Interaction of Solids with EM Field

Learning Objectives:

Understand how solids interact with electromagnetic (EM) waves.
Introduce classical free-electron models of electrical conduction.
Derive electrical conductivity using the Drude model.
Identify the limitations of the Drude model.

When an electromagnetic field interacts with a solid, it can:

Induce currents (conductivity)
Polarize the material (dielectric response)
Be reflected, transmitted, or absorbed depending on the material properties

The interaction depends on:

The electronic structure of the solid
The frequency of the electromagnetic radiation
Scattering mechanisms inside the solid

Classical Free Electron Theory (Drude Model)

The Drude Model (1900) is the earliest attempt to explain electrical and thermal conductivity in metals using classical physics.

Assumptions:

Electrons behave like classical particles.
Electrons undergo random collisions (scattering) with fixed positive ions.
Between collisions, electrons accelerate freely under the influence of electric field E.
The average time between collisions is called the relaxation time $ \tau $.

An external electric field $ \mathbf{E} $ applies a force $ \mathbf{F} = -e\mathbf{E} $ on each electron.

The equation of motion becomes:

\[m \frac{d\mathbf{v}}{dt} = -e\mathbf{E} - \frac{m\mathbf{v}}{\tau}\]

In steady state ($ \frac{d\mathbf{v}}{dt} = 0 $):

\[\mathbf{v}_{\text{avg}} = -\frac{e\tau}{m} \mathbf{E}\]

Electrical Conductivity

Current density $ \mathbf{J} $ is given by:

\[\mathbf{J} = -ne\mathbf{v}_{\text{avg}} = \frac{ne^2\tau}{m} \mathbf{E}\]

Hence, electrical conductivity $ \sigma $ is:

\[\sigma = \frac{ne^2\tau}{m}\]

$ n $: Number of free electrons per unit volume
$ e $: Charge of electron
$ \tau $: Relaxation time
$ m $: Mass of electron

So:

\[\mathbf{J} = \sigma \mathbf{E}\]

Limitations of the Drude Model

Fails to explain temperature dependence of conductivity accurately.
Cannot explain positive Hall coefficient in some metals.
Does not account for quantum statistics (Fermi-Dirac distribution).
Over-simplifies electron-lattice interactions.

Macroscopic Theory of Optical Constants, Dispersion, and Absorption

Learning Objectives:

Understand how electromagnetic waves propagate in a medium.
Define key optical constants: refractive index, absorption coefficient, and dielectric function.
Relate microscopic material properties to macroscopic electromagnetic behavior.
Derive expressions for the complex dielectric function and refractive index.
Explore the Lorentz oscillator model and dispersion formulas.
Connect theory with observable properties of materials.

Electromagnetic Wave in Matter

Maxwell’s equations in a linear, isotropic, homogeneous medium (no free charge/current):

\[\nabla \cdot \mathbf{D} = 0, \quad \nabla \cdot \mathbf{B} = 0\] \[\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, \quad \nabla \times \mathbf{H} = \frac{\partial \mathbf{D}}{\partial t}\]

Constitutive relations:

\[\mathbf{D} = \varepsilon \mathbf{E}, \quad \mathbf{B} = \mu \mathbf{H}\]

For optical frequencies, $ \mu \approx \mu_0 $, so we focus on dielectric function $ \varepsilon(\omega) $.

Wave Equation in Dielectric Medium

The wave equation for the electric field becomes:

\[\nabla^2 \mathbf{E} - \mu_0 \varepsilon \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0\]

Assume a plane wave solution:

\[\mathbf{E}(z,t) = \mathbf{E}_0 e^{i(kz - \omega t)}\]

With:

\[k = \omega \sqrt{\mu_0 \varepsilon(\omega)} = \frac{\omega n(\omega)}{c}\]

Complex Dielectric Function and Optical Constants

We define a complex dielectric function:

\[\varepsilon(\omega) = \varepsilon_1(\omega) + i \varepsilon_2(\omega)\]

$ \varepsilon_1(\omega) $: describes dispersion
$ \varepsilon_2(\omega) $: describes absorption

The complex refractive index is:

\[\tilde{n}(\omega) = n(\omega) + i\kappa(\omega)\]

where:

$n(\omega)$: refractive index (phase velocity)
$\kappa(\omega)$: extinction coefficient (attenuation)

Relationship to dielectric function:

\[\varepsilon(\omega) = \tilde{n}^2(\omega) = (n + i\kappa)^2\]

Expanding:

\[\varepsilon_1 = n^2 - \kappa^2, \quad \varepsilon_2 = 2n\kappa\]

Absorption Coefficient

The wave propagates as:

\[E(z) = E_0 e^{i(kz - \omega t)} = E_0 e^{-\alpha z/2} e^{i(k' z - \omega t)}\]

The absorption coefficient $ \alpha $ is related to $ \kappa $ by:

\[\alpha = \frac{4\pi \kappa}{\lambda}\]

This describes how the wave amplitude decays exponentially inside the material.

Dispersion and Absorption in Solids

What is Dispersion?

Dispersion refers to the frequency dependence of the refractive index $n(\omega)$ or dielectric function $\varepsilon(\omega)$.

It causes:

Wavelength-dependent phase velocity
Color separation in a prism

Mathematically:

\[n(\omega) = \text{Re}[\tilde{n}(\omega)] = \sqrt{\frac{\sqrt{\varepsilon_1^2 + \varepsilon_2^2} + \varepsilon_1}{2}}\]

What is Absorption?

Absorption transfers EM energy to internal degrees of freedom (electrons/phonons).

Represented by $ \varepsilon_2(\omega) $ or $\kappa$
Intensity decay:

\[I(z) = I_0 e^{-\alpha z}, \quad \text{where} \quad \alpha = \frac{4\pi \kappa}{\lambda}\]

Lorentz Oscillator Model

Bound electrons behave like damped harmonic oscillators:

\[m \frac{d^2 x}{dt^2} + m\gamma \frac{dx}{dt} + m\omega_0^2 x = -eE e^{-i\omega t}\]

Solution gives displacement:

\[x(\omega) = \frac{-eE_0}{m(\omega_0^2 - \omega^2 - i\gamma \omega)}\]

Polarization:

\[P(\omega) = N e x(\omega) = \frac{Ne^2}{m} \cdot \frac{1}{\omega_0^2 - \omega^2 - i\gamma \omega} E(\omega)\]

Dielectric function:

\[\varepsilon(\omega) = 1 + \frac{Ne^2}{\varepsilon_0 m} \cdot \frac{1}{\omega_0^2 - \omega^2 - i\gamma \omega}\]

Dispersion Formula and Applications

We typically write the dielectric function as:

\[\varepsilon(\omega) = \varepsilon_\infty + \frac{f}{\omega_0^2 - \omega^2 - i\gamma \omega}\]

Where:

$\varepsilon_\infty$: high-frequency contribution
$f = \frac{Ne^2}{\varepsilon_0 m}$: oscillator strength

Splitting into Real and Imaginary Parts

\[\varepsilon(\omega) = \varepsilon_1(\omega) + i \varepsilon_2(\omega)\]

Real part (dispersion):

\[\varepsilon_1(\omega) = \varepsilon_\infty + \frac{f(\omega_0^2 - \omega^2)}{(\omega_0^2 - \omega^2)^2 + (\gamma \omega)^2}\]

Imaginary part (absorption):

\[\varepsilon_2(\omega) = \frac{f \gamma \omega}{(\omega_0^2 - \omega^2)^2 + (\gamma \omega)^2}\]

Physical Interpretation

$\varepsilon_1(\omega)$ shows strong frequency dependence near $ \omega_0 $
$\varepsilon_2(\omega)$ peaks at $ \omega = \omega_0 $

Frequency Behavior:

$ \omega \ll \omega_0 \Rightarrow \varepsilon_1 > \varepsilon_\infty $
$ \omega \gg \omega_0 \Rightarrow \varepsilon_1 \rightarrow \varepsilon_\infty $
Peak in $ \varepsilon_2(\omega) $ at resonance $ \omega = \omega_0 $