Categories: Electromagnetism, Optics, Physics.

# Drude model

The Drude model classically predicts the dielectric function and electric conductivity of a gas of free charge carriers, as found in metals and doped semiconductors.

## Metals

An electromagnetic wave has an oscillating electric field $$E(t) = E_0 \exp\!(- i \omega t)$$ that exerts a force on the charge carriers, which have mass $$m$$ and charge $$q$$. They thus obey the following equation of motion, where $$\gamma$$ is a frictional damping coefficient:

\begin{aligned} m \dv[2]{x}{t} + m \gamma \dv{x}{t} = q E_0 \exp\!(- i \omega t) \end{aligned}

Inserting the ansatz $$x(t) = x_0 \exp\!(- i \omega t)$$ and isolating for the displacement $$x_0$$ yields:

\begin{aligned} - x_0 m \omega^2 - i x_0 m \gamma \omega = q E_0 \quad \implies \quad x_0 = - \frac{q E_0}{m (\omega^2 + i \gamma \omega)} \end{aligned}

The polarization density $$P(t)$$ is therefore as shown below. Note that the dipole moment $$p$$ goes from negative to positive, and the electric field $$E$$ from positive to negative. Let $$N$$ be the density of carriers in the gas, then:

\begin{aligned} P(t) = N p(t) = N q x(t) = - \frac{N q^2}{m (\omega^2 + i \gamma \omega)} E(t) \end{aligned}

The electric displacement field $$D$$ is thus as follows, where $$\varepsilon_r$$ is the unknown relative permittivity of the gas, which we will find shortly:

\begin{aligned} D = \varepsilon_0 \varepsilon_r E = \varepsilon_0 E + P = \varepsilon_0 \bigg( 1 - \frac{N q^2}{\varepsilon_0 m} \frac{1}{\omega^2 + i \gamma \omega} \bigg) E \end{aligned}

The parenthesized expression is the desired dielectric function $$\varepsilon_r$$, which depends on $$\omega$$:

\begin{aligned} \boxed{ \varepsilon_r(\omega) %= 1 - \frac{N q^2}{\varepsilon_0 m} \frac{1}{\omega^2 + i \gamma \omega} = 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega} } \end{aligned}

Where we have defined the important so-called plasma frequency like so:

\begin{aligned} \boxed{ \omega_p \equiv \sqrt{\frac{N q^2}{\varepsilon_0 m}} } \end{aligned}

If $$\gamma = 0$$, then $$\varepsilon_r$$ is negative $$\omega < \omega_p$$, positive for $$\omega > \omega_p$$, and zero for $$\omega = \omega_p$$. Respectively, this leads to an imaginary index $$\sqrt{\varepsilon_r}$$ (high absorption), a real index tending to $$1$$ (transparency), and the possibility of self-sustained plasma oscillations. For metals, $$\omega_p$$ lies in the UV.

We can refine this result for $$\varepsilon_r$$, by recognizing the (mean) velocity $$v = \dv*{x}{t}$$, and rewriting the equation of motion accordingly:

\begin{aligned} m \dv{v}{t} + m \gamma v = q E(t) \end{aligned}

Note that $$m v$$ is simply the momentum $$p$$. We define the momentum scattering time $$\tau \equiv 1 / \gamma$$, which represents the average time between collisions, where each collision resets the involved particles’ momentums to zero. Or, more formally:

\begin{aligned} \dv{p}{t} = - \frac{p}{\tau} + q E \end{aligned}

Returning to the equation for the mean velocity $$v$$, we insert the ansatz $$v(t) = v_0 \exp\!(- i \omega t)$$, for the same electric field $$E(t) = E_0 \exp\!(-i \omega t)$$ as before:

\begin{aligned} - i m \omega v_0 + \frac{m}{\tau} v_0 = q E_0 \quad \implies \quad v_0 = \frac{q \tau}{m (1 - i \omega \tau)} E_0 \end{aligned}

From $$v(t)$$, we find the resulting average current density $$J(t)$$ to be as follows:

\begin{aligned} J(t) = - N q v(t) = \sigma E(t) \end{aligned}

Where $$\sigma(\omega)$$ is the AC conductivity, which depends on the DC conductivity $$\sigma_0$$:

\begin{aligned} \boxed{ \sigma = \frac{\sigma_0}{1 - i \omega \tau} } \qquad \quad \boxed{ \sigma_0 = \frac{N q^2 \tau}{m} } \end{aligned}

We can use these quantities to rewrite the dielectric function $$\varepsilon_r$$ from earlier:

\begin{aligned} \boxed{ \varepsilon_r(\omega) = 1 + \frac{i \sigma(\omega)}{\varepsilon_0 \omega} } \end{aligned}

## Doped semiconductors

Doping a semiconductor introduces free electrons (n-type) or free holes (p-type), which can be treated as free particles moving in the bands of the material.

The Drude model can also be used in this case, by replacing the actual carrier mass $$m$$ by the effective mass $$m^*$$. Furthermore, semiconductors already have a high intrinsic permittivity $$\varepsilon_{\mathrm{int}}$$ before the dopant is added, so the diplacement field $$D$$ is:

\begin{aligned} D = \varepsilon_0 E + P_{\mathrm{int}} + P_{\mathrm{free}} = \varepsilon_{\mathrm{int}} \varepsilon_0 E - \frac{N q^2}{m^* (\omega^2 + i \gamma \omega)} E \end{aligned}

Where $$P_{\mathrm{int}}$$ is the intrinsic undoped polarization, and $$P_{\mathrm{free}}$$ is the contribution of the free carriers. The dielectric function $$\varepsilon_r(\omega)$$ is therefore given by:

\begin{aligned} \boxed{ \varepsilon_r(\omega) %= \varepsilon_{\mathrm{int}}(\omega) - \frac{N q^2}{\varepsilon_0 m^*} \frac{1}{\omega^2 + i \gamma \omega} = \varepsilon_{\mathrm{int}} \Big( 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega} \Big) } \end{aligned}

Where the plasma frequency $$\omega_p$$ has been redefined as follows to include $$\varepsilon_\mathrm{int}$$:

\begin{aligned} \boxed{ \omega_p = \sqrt{\frac{N q^2}{\varepsilon_{\mathrm{int}} \varepsilon_0 m^*}} } \end{aligned}

The meaning of $$\omega_p$$ is the same as for metals, with high absorption for $$\omega < \omega_p$$. However, due to the lower carrier density $$N$$ in a semiconductor, $$\omega_p$$ lies in the IR rather than UV.

However, instead of asymptotically going to $$1$$ for $$\omega > \omega_p$$ like a metal, $$\varepsilon_r$$ tends to $$\varepsilon_\mathrm{int}$$ instead, and crosses $$1$$ along the way, at which point the reflectivity is zero. This occurs at:

\begin{aligned} \omega^2 = \frac{\varepsilon_{\mathrm{int}}}{\varepsilon_{\mathrm{int}} - 1} \omega_p^2 \end{aligned}

This is used to experimentally determine the effective mass $$m^*$$ of the doped semiconductor, by finding which value of $$m^*$$ gives the measured $$\omega$$.

## References

1. M. Fox, Optical properties of solids, 2nd edition, Oxford.
2. S.H. Simon, The Oxford solid state basics, Oxford.

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.