path: root/source/know/concept/drude-model/index.md

---
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.