:
The levels are evenly spaced at:
where 1/2ħω is the zero-point energy of a quantum harmonic oscillator.
An exact amount of energy ħω must be supplied to the harmonic oscillator lattice to push it to the next energy level. In comparison to the photon 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 and operator techniques described later.[8]
Three-dimensional lattice
This may be generalized to a three-dimensional lattice. The wavenumber k is replaced by a three-dimensional
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 waves. In three dimensions, vibration is not restricted to the direction of propagation, and can also occur in the perpendicular planes, like transverse waves. 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 m1, m2 repeated periodically at a distance a, connected by springs of spring constant K, two modes of vibration result:[10]
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. 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, ∂ωk/∂k (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
The modes are also referred to as the branches of phonon dispersion. In general, if there are p atoms (denoted by N earlier) in the primitive unit cell, there will be 3p branches of phonon dispersion in a 3-dimensional crystal. Out of these, 3 branches correspond to acoustic modes and the remaining 3p-3 branches will correspond to optical modes. In some special directions, some branches coincide due to symmetry. These branches are called degenerate. In acoustic modes, all the p atoms vibrate in phase. So there is no change in the relative displacements of these atoms during the wave propagation.
Study of phonon dispersion is useful for modeling propagation of sound waves in solids, which is characterized by phonons. The energy of each phonon, as given earlier, is ħω. The velocity of the wave also is given in terms of ω and k . The direction of the wave vector is the direction of the wave propagation and the phonon polarization vector gives the direction in which the atoms vibrate. Actually, in general, the wave velocity in a crystal is different for different directions of k. In other words, most crystals are anisotropic for phonon proagation.
A wave is longitudinal if the atoms vibrate in the same direction as the wave propagation. In a transverse wave, the atoms vibrate perpendicular to the wave propagation. However, except for isotropic crystals, waves in a crystal are not exactly longitudinal or transverse. For general anisotropic crystals, the phonon waves are longitudinal or transverse only in certain special symmetry directions. In other directions, they can be nearly longitudinal or nearly transverse. It is only for labeling convenience, that they are often called longitudinal or transverse but are actually quasi-longitudinal or quasi-transverse. Note that in the three-dimensional case, there are two directions perpendicular to a straight line at each point on the line. Hence, there are always two (quasi) transverse waves for each (quasi) longitudinal wave.
Many phonon dispersion curves have been measured by
The physics of sound in
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 bosons.[2] The
- and
The following commutators can be easily obtained by substituting in the canonical commutation relation:
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:[2]
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
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.[11][2]
Three important properties of phonons may be deduced from this technique. First, phonons are
This technique is readily generalized to three dimensions, where the Hamiltonian takes the form:[11][2]
Which can be interpreted as the sum of 3N independent oscillator Hamiltonians, one for each wave vector and polarization.[11]
Acoustic and optical phonons
Solids with more than one atom in the smallest unit cell 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
Optical phonons have a non-zero frequency at the
When measuring optical phonon energy experimentally, optical phonon frequencies are sometimes given in spectroscopic
Crystal momentum
By analogy to
where
for any integer n. A phonon with wavenumber k is thus equivalent to an infinite family of phonons with wavenumbers k ± 2π/a, k ± 4π/a, and so forth. Physically, the reciprocal lattice vectors act as additional chunks of momentum which the lattice can impart to the phonon.
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. Additional Brillouin zones may be defined as copies of the first zone, shifted by some reciprocal lattice vector.
Thermodynamics
The thermodynamic 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 which determines the 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.[citation needed]
At
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.[14] This behavior is an extension of the harmonic potential into the anharmonic regime. The behavior of thermal phonons is similar to the photon gas produced by an electromagnetic cavity, 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 thermal equilibrium and within the harmonic regime, the probability of finding phonons or photons in a given state with a given angular frequency is:[15]
where ωk,s is the frequency of the phonons (or photons) in the state, kB is the Boltzmann constant, and T is the temperature.
Phonon tunneling
Phonons have been shown to exhibit
Operator formalism
The phonon Hamiltonian is given by
In terms of the creation and annihilation operators, these are given by
Here, in expressing the Hamiltonian in operator formalism, we have not taken into account the 1/2ħωq term as, given a continuum or infinite lattice, the 1/2ħωq terms will add up yielding an infinite term. Because the difference in energy is what we measure and not the absolute value of it, the constant term 1/2ħωq can be ignored without changing the equations of motion. Hence, the 1/2ħωq factor is absent in the operator formalized expression for the Hamiltonian.
The ground state, also called the "
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 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. |α,β⟩ = |β,α⟩), so therefore they are considered
Nonlinearity
As well as
Predicted properties
Recent research has shown that phonons and rotons may have a non-negligible mass and be affected by gravity just as standard particles are.[20] In particular, phonons are predicted to have a kind of negative mass and negative gravity.[21] 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.[22] Although the effect would be too small to measure, it is possible that future equipment could lead to successful results.
Superconductivity
Superconductivity is a state of electronic matter in which electrical resistance vanishes and magnetic fields are expelled from the material. In a superconductor, electrons are bound together into Cooper pairs by a weak attractive force. In a conventional superconductor, this attraction is caused by an exchange of phonons between the electrons.[23] The evidence that phonons, the vibrations of the ionic lattice, are relevant for superconductivity is provided by the isotope effect, the dependence of the superconducting critical temperature on the mass of the ions.
Other research
In 2019, researchers were able to isolate individual phonons without destroying them for the first time.[24]
They have been also shown to form “phonon winds” where an electric current in a graphene surface is generated by a liquid flow above it due to the viscous forces at the liquid–solid interface.[25][26]
See also
- Boson
- Brillouin scattering
- Fracton
- Linear elasticity
- Mechanical wave
- Phonon scattering
- Carrier scattering
- Phononic crystal
- Rayleigh wave
- Relativistic heat conduction
- Rigid unit modes
- SASER
- Second sound
- Surface acoustic wave
- Surface phonon
- Thermal conductivity
- Vibration
References
- ISBN 978-3-540-85062-5.
- ^ ISBN 978-1-107-13739-4.
- ISBN 978-0-19-968077-1.
- ISBN 978-0-19-851536-4.
- ^ a b Maradudin, A.; Montroll, E.; Weiss, G.; Ipatova, I. (1971). Theory of lattice dynamics in the harmonic approximation. Solid State Physics. Vol. Supplement 3 (Second ed.). New York: Academic Press.
- ISBN 9780070409545.
- ISBN 978-0486432618.
- ISBN 978-0-306-46338-9.
- ^ a b
Yu, Peter Y.; Cardona, Manuel (2010). "Fig. 3.2: Phonon dispersion curves in GaAs along high-symmetry axes". Fundamentals of Semiconductors. Physics and Materials Properties (4th ed.). Springer. p. 111. ISBN 978-3-642-00709-5.
- ^ ISBN 978-0-12-384954-0.
- ^ ISBN 0-03-083993-9.
- ISBN 978-0-691-14016-2.
- ISBN 978-0-471-41526-8.
- ^ a b "Non-metals: thermal phonons". University of Cambridge Teaching and Learning Packages Library. Retrieved 15 August 2020.
- ISBN 978-93-80931-89-0.
- ^ a b "Tunneling across a tiny gap". News.mit.edu. 7 April 2015. Retrieved 13 August 2019.
- ISBN 978-0-8053-2508-9.
- S2CID 17019967.
- .
- ^ Alberto Nicolis and Riccardo Penco. (2017). Mutual Interactions of Phonons, Rotons, and Gravity, Arxiv.org, Retrieved November 27, 2018
- ^ Angelo Esposito, Rafael Krichevsky, and Alberto Nicolis. (2018). The mass of sound Retrieved November 11, 2018
- ^ "Researchers suggest phonons may have mass and perhaps negative gravity". Phys.org. Retrieved 13 August 2019.
- ISBN 0486435032.
- S2CID 195774243.
- S2CID 248665478.
- S2CID 248665478.
External links
- Quotations related to Phonon at Wikiquote
- Explained: Phonons, MIT News, 2010.
- Optical and acoustic modes
- Phonons in a One Dimensional Microfluidic Crystal [1] and [2] with movies in [3].