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A phonon is a quasiparticle or collective excitation associated with mechanical vibrations in a periodic, elastic arrangement of atoms or molecules in condensed matter, especially in solids and some liquids. In quantum mechanics, a phonon is the quantized excitation of a normal mode of vibration in an interacting lattice.^[1] Phonons can be regarded as quantized sound waves, in close analogy to photons as the quanta of light.^[2]

The study of phonons is a central part of condensed matter physics because phonons play a major role in the thermal, elastic, optical, and electronic properties of materials. They are essential in understanding thermal conductivity, aspects of electrical conductivity, lattice heat capacity, neutron scattering, and important interactions in solids such as electron–phonon coupling.

The concept of the phonon was introduced in 1930 by the Soviet physicist Igor Tamm, and the name phonon was suggested by Yakov Frenkel.^[3] The word derives from the Greek φωνή (phonē), meaning sound or voice, reflecting the fact that long-wavelength phonons correspond to sound propagation in matter. The analogy with photons emphasizes the wave–particle duality of lattice vibrations.

Acoustic and optical lattice vibrations in a diatomic chain. In acoustic modes neighboring atoms move largely in phase, while in optical modes they move out of phase.

Definition

A phonon is the quantum-mechanical description of an elementary vibrational motion in which a crystal lattice oscillates at a single frequency.^[4] In classical mechanics, this corresponds to a normal mode of vibration. Any general lattice vibration may be decomposed into a superposition of such normal modes, in much the same way that an arbitrary periodic signal can be decomposed into Fourier components.

In the classical description these modes are wave-like. In the quantum description they also have particle-like properties, and the discrete packet of vibrational energy associated with a given mode is called a phonon.

Lattice dynamics

In a crystalline solid, atoms occupy equilibrium positions in a regular lattice. Since the atoms remain near these positions, they must exert forces on one another. These restoring forces may arise from covalent bonds, electrostatic attraction, Van der Waals forces, and related interactions. The full many-body problem is complicated, so in lattice dynamics two common approximations are often introduced.

First, one usually considers mainly interactions between neighboring atoms. Second, the interaction potential is approximated as harmonic near equilibrium. This means the potential is expanded to quadratic order, so the restoring force is proportional to displacement. In that approximation, the lattice behaves like a collection of masses connected by springs.

For a regular lattice, the potential energy can be written schematically in terms of atomic displacements as a sum over neighboring pairs. Although highly simplified, this harmonic description is sufficient to explain the existence of phonons and their essential properties.^[5]^[6]

Lattice waves

Because atoms in a lattice are coupled, displacing one atom from equilibrium affects its neighbors and gives rise to propagating vibrational waves through the crystal. These are lattice waves. Their amplitudes are given by the atomic displacements, and for periodic normal modes the waves have definite wavelengths and frequencies.

There is a shortest distinct wavelength, on the order of twice the lattice spacing. Wavelengths shorter than this are not independent because of the periodicity of the lattice. This restriction is reflected in the structure of reciprocal space and the first Brillouin zone.

Not every arbitrary lattice motion has a single wavelength and frequency, but the normal modes do. These normal modes are the basis from which phonons arise upon quantization.

One-dimensional lattice

To understand the essential physics, it is useful to begin with a one-dimensional chain of identical atoms connected by springs. Although simplified, this model already shows the key features of phonons.

Classical treatment

Suppose each atom has mass m and is connected to its nearest neighbors by springs of elastic constant C. If u_n denotes the displacement of the nth atom from equilibrium, then its equation of motion is

- 2 C u_{n} + C (u_{n + 1} + u_{n - 1}) = m \frac{d^{2} u_{n}}{d t^{2}} .

Because the atoms are coupled, these equations are not independent. They can be decoupled by expressing the displacements as a discrete Fourier series in terms of normal coordinates. This leads to independent oscillators labeled by wave number k, each with a frequency determined by the dispersion relation

ω_{k} = \sqrt{\frac{2 C}{m} (1 - \cos k a)} .

Each normal coordinate corresponds to a vibrational normal mode of the lattice. The relation between ω and k is called the dispersion relation. In the long-wavelength limit, the dispersion becomes approximately linear and the resulting wave behaves like an ordinary sound wave.

Quantum treatment

In the quantum version of the harmonic chain, the positions and momenta of the atoms become operators, and the lattice is described by a Hamiltonian of coupled harmonic oscillators:

ℋ = \sum_{i = 1}^{N} \frac{p_{i}^{2}}{2 m} + \frac{1}{2} m ω^{2} \sum_{{i j} (n n)} {(x_{i} - x_{j})}^{2} .

By transforming to normal coordinates in reciprocal space, the Hamiltonian becomes a sum of independent harmonic oscillators, one for each allowed wave number. The energy levels for each mode are then

E_{n} = (\frac{1}{2} + n) ℏ ω_{k} n = 0, 1, 2, 3 \dots

This shows that vibrational energy is quantized. The amount of energy required to raise a mode by one step is exactly ħω_k. This discrete excitation is the phonon.

Thus, a phonon is not a single atom vibrating on its own, but a quantized collective mode involving the lattice as a whole.

Three-dimensional lattice

The analysis generalizes naturally to three-dimensional crystals. In that case the scalar wave number is replaced by a wavevector k, and each allowed k is associated with multiple polarization directions.

In one dimension only longitudinal motion exists. In three dimensions, atomic displacements can occur parallel or perpendicular to the direction of wave propagation. This produces one longitudinal branch and two transverse branches. These different polarization directions are central to the classification of phonons in real crystals.

Dispersion relation

The phonon dispersion relation gives the dependence of phonon frequency on wavevector, ω = ω(k). It is one of the key tools in lattice dynamics because it determines the phonon spectrum and therefore many thermodynamic and transport properties of a crystal.

In a crystal with more than one atom in its primitive cell, there are multiple branches of the dispersion relation. For a simple diatomic chain, two branches appear: an acoustic branch and an optical branch. Their frequencies are given by

ω_{\pm}^{2} = K (\frac{1}{m_{1}} + \frac{1}{m_{2}}) \pm K \sqrt{{(\frac{1}{m_{1}} + \frac{1}{m_{2}})}^{2} - \frac{4 \sin^{2} \frac{k a}{2}}{m_{1} m_{2}}} .

The lower branch corresponds to acoustic modes and the upper branch to optical modes.^[7]

For long wavelengths, the acoustic branch is nearly linear. Its slope gives the group velocity and therefore the speed of sound in the crystal. At larger wavevectors, the dispersion deviates from linear behavior because of the underlying lattice structure.

Crystals with p atoms in the primitive cell have 3p phonon branches in three dimensions: three acoustic branches and 3p − 3 optical branches. Measured phonon dispersion curves, often obtained by inelastic neutron scattering, provide detailed information about the dynamics of real materials.^[8]

Acoustic and optical phonons

Solids with more than one atom in the smallest unit cell exhibit both acoustic phonons and optical phonons.

Acoustic phonons correspond to modes in which neighboring atoms move approximately in phase. In the long-wavelength limit this amounts to a uniform displacement of the crystal, which costs no restoring energy, so the frequency tends to zero as the wavelength becomes very large. Acoustic phonons are responsible for ordinary sound propagation in solids. Longitudinal and transverse acoustic phonons are often abbreviated LA and TA.

Optical phonons correspond to out-of-phase motion of atoms in the basis. In ionic crystals, these oscillations produce an oscillating dipole moment and can couple to electromagnetic radiation, especially in the infrared. For this reason they are called optical phonons.^[2] Longitudinal and transverse optical phonons are often labeled LO and TO. Optical phonons have a nonzero frequency at the center of the Brillouin zone.

Optical phonons may be infrared active or Raman active, making them important in vibrational spectroscopy and in the optical characterization of crystals.

Crystal momentum

Phonons are commonly assigned a quantity ħk analogous to momentum, but this is not ordinary mechanical momentum. It is more precisely called crystal momentum or pseudomomentum.^[9]

Because of lattice periodicity, wavevectors that differ by a reciprocal lattice vector describe equivalent states. This is why phonon wavevectors are usually taken inside the first Brillouin zone. The concept of crystal momentum is essential for analyzing scattering processes involving phonons, electrons, and photons in solids.

Operator formalism and second quantization

The phonon formalism becomes especially powerful when expressed using creation and annihilation operators. After transforming the lattice Hamiltonian into normal modes, one may define bosonic operators that create or destroy one quantum of vibrational energy in a given mode. The Hamiltonian then takes the form

ℋ = \sum_{k} ℏ ω_{k} ({b_{k}}^{†} b_{k} + \frac{1}{2})

or, more generally,

ℋ = \sum_{α} ℏ ω_{α} {a_{α}}^{†} a_{α} .

In this description, each phonon mode behaves like an independent quantum harmonic oscillator. The operators $a †$ and $a$ create and annihilate phonons respectively. The occupation number gives how many phonons are present in a given mode. This shows directly that phonons obey Bose–Einstein statistics and are therefore bosons.^[10]^[11]

The vacuum state is the state with no phonons. Excited states correspond to one or more phonons occupying one or more modes.

Thermodynamics

The thermodynamic properties of a crystal are strongly shaped by its phonon spectrum. The full set of allowed phonon states defines the phonon density of states, which determines how vibrational modes contribute to heat capacity, internal energy, and thermal transport.

At absolute zero, a harmonic crystal is in its ground state and contains no thermal phonons. At nonzero temperature, random lattice vibrations populate the available phonon states. These thermal phonons can be described statistically using the Bose–Einstein distribution:

n (ω_{k, s}) = \frac{1}{\exp (\frac{ℏ ω_{k, s}}{k_{B} T}) - 1}

where ω_k,s is the mode frequency, k_B is the Boltzmann constant, and T is the temperature.^[12]

High-frequency phonons contribute strongly to heat capacity, while low-frequency phonons are especially important in determining thermal conductivity. The resulting thermal phonon gas is analogous in many ways to a photon gas in black-body radiation.^[13]

Phonon tunneling

Phonons have also been shown to exhibit quantum tunneling behavior. Across nanometre-scale gaps, heat can be transferred by phonons that effectively tunnel between materials.^[14] This effect occurs over distances too large for normal contact conduction and too small for ordinary thermal radiation, making it a distinctly nanoscale quantum heat-transfer mechanism.

Superconductivity

Phonons play a fundamental role in conventional superconductivity. In such materials, electrons form Cooper pairs due to an effective attractive interaction mediated by phonons. This interaction, treated in the BCS theory framework, explains why superconductors can exhibit zero electrical resistance and expel magnetic fields below a critical temperature. Evidence for the importance of phonons is provided by the isotope effect, in which the superconducting critical temperature depends on ionic mass.^[15]

Other properties and research

Phonons continue to be an active topic of research in condensed matter and quantum physics. Nonlinear phonon processes, including phonon–phonon interactions and squeezed phonon states, have been studied theoretically and experimentally.^[16]^[17]

Recent theoretical work has also explored whether phonons and related excitations may be associated with effective mass and even unusual gravitational behavior in appropriate effective descriptions.^[18]^[19] Experimental advances have further enabled increasingly direct control and detection of phononic excitations in nanoscale and quantum systems.^[20]