Phonon

This article covers the phonon in solid state physics; for multimedia API Phonon KDE see Phonon (KDE) .

A phonon is the elementary excitation ( quantum ) of the elastic field. In solid-state physics , phonons describe elementary or collective excitations of the lattice vibrations of a solid and can be understood as bosonic quasiparticles .

The term phonon (from Greek φωνή phonē , German 'sound' ) was chosen in analogy to the oscillation quanta of the electromagnetic field , the photons , and was first used by JI Frenkel in 1932 in his book  Wave Mechanics, Elementary Theory .

Modes of vibration

Comparison of optical and acoustic transverse waves of phonons on a 2-atom basis for small k.

longitudinal acoustic mode

In a three-dimensional crystal with atoms in the primitive base , possible vibration modes exist for every wave vector compatible with the crystal symmetry : acoustic (one longitudinal and two transversal ) and optical: ${\displaystyle N}$ ${\displaystyle 3N}$ ${\displaystyle 3}$ ${\displaystyle (3N-3)}$

• Acoustic phonons (also known as sound quanta) are the quanta of sound waves that travel through the crystal lattice . In the center of the Brillouin zone , neighboring atoms move in the same direction.
• In contrast, in optical phonons the atoms within the base move against each other. The term “optical” is based on the fact that the oscillation frequencies of optical phonons are often in the range of infrared or visible light .

The designation of optical phonons takes place regardless of whether the phonons are actually optically active in the sense that phonons interact with a photon : Interactions with photons are not just that a phonon can be generated by absorbing a photon , or that conversely, a photon can be emitted by annihilating a phonon. Rather, there are also interactions of a photon with two phonons and an electron-photon-phonon interaction. Phonons can only be optically active when there is electrical polarization within the base , which is generally the case when the base is made up of different atoms. Crystals that interact with infrared photons are called infrared-active . Examples of such lattices are ion lattices , for example in sodium chloride crystals .

The model of the lattice vibrations assumes a crystalline order. Also amorphous solids such as glasses show oscillations of atoms to each other, but it is designated not as phonon vibrations. The influence of the disorder is small for long-wave acoustic vibrations .

Excitation energy and statistics

If one considers harmonic lattice oscillations in reciprocal space , one obtains decoupled oscillations in the momentum space ( normal oscillations ). The energy states of these oscillations are according to the levels of a harmonic oscillator ${\displaystyle \varepsilon _{n}}$

${\displaystyle \varepsilon _{n}(\mathbf {k} )=\hbar \cdot \omega (\mathbf {k} )\cdot \left(n+{\frac {1}{2}}\right)}$.

The frequency depends on the oscillation mode and the wave vector , see dispersion . ${\displaystyle \omega }$ ${\displaystyle n}$ ${\displaystyle \mathbf {k} }$

Since phonons belong to the bosons , the mean occupation number in thermal equilibrium is calculated according to the Bose-Einstein distribution as ${\displaystyle \langle n\rangle }$

${\displaystyle \langle n\rangle ={\frac {1}{e^{\hbar \cdot \omega /(k_{\mathrm {B} }\cdot T)}-1}}}$

With

• ${\displaystyle \hbar }$: reduced Planck's quantum of action
• ${\displaystyle k_{\mathrm {B} }}$: Boltzmann constant
• ${\displaystyle T}$: absolute temperature .

The occupation statistics are independent of the chemical potential , because the particle number of the phonons is not a conserved quantity . ${\displaystyle \mu }$

Usually (as above) statistical mixtures of states with a certain number of phonons ( Fock states ) are used. As Roy J. Glauber showed for photons in 1963, there are also so-called coherent states with an indefinite number of particles for phonons , which are very similar to classical lattice vibrations. While the expected value of the deflection is  0 in Fock states , it satisfies the classical time dependence of lattice vibrations in coherent phonon states.

proof

The phonon dispersion, i.e. H. The relationship between the energy and the momentum of the lattice vibrations can be examined using inelastic neutron scattering , inelastic X-ray scattering and high-resolution electron energy loss spectroscopy ( HREELS ). Phonons with small momentum, i.e. H. in the center of the Brillouin zone , can be detected by Raman , infrared spectroscopy or Brillouin scattering . The first phonon dispersion curve was recorded in 1955 at the Chalk River reactor by Bertram Brockhouse with neutron scattering on an aluminum single crystal .

Dispersion

The dispersion relation indicates the dependence of the energy or angular frequency on the pulse or wave number . For phonons, this relationship results from Newton's equation of motion . For this purpose it is assumed that the atoms are in a periodic potential in which they oscillate . ${\displaystyle \omega }$ ${\displaystyle k}$ ${\displaystyle V}$

Two neighboring atoms have a phase difference of , whereby the distance between two neighboring atoms is in the rest position. A phase difference of corresponds to one of zero; higher phase differences are accordingly equivalent to a value between and . For reasons of symmetry, consider the interval between and . The corresponding values from the first Brillouin zone , ie . This means that all physically relevant wave numbers have been covered. ${\displaystyle ka}$ ${\displaystyle a}$ ${\displaystyle 2\pi }$ ${\displaystyle 0}$ ${\displaystyle 2\pi }$ ${\displaystyle -\pi }$ ${\displaystyle \pi }$ ${\displaystyle k}$ ${\displaystyle k\in \lbrack -\pi /a,\pi /a\rbrack }$

Acoustic modes

Dispersion relation ${\displaystyle \omega (k)}$

For the simple model of a linear chain of atoms connected by springs, the dispersion relation is as a first approximation

${\displaystyle \omega (k)=2{\sqrt {\frac {C}{m}}}\left|\sin \left({\frac {ka}{2}}\right)\right|}$,

where C (in kg / s ^ 2) is the spring constant between the two adjacent planes and m is the mass of the atom.

For low values ​​of the expression is approximate ${\displaystyle k\left(ak\ll 1\right)}$

${\displaystyle \omega (k)\approx a{\sqrt {\frac {C}{m}}}|k|=c_{\mathrm {s} }|k|}$.

${\displaystyle c_{\mathrm {s} }}$is the speed of sound. The following applies at the zone boundaries

${\displaystyle \omega =2{\sqrt {\frac {C}{m}}}={\text{const}}.}$

The group speed , i.e. the speed of energy transport in the medium, results in

${\displaystyle v_{\mathrm {g} }={\frac {\mathrm {d} \omega }{\mathrm {d} k}}={\sqrt {\frac {Ca^{2}}{m}}}\cos \left({\frac {ka}{2}}\right)}$.

At the edge of the zone, the group speed is zero: the wave behaves like a standing wave.

Optical modes

Optical branches only exist on a polyatomic basis . The formula describes the dispersion relation for the model of a linear chain with two different atoms, which have the masses and . The force constant remains constant. It turns out ${\displaystyle m_{1}}$ ${\displaystyle m_{2}}$ ${\displaystyle C}$

${\displaystyle \omega ^{2}(k)={\frac {C(m_{1}+m_{2})}{m_{1}\cdot m_{2}}}\left[1+{\sqrt {1-{\frac {4m_{1}\cdot m_{2}}{(m_{1}+m_{2})^{2}}}\cdot \sin ^{2}\left({\frac {ka}{2}}\right)}}\,\right]}$

and thus approximately for the optical branch. ${\displaystyle \omega ^{2}\approx 2\,{\frac {C(m_{1}+m_{2})}{m_{1}\cdot m_{2}}}}$

The optical branch normally has a higher frequency than the acoustic branch and almost no dispersion. In the above formula, the acoustic branch corresponds to a minus sign in front of the root.

See also

• Kohn anomaly

literature

• Charles Kittel: Introduction to Solid State Physics . Oldenbourg, 2002
• Michael A. Stroscio, Mitra Dutta: Phonons in nanostructures. Cambridge Univ. Press, Cambridge 2005, ISBN 978-0-521-01805-0 .

Web links

Commons : Phonon  - collection of pictures, videos and audio files

• Introductory presentation of phonons and coherent phonons in the linear chain

Individual evidence

1. ↑ Jakow Ilyich Frenkel: Wave Mechanics. Elementary Theory. Clarendon Press, Oxford 1932.
2. ↑ The term optically active phonons must also be differentiated from the term optical activity of transparent materials.
3. ↑ Udo Scherz: Lecture notes "Theoretical Optics", WS 2012, Chapter 6.3 (PDF)
4. ^ BN Brockhouse, AT Stewart: Scattering of Neutrons by Phonons in an Aluminum Single Crystal . In: Physical Review . 100, 1955, p. 756. doi : 10.1103 / PhysRev.100.756 .
5. ^ Siegfried Hunklinger: Solid State Physics . 5th edition. De Gruyter, Berlin / Boston 2018, ISBN 978-3-11-056774-8 , pp. 187-192 .
6. ↑ Kittel: Introduction to Solid State Physics . 5th edition. Oldenbourg, 1980, p. 134 ff.
7. ^ Ibach, Lüth: Solid State Physics . Springer 1990, p. 57