From f7c7464e29cb19083a2488c393f3707e97248c4f Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sat, 25 Sep 2021 10:26:44 +0200
Subject: Expand knowledge base

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+---
+title: "Drude model"
+firstLetter: "D"
+publishDate: 2021-09-23
+categories:
+- Physics
+- Electromagnetism
+- Optics
+
+date: 2021-09-23T16:22:51+02:00
+draft: false
+markup: pandoc
+---
+
+# 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](/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:
+
+$$\begin{aligned}
+    m \dv[2]{x}{t} + m \gamma \dv{x}{t}
+    = q E_0 \exp\!(- i \omega t)
+\end{aligned}$$
+
+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.
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