---
title: "Drude model"
sort_title: "Drude model"
date: 2021-09-23
categories:
- Physics
- Electromagnetism
- Optics
layout: "concept"
---

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](/know/concept/electromagnetic-wave-equation/)
has an oscillating [electric field](/know/concept/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:

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

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 $$\omega$$:

$$\begin{aligned}
    \boxed{
        \varepsilon_r(\omega)
        = 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega}
    }
\end{aligned}$$

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 $$v = \idv{x}{t}$$,
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:

$$\begin{aligned}
    \boxed{
        \varepsilon_r(\omega)
        = \varepsilon_{\mathrm{int}} \Big( 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega} \Big)
    }
\end{aligned}$$

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.