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| -rw-r--r-- | source/know/concept/clausius-mossotti-relation/index.md | 6 | ||||
| -rw-r--r-- | source/know/concept/lyddane-sachs-teller-relation/index.md | 250 |
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diff --git a/source/know/concept/clausius-mossotti-relation/index.md b/source/know/concept/clausius-mossotti-relation/index.md index a0f4916..03bdcac 100644 --- a/source/know/concept/clausius-mossotti-relation/index.md +++ b/source/know/concept/clausius-mossotti-relation/index.md @@ -55,7 +55,8 @@ the dipole term will be dominant in that case, given by: $$\begin{aligned} V_i(\vb{r}) - \approx \frac{1}{4 \pi \varepsilon_0} \frac{1}{|\vb{r}|^2} \int \rho_i(\vb{r}') \: |\vb{r}'| \cos{\theta} \dd{\vb{r}'} + \approx \frac{1}{4 \pi \varepsilon_0} \frac{1}{|\vb{r}|^2} + \int_{-\infty}^\infty \rho_i(\vb{r}') \: |\vb{r}'| \cos{\theta} \dd{\vb{r}'} \end{aligned}$$ Where $$\theta$$ is the angle between $$\vb{r}$$ and $$\vb{r}'$$, @@ -64,7 +65,8 @@ with the unit vector $$\vu{r}$$, normalized from $$\vb{r}$$: $$\begin{aligned} V_i(\vb{r}) - = \frac{1}{4 \pi \varepsilon_0} \frac{1}{|\vb{r}|^2} \: \vu{r} \cdot \!\!\int \vb{r}' \rho_i(\vb{r}') \dd{\vb{r}'} + = \frac{1}{4 \pi \varepsilon_0} \frac{1}{|\vb{r}|^2} + \: \vu{r} \cdot \!\!\int_{-\infty}^\infty \vb{r}' \rho_i(\vb{r}') \dd{\vb{r}'} \end{aligned}$$ The integral is a more general definition of the dipole moment $$\vb{p}_i$$. diff --git a/source/know/concept/lyddane-sachs-teller-relation/index.md b/source/know/concept/lyddane-sachs-teller-relation/index.md new file mode 100644 index 0000000..e80bf00 --- /dev/null +++ b/source/know/concept/lyddane-sachs-teller-relation/index.md @@ -0,0 +1,250 @@ +--- +title: "Lyddane-Sachs-Teller relation" +sort_title: "Lyddane-Sachs-Teller relation" +date: 2024-04-15 +categories: +- Physics +layout: "concept" +--- + +While the [Lorentz oscillator model](/know/concept/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](/know/concept/electromagnetic-wave-equation/) +passes by, its [electric field](/know/concept/electric-field/) +$$\vb{E}(t)$$ 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 $$\omega$$ and wavenumber $$k$$, +and the phonon $$\Omega$$ and $$K$$, then: + +$$\begin{aligned} + \hbar \omega + = \hbar \Omega + \qquad \qquad + \hbar k + = \hbar K +\end{aligned}$$ + +In other words, such a conversion can only take place +at intersections of the dispersion relations $$\omega(k)$$ and $$\Omega(K)$$. +The latter consists of two branches: +low-frequency *acoustic* modes and higher-frequency *optical* modes. +Meanwhile, the photon dispersion is simply $$\omega = c k / n$$, +where $$n$$ is the medium's refractive index. + +For acoustic phonons, the dispersions only intersect at $$k = K = 0$$, +which is simply a static solid in a static electric field. +For optical phonons, the intersection is at a nonzero $$k$$. +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 $$\ipdv{\omega}{k}$$ and $$\ipdv{\Omega}{K}$$ in this case. +Clearly, light is much faster than sound, +so $$\omega(k)$$ is much steeper than $$\Omega(K)$$, +meaning that the photon-phonon conversion +will happen at relatively low $$k$$. +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 $$z$$-axis, +each containing a positive and a negative ion +oscillating transversely along the $$x$$-axis. +For optical phonon modes, the ions always move in opposite directions. +Let the ions have masses $$m_{-}$$ and $$m_{+}$$, +then the Lorentz oscillator model tells us +that the displacements $$\vb{x}_{+}(t)$$ and $$\vb{x}_{-}(t)$$ are governed by: + +$$\begin{aligned} + m_{+} \dvn{2}{\vb{x}_{+}}{t} + &= - \kappa (\vb{x}_{+} - \vb{x}_{-}) + q \vb{E} + \\ + m_{-} \dvn{2}{\vb{x}_{-}}{t} + &= - \kappa (\vb{x}_{-} - \vb{x}_{+}) - q \vb{E} +\end{aligned}$$ + +Where $$\vb{E}(t) = \vb{E}_0 e^{- i \omega t}$$ represents the light, +and $$\kappa$$ 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 $$\vb{x}_{+} - \vb{x}_{-}$$ instead of $$\vb{x}_{+}$$. + +Respectively dividing the equations by $$m_{+}$$ and $$m_{-}$$ +and subtracting the latter from the former, +we arrive at the following combined equation, +where $$m$$ is the [reduced mass](/know/concept/reduced-mass/): + +$$\begin{aligned} + \dvn{2}{}{t} (\vb{x}_{+} - \vb{x}_{-}) + = - \frac{\kappa}{m} (\vb{x}_{+} - \vb{x}_{-}) + \frac{q}{m} \vb{E} +\end{aligned}$$ + +Defining the relative displacement $$\vb{x} \equiv \vb{x}_{+} \!-\! \vb{x}_{-}$$, +and recognizing that $$\kappa / m$$ is the TO phonons' +natural resonance frequency $$\Omega_\mathrm{TO}^2$$: + +$$\begin{aligned} + \dvn{2}{\vb{x}}{t} + \Omega_\mathrm{TO}^2 \vb{x} + = \frac{q}{m} \vb{E} +\end{aligned}$$ + +Note that $$\Omega_\mathrm{TO}$$ is the phonon frequency for $$K = 0$$. +This is because IR light waves are much larger than the crystal's unit cell, +so we are ignoring all spatial variation in $$\vb{E}$$ +(i.e. the [electric dipole approximation](/know/concept/electric-dipole-approximation/)). +This is equivalent to assuming that $$K \approx 0$$. + +For the sake of generality, +we also introduce an empirical damping rate $$\gamma$$, +like in the original Lorentz oscillator model: + +$$\begin{aligned} + \dvn{2}{\vb{x}}{t} + \gamma \dv{\vb{x}}{t} + \Omega_\mathrm{TO}^2 \vb{x} + = \frac{q}{m} \vb{E} +\end{aligned}$$ + +Inserting the ansatz $$\vb{x}(t) = \vb{x}_0 e^{- i \omega t}$$ +and isolating for the amplitude $$\vb{x}_0$$, we find: + +$$\begin{aligned} + \vb{x}_0 + = \frac{q \vb{E}_0}{m (\Omega_\mathrm{TO}^2 - \omega^2 - i \gamma \omega)} +\end{aligned}$$ + +The induced polarization density $$\vb{P}$$ is then the sum +of the electrons' and ions' contributions $$\vb{P}_e$$ and $$\vb{P}_i$$. +The former is described by a background susceptibility $$\chi$$, +and the latter by each unit cell's dipole moment $$\vb{p} = q \vb{x}$$ +multiplied by the number of cells per unit volume $$N$$: + +$$\begin{aligned} + \vb{P} + \approx \varepsilon_0 \chi \vb{E} + N q \vb{x} + = \bigg( \varepsilon_0 \chi + \frac{N q^2}{m (\Omega_\mathrm{TO}^2 - \omega^2 - i \gamma \omega)} \bigg) \vb{E} +\end{aligned}$$ + +Note that we are neglecting how each dipole shields its neighbors. +This approximation can be improved afterwards by using +the [Clausius-Mossotti relation](/know/concept/clausius-mossotti-relation/). + +With our expression for $$\vb{P}$$, we can find +the [dielectric function](/know/concept/dielectric-function/) $$\varepsilon_r(\omega)$$ +using the definition of the electric displacement field +$$\vb{D} = \varepsilon_0 \vb{E} + \vb{P} = \varepsilon_0 \varepsilon_r \vb{E}$$, +yielding: + +$$\begin{aligned} + \boxed{ + \varepsilon_r(\omega) + = 1 + \chi(\omega) + \frac{N q^2}{\varepsilon_0 m (\Omega_\mathrm{TO}^2 - \omega^2 - i \gamma \omega)} + } +\end{aligned}$$ + +In the limits of low and high frequencies $$\omega$$, +we see that $$\varepsilon_r$$ is higher in the former: + +$$\begin{aligned} + \varepsilon_{\mathrm{low}} + &= \, \lim_{\omega \to 0} \, \varepsilon_r(\omega) + = 1 + \chi_\mathrm{low} + \frac{N q^2}{\varepsilon_0 m \Omega_\mathrm{TO}^2} + \\ + \varepsilon_{\mathrm{high}} + &= \lim_{\omega \to \infty} \varepsilon_r(\omega) + = 1 + \chi_\mathrm{high} +\end{aligned}$$ + +We can use these quantities to rewrite the relative permittivity $$\varepsilon_r$$ as follows: + +$$\begin{aligned} + \varepsilon_r(\omega) + = \varepsilon_{\mathrm{high}} + (\varepsilon_{\mathrm{low}} - \varepsilon_{\mathrm{high}}) + \frac{\Omega_\mathrm{TO}^2}{\Omega_\mathrm{TO}^2 - \omega^2 - i \gamma \omega} +\end{aligned}$$ + +For weak damping $$\gamma \approx 0$$, there exists a frequency, +which we will call $$\Omega_\mathrm{LO}$$ in anticipation, +where the dielectric function is zero: + +$$\begin{aligned} + 0 + = \varepsilon_r(\Omega_\mathrm{LO}) + = \varepsilon_{\mathrm{high}} + + (\varepsilon_{\mathrm{low}} - \varepsilon_{\mathrm{high}}) \frac{\Omega_\mathrm{TO}^2}{\Omega_\mathrm{TO}^2 - \Omega_\mathrm{LO}^2} +\end{aligned}$$ + +The physical significance of $$\varepsilon_r = 0$$ can be +seen from [Gauss' law](/know/concept/maxwells-equations), under the assumption that there is +no net charge density: + +$$\begin{aligned} + \nabla \cdot \vb{D} + = \varepsilon_0 \varepsilon_r \nabla \cdot \vb{E} + = 0 +\end{aligned}$$ + +If $$\varepsilon_r \neq 0$$, then $$\nabla \cdot \vec{E} = 0$$, +corresponding to a transverse light wave as usual. +However, if $$\varepsilon_r = 0$$, then $$\nabla \cdot \vec{E} \neq 0$$, +representing a longitudinal electric wave, like a plasmon in metal. +Rearranging the equation for $$\Omega_\mathrm{LO}$$ +gives us the **Lyddane-Sachs-Teller (LST) relation**: + +$$\begin{aligned} + \boxed{ + \frac{\Omega_\mathrm{LO}^2}{\Omega_\mathrm{TO}^2} + = \frac{\varepsilon_{\mathrm{low}}}{\varepsilon_{\mathrm{high}}} + } +\end{aligned}$$ + +$$\Omega_\mathrm{LO}$$ is the natural frequency +of such **longitudinal optical (LO) phonons** for $$K = 0$$. +Recall that only transverse phonons interact with light: +the significance of this result is that we can measure +$$\varepsilon_\mathrm{low}$$, $$\varepsilon_\mathrm{high}$$, +and $$\Omega_\mathrm{TO}$$ 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 $$\omega$$-values between $$\Omega_\mathrm{TO}$$ and $$\Omega_\mathrm{LO}$$, +the permittivity $$\varepsilon_r$$ is negative, +meaning the reflectivity $$R$$ equals $$1$$, +i.e. the material becomes a perfect reflector: + +$$\begin{aligned} + R + = \bigg| \frac{i \sqrt{|\varepsilon_r|} - 1}{i \sqrt{|\varepsilon_r|} + 1} \bigg|^2 + = \frac{|\varepsilon_r|^2 + 1^2}{|\varepsilon_r|^2 + 1^2} + = 1 +\end{aligned}$$ + +This region of 100% reflectivity is called the **Reststrahlen band**. +In practice, real materials have $$\gamma > 0$$, which reduces $$R$$ somewhat. + +Because the photons and TO phonons interact so strongly +for $$\omega \approx \Omega_\mathrm{TO}$$, +they can be treated as a single **phonon polariton** there, +with a dispersion relation given by: + +$$\begin{aligned} + \omega_\mathrm{pp}(K) + = \frac{c}{\sqrt{\varepsilon_r(\omega_\mathrm{pp})}} K +\end{aligned}$$ + +Earlier, when treating the photon and phonon separately, +we wanted the intersection between $$\omega(k)$$ and $$\Omega(K)$$. +But now, for $$\omega_\mathrm{pp}(K)$$, there is none! This is a good example +of the typical *anti-crossing* behavior of strongly coupled systems. + + + +## References +1. M. Fox, + *Optical properties of solids*, 2nd edition, + Oxford. |