In solid-state physics, polaritons are quasiparticles arising from the strong coupling of photons with optical phonons in a crystal. These coupled modes play a central role in understanding the optical properties of ionic crystals, particularly in the infrared frequency range.

What are Polaritons?

• Polaritons are quanta of a coupled electromagnetic field and lattice vibration (phonon).
• They arise when transverse optical (TO) phonons interact with transverse electromagnetic waves (photons).

• This interaction modifies the propagation of waves through the crystal and leads to mixed electric-mechanical excitations.

• Resonance occurs when:
  • Frequencies match: $ \omega_{\text{photon}} \approx \omega_{\text{phonon}} $
  • Wavevectors match: $ k_{\text{photon}} \approx k_{\text{phonon}} $
• This coupling introduces new dispersion relations that deviate from the uncoupled phonon and photon dispersions.

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Electromagnetic Wave Equation with Polarization

In the presence of polarization $ \vec{P} $, Maxwell’s equations in CGS units give:

\[\frac{1}{c^2} \frac{\partial^2 \vec{E}}{\partial t^2} = \nabla^2 (\vec{E} + 4\pi \vec{P})\]

• The polarization $ \vec{P} $ is induced by displacement of ions: \(\vec{P} = N q \vec{u}\) where:
  • $ N $: number of ion pairs per unit volume
  • $ q $: effective charge
  • $ \vec{u} $: relative ionic displacement

The equation of motion for $ \vec{P} $ is that of a driven harmonic oscillator:

\[\frac{d^2 \vec{P}}{dt^2} + \omega_T^2 \vec{P} = \frac{N q^2}{M} \vec{E}\]

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Derivation:

The polarization is the dipole moment per unit volume:

\[\vec{P} = N q \vec{u} \tag{2}\]

The ions obey Newton’s second law under a restoring force and the applied electric field:

\[M \frac{d^2 \vec{u}}{dt^2} = -M \omega_T^2 \vec{u} + q \vec{E} \tag{3}\]

• The $-\omega_T^2$ term represents the restoring force due to the lattice (transverse optical phonon).
• The $q \vec{E}$ term represents the force from the external electric field.

From equation (2):

\[\vec{u} = \frac{\vec{P}}{N q}\]

Differentiate twice with respect to time:

\[\frac{d^2 \vec{u}}{dt^2} = \frac{1}{N q} \frac{d^2 \vec{P}}{dt^2}\]

Substitute into Newton’s law (3):

\[M \cdot \frac{1}{N q} \cdot \frac{d^2 \vec{P}}{dt^2} = -M \omega_T^2 \cdot \frac{\vec{P}}{N q} + q \vec{E}\]

Multiply both sides by $\frac{N q}{M}$:

\[\frac{d^2 \vec{P}}{dt^2} = -\omega_T^2 \vec{P} + \frac{N q^2}{M} \vec{E}\]

Rearranged:

\[\frac{d^2 \vec{P}}{dt^2} + \omega_T^2 \vec{P} = \frac{N q^2}{M} \vec{E} \tag{4}\]

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Polariton Dispersion Relation

Combining the above equations, the dispersion relation for polaritons becomes:

\[\varepsilon(\omega) = \varepsilon(\infty) + \frac{4\pi N q^2 / M}{\omega_T^2 - \omega^2}\]

At $ K = 0 $, two solutions emerge:

• $ \omega = 0 $: corresponds to a free photon
• $ \omega = \omega_T $: transverse optical phonon frequency in absence of coupling

The above equation can be obtained under the assumption of time-harmonic (monochromatic) solutions of the form:

• $\vec{E}(\vec{r}, t) = \vec{E}_0 e^{i(\vec{k} \cdot \vec{r} - \omega t)}$
• $\vec{P}(\vec{r}, t) = \vec{P}_0 e^{i(\vec{k} \cdot \vec{r} - \omega t)}$

Then:

• $\frac{\partial^2}{\partial t^2} \rightarrow -\omega^2$
• $\nabla^2 \rightarrow -k^2$

when plugged into the equation for $\vec{P}$ we get $\vec{P}_0$ as:

\[\vec{P}_0 = \frac{N q^2 / M}{\omega_T^2 - \omega^2} \vec{E}_0 \tag{3}\]

Finally from Maxwell’s Equation we get

\[-\frac{\omega^2}{c^2} \vec{E}_0 = -k^2 \left( \vec{E}_0 + 4\pi \vec{P}_0 \right)\]

Cancel negative signs and factor: \(\frac{\omega^2}{c^2} = k^2 \left(1 + \frac{4\pi N q^2 / M}{\omega_T^2 - \omega^2} \right)\)

Now define the frequency-dependent dielectric function:

\[\varepsilon(\omega) = 1 + \frac{4\pi N q^2 / M}{\omega_T^2 - \omega^2}\]