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