In condensed matter physics, a phonon polariton is a type of quasiparticle that can form in a diatomic ionic crystal due to coupling of transverse optical phonons and photons. They are particular type of polariton, which behave like bosons. Phonon polaritons occur in the region where the wavelength and energy of phonons and photons are similar, as to adhere to the avoided crossing principle. Phonon polariton spectra have traditionally been studied using Raman spectroscopy. The recent advances in (scattering-type) scanning near-field optical microscopy [(s-)SNOM] and atomic force microscopy (AFM) have made it possible to observe the polaritons in a more direct way.
Theory
A phonon polariton is a type of quasiparticle that can form in some crystals due to the coupling of photons and lattice vibrations. They have properties of both light and sound waves, and can travel at very slow speeds in the material. They are useful for manipulating electromagnetic fields at nanoscale and enhancing optical phenomena. Phonon polaritons only result from coupling of transverse optical phonons, this is due to the particular form of the dispersion relation of the phonon and photon and their interaction. Photons consist of electromagnetic waves, which are always transverse. Therefore, they can only couple with transverse phonons in crystals. Near k = 0 {\displaystyle \mathbf {k} =0} the dispersion relation of an acoustic phonon can be approximated as being linear, with a particular gradient giving a dispersion relation of the form ω a c = v a c k {\displaystyle \omega _{\rm {ac}}=v_{\rm {ac}}k} , with v a c {\displaystyle v_{\rm {ac}}} the speed of the wave, ω a c {\displaystyle \omega _{\rm {ac}}} the angular frequency and k the absolute value of the wave vector k {\displaystyle \mathbf {k} } . The dispersion relation of photons is also linear, being also of the form ω p = c k {\displaystyle \omega _{\rm {p}}=ck} , with c being the speed of light in vacuum. The difference lies in the magnitudes of their speeds, the speed of photons is many times larger than the speed for the acoustic phonons. The dispersion relations will therefore never cross each other, resulting in a lack of coupling. The dispersion relations touch at k = 0 {\displaystyle \mathbf {k} =0} , but since the waves have no energy, no coupling will occur. Optical phonons, by contrast, have a non-zero angular frequency at k = 0 {\displaystyle \mathbf {k} =0} and have a negative slope, which is also much smaller in magnitude to that of photons. This will result in the crossing of the optical phonon branch and the photon dispersion, leading to their coupling and the forming of a phonon polariton.
Dispersion relation The behavior of the phonon polaritons can be described by the dispersion relation. This dispersion relation is most easily derived for diatomic ion crystals with optical isotropy, for example sodium chloride and zinc sulfide. Since the atoms in the crystal are charged, any lattice vibration which changes the relative distance between the two atoms in the unit cell will change the dielectric polarization of the material. To describe these vibrations, it is useful to introduce the parameter w, which is given by:
w = q μ V {\displaystyle \mathbf {w} =\mathbf {q} {\sqrt {\frac {\mu }{V}}}}
Where
q {\displaystyle \mathbf {q} } is the displacement of the positive atom relative to the negative atom; μ is the reduced mass of the two atoms; V is the volume of the unit cell. Using this parameter, the behavior of the lattice vibrations for long waves can be described by the following equations:
w ¨ = − ω 0 2 w + ( ϵ 0 − ϵ ∞ 4 π ) 1 / 2 ω 0 E {\displaystyle {\ddot {\mathbf {w} }}=-{\omega _{0}}^{2}\mathbf {w} +({\frac {\epsilon _{0}-\epsilon _{\infty }}{4\pi }})^{1/2}\omega _{0}\mathbf {E} }
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