Categories: Physics.

Lyddane-Sachs-Teller relation

While the Lorentz oscillator model originally studied the electric dipole formed by an electron and its nucleus, it can also be applied to the nuclei of polar crystals, i.e. crystals held together by polar bonds between ions. When an electromagnetic wave passes by, its electric field E(t)\vb{E}(t) exerts a force on the ions, leading to an optical response.

We are talking about light waves (photons) creating lattice vibrations (phonons), i.e. a photon-phonon conversion, where the total energy and momentum must be conserved. If the photon has frequency ω\omega and wavenumber kk, and the phonon Ω\Omega and KK, then:

ω=Ωk=K\begin{aligned} \hbar \omega = \hbar \Omega \qquad \qquad \hbar k = \hbar K \end{aligned}

In other words, such a conversion can only take place at intersections of the dispersion relations ω(k)\omega(k) and Ω(K)\Omega(K). The latter consists of two branches: low-frequency acoustic modes and higher-frequency optical modes. Meanwhile, the photon dispersion is simply ω=ck/n\omega = c k / n, where nn is the medium’s refractive index.

For acoustic phonons, the dispersions only intersect at k=K=0k = K = 0, which is simply a static solid in a static electric field. For optical phonons, the intersection is at a nonzero kk. In addition, light is a transverse wave, so it can only interact with transverse phonons, meaning that we must only consider transverse optical (TO) phonons.

A wave’s group velocity is the slope of its dispersion, so ω/k\ipdv{\omega}{k} and Ω/K\ipdv{\Omega}{K} in this case. Clearly, light is much faster than sound, so ω(k)\omega(k) is much steeper than Ω(K)\Omega(K), meaning that the photon-phonon conversion will happen at relatively low kk. In practice, the intersection is in the infrared (IR), hence TO phonons are sometimes called IR active.

We consider a 1D chain of unit cells along the zz-axis, each containing a positive and a negative ion oscillating transversely along the xx-axis. For optical phonon modes, the ions always move in opposite directions. Let the ions have masses mm_{-} and m+m_{+}, then the Lorentz oscillator model tells us that the displacements x+(t)\vb{x}_{+}(t) and x(t)\vb{x}_{-}(t) are governed by:

m+d2x+dt2=κ(x+x)+qEmd2xdt2=κ(xx+)qE\begin{aligned} m_{+} \dvn{2}{\vb{x}_{+}}{t} &= - \kappa (\vb{x}_{+} - \vb{x}_{-}) + q \vb{E} \\ m_{-} \dvn{2}{\vb{x}_{-}}{t} &= - \kappa (\vb{x}_{-} - \vb{x}_{+}) - q \vb{E} \end{aligned}

Where E(t)=E0eiωt\vb{E}(t) = \vb{E}_0 e^{- i \omega t} represents the light, and κ\kappa is the spring constant of the polar bonds’ restoring force. Note that the latter depends on the displacement between the ions, instead of from their equilibrium position, so we need to write x+x\vb{x}_{+} - \vb{x}_{-} instead of x+\vb{x}_{+}.

Respectively dividing the equations by m+m_{+} and mm_{-} and subtracting the latter from the former, we arrive at the following combined equation, where mm is the reduced mass:

d2dt2(x+x)=κm(x+x)+qmE\begin{aligned} \dvn{2}{}{t} (\vb{x}_{+} - \vb{x}_{-}) = - \frac{\kappa}{m} (\vb{x}_{+} - \vb{x}_{-}) + \frac{q}{m} \vb{E} \end{aligned}

Defining the relative displacement xx+ ⁣ ⁣x\vb{x} \equiv \vb{x}_{+} \!-\! \vb{x}_{-}, and recognizing that κ/m\kappa / m is the TO phonons’ natural resonance frequency ΩTO2\Omega_\mathrm{TO}^2:

d2xdt2+ΩTO2x=qmE\begin{aligned} \dvn{2}{\vb{x}}{t} + \Omega_\mathrm{TO}^2 \vb{x} = \frac{q}{m} \vb{E} \end{aligned}

Note that ΩTO\Omega_\mathrm{TO} is the phonon frequency for K=0K = 0. This is because IR light waves are much larger than the crystal’s unit cell, so we are ignoring all spatial variation in E\vb{E} (i.e. the electric dipole approximation). This is equivalent to assuming that K0K \approx 0.

For the sake of generality, we also introduce an empirical damping rate γ\gamma, like in the original Lorentz oscillator model:

d2xdt2+γdxdt+ΩTO2x=qmE\begin{aligned} \dvn{2}{\vb{x}}{t} + \gamma \dv{\vb{x}}{t} + \Omega_\mathrm{TO}^2 \vb{x} = \frac{q}{m} \vb{E} \end{aligned}

Inserting the ansatz x(t)=x0eiωt\vb{x}(t) = \vb{x}_0 e^{- i \omega t} and isolating for the amplitude x0\vb{x}_0, we find:

x0=qE0m(ΩTO2ω2iγω)\begin{aligned} \vb{x}_0 = \frac{q \vb{E}_0}{m (\Omega_\mathrm{TO}^2 - \omega^2 - i \gamma \omega)} \end{aligned}

The induced polarization density P\vb{P} is then the sum of the electrons’ and ions’ contributions Pe\vb{P}_e and Pi\vb{P}_i. The former is described by a background susceptibility χ\chi, and the latter by each unit cell’s dipole moment p=qx\vb{p} = q \vb{x} multiplied by the number of cells per unit volume NN:

Pε0χE+Nqx=(ε0χ+Nq2m(ΩTO2ω2iγω))E\begin{aligned} \vb{P} \approx \varepsilon_0 \chi \vb{E} + N q \vb{x} = \bigg( \varepsilon_0 \chi + \frac{N q^2}{m (\Omega_\mathrm{TO}^2 - \omega^2 - i \gamma \omega)} \bigg) \vb{E} \end{aligned}

Note that we are neglecting how each dipole shields its neighbors. This approximation can be improved afterwards by using the Clausius-Mossotti relation.

With our expression for P\vb{P}, we can find the dielectric function εr(ω)\varepsilon_r(\omega) using the definition of the electric displacement field D=ε0E+P=ε0εrE\vb{D} = \varepsilon_0 \vb{E} + \vb{P} = \varepsilon_0 \varepsilon_r \vb{E}, yielding:

εr(ω)=1+χ(ω)+Nq2ε0m(ΩTO2ω2iγω)\begin{aligned} \boxed{ \varepsilon_r(\omega) = 1 + \chi(\omega) + \frac{N q^2}{\varepsilon_0 m (\Omega_\mathrm{TO}^2 - \omega^2 - i \gamma \omega)} } \end{aligned}

In the limits of low and high frequencies ω\omega, we see that εr\varepsilon_r is higher in the former:

εlow=limω0εr(ω)=1+χlow+Nq2ε0mΩTO2εhigh=limωεr(ω)=1+χhigh\begin{aligned} \varepsilon_{\mathrm{low}} &= \, \lim_{\omega \to 0} \, \varepsilon_r(\omega) = 1 + \chi_\mathrm{low} + \frac{N q^2}{\varepsilon_0 m \Omega_\mathrm{TO}^2} \\ \varepsilon_{\mathrm{high}} &= \lim_{\omega \to \infty} \varepsilon_r(\omega) = 1 + \chi_\mathrm{high} \end{aligned}

We can use these quantities to rewrite the relative permittivity εr\varepsilon_r as follows:

εr(ω)=εhigh+(εlowεhigh)ΩTO2ΩTO2ω2iγω\begin{aligned} \varepsilon_r(\omega) = \varepsilon_{\mathrm{high}} + (\varepsilon_{\mathrm{low}} - \varepsilon_{\mathrm{high}}) \frac{\Omega_\mathrm{TO}^2}{\Omega_\mathrm{TO}^2 - \omega^2 - i \gamma \omega} \end{aligned}

For weak damping γ0\gamma \approx 0, there exists a frequency, which we will call ΩLO\Omega_\mathrm{LO} in anticipation, where the dielectric function is zero:

0=εr(ΩLO)=εhigh+(εlowεhigh)ΩTO2ΩTO2ΩLO2\begin{aligned} 0 = \varepsilon_r(\Omega_\mathrm{LO}) = \varepsilon_{\mathrm{high}} + (\varepsilon_{\mathrm{low}} - \varepsilon_{\mathrm{high}}) \frac{\Omega_\mathrm{TO}^2}{\Omega_\mathrm{TO}^2 - \Omega_\mathrm{LO}^2} \end{aligned}

The physical significance of εr=0\varepsilon_r = 0 can be seen from Gauss’ law, under the assumption that there is no net charge density:

D=ε0εrE=0\begin{aligned} \nabla \cdot \vb{D} = \varepsilon_0 \varepsilon_r \nabla \cdot \vb{E} = 0 \end{aligned}

If εr0\varepsilon_r \neq 0, then E=0\nabla \cdot \vec{E} = 0, corresponding to a transverse light wave as usual. However, if εr=0\varepsilon_r = 0, then E0\nabla \cdot \vec{E} \neq 0, representing a longitudinal electric wave, like a plasmon in metal. Rearranging the equation for ΩLO\Omega_\mathrm{LO} gives us the Lyddane-Sachs-Teller (LST) relation:

ΩLO2ΩTO2=εlowεhigh\begin{aligned} \boxed{ \frac{\Omega_\mathrm{LO}^2}{\Omega_\mathrm{TO}^2} = \frac{\varepsilon_{\mathrm{low}}}{\varepsilon_{\mathrm{high}}} } \end{aligned}

ΩLO\Omega_\mathrm{LO} is the natural frequency of such longitudinal optical (LO) phonons for K=0K = 0. Recall that only transverse phonons interact with light: the significance of this result is that we can measure εlow\varepsilon_\mathrm{low}, εhigh\varepsilon_\mathrm{high}, and ΩTO\Omega_\mathrm{TO} with light, and use that to calculate a quantity for an effect that we cannot interact with directly. The caveat is that this is only valid for simple polar crystals.

For ω\omega-values between ΩTO\Omega_\mathrm{TO} and ΩLO\Omega_\mathrm{LO}, the permittivity εr\varepsilon_r is negative, meaning the reflectivity RR equals 11, i.e. the material becomes a perfect reflector:

R=iεr1iεr+12=εr2+12εr2+12=1\begin{aligned} R = \bigg| \frac{i \sqrt{|\varepsilon_r|} - 1}{i \sqrt{|\varepsilon_r|} + 1} \bigg|^2 = \frac{|\varepsilon_r|^2 + 1^2}{|\varepsilon_r|^2 + 1^2} = 1 \end{aligned}

This region of 100% reflectivity is called the Reststrahlen band. In practice, real materials have γ>0\gamma > 0, which reduces RR somewhat.

Because the photons and TO phonons interact so strongly for ωΩTO\omega \approx \Omega_\mathrm{TO}, they can be treated as a single phonon polariton there, with a dispersion relation given by:

ωpp(K)=cεr(ωpp)K\begin{aligned} \omega_\mathrm{pp}(K) = \frac{c}{\sqrt{\varepsilon_r(\omega_\mathrm{pp})}} K \end{aligned}

Earlier, when treating the photon and phonon separately, we wanted the intersection between ω(k)\omega(k) and Ω(K)\Omega(K). But now, for ωpp(K)\omega_\mathrm{pp}(K), there is none! This is a good example of the typical anti-crossing behavior of strongly coupled systems.


  1. M. Fox, Optical properties of solids, 2nd edition, Oxford.