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Diffstat (limited to 'source/know/concept/drude-model')
| -rw-r--r-- | source/know/concept/drude-model/index.md | 98 |
1 files changed, 49 insertions, 49 deletions
diff --git a/source/know/concept/drude-model/index.md b/source/know/concept/drude-model/index.md index 28e6dc2..b175e64 100644 --- a/source/know/concept/drude-model/index.md +++ b/source/know/concept/drude-model/index.md @@ -18,19 +18,19 @@ as found in metals and doped semiconductors. An [electromagnetic wave](/know/concept/electromagnetic-wave-equation/) has an oscillating [electric field](/know/concept/electric-field/) -$E(t) = E_0 \exp(- i \omega t)$ +$$E(t) = E_0 \exp(- i \omega t)$$ that exerts a force on the charge carriers, -which have mass $m$ and charge $q$. +which have mass $$m$$ and charge $$q$$. They thus obey the following equation of motion, -where $\gamma$ is a frictional damping coefficient: +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: +Inserting the ansatz $$x(t) = x_0 \exp(- i \omega t)$$ +and isolating for the displacement $$x_0$$ yields: $$\begin{aligned} - x_0 m \omega^2 - i x_0 m \gamma \omega @@ -40,10 +40,10 @@ $$\begin{aligned} = - \frac{q E_0}{m (\omega^2 + i \gamma \omega)} \end{aligned}$$ -The polarization density $P(t)$ is therefore as shown below. -Note that the dipole moment $p$ goes from negative to positive, -and the electric field $E$ from positive to negative. -Let $N$ be the density of carriers in the gas, then: +The polarization density $$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) @@ -52,8 +52,8 @@ $$\begin{aligned} = - \frac{N q^2}{m (\omega^2 + i \gamma \omega)} E(t) \end{aligned}$$ -The electric displacement field $D$ is thus as follows, -where $\varepsilon_r$ is the unknown relative permittivity of the gas, +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: $$\begin{aligned} @@ -63,8 +63,8 @@ $$\begin{aligned} = \varepsilon_0 \bigg( 1 - \frac{N q^2}{\varepsilon_0 m} \frac{1}{\omega^2 + i \gamma \omega} \bigg) E \end{aligned}$$ -The parenthesized expression is the desired dielectric function $\varepsilon_r$, -which depends on $\omega$: +The parenthesized expression is the desired dielectric function $$\varepsilon_r$$, +which depends on $$\omega$$: $$\begin{aligned} \boxed{ @@ -82,26 +82,26 @@ $$\begin{aligned} } \end{aligned}$$ -If $\gamma = 0$, then $\varepsilon_r$ is -negative $\omega < \omega_p$, -positive for $\omega > \omega_p$, -and zero for $\omega = \omega_p$. +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), +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. +For metals, $$\omega_p$$ lies in the UV. -We can refine this result for $\varepsilon_r$, -by recognizing the (mean) velocity $v = \idv{x}{t}$, +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** $\tau \equiv 1 / \gamma$, +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: @@ -111,9 +111,9 @@ $$\begin{aligned} = - \frac{p}{\tau} + q E \end{aligned}$$ -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: +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: $$\begin{aligned} - i m \omega v_0 + \frac{m}{\tau} v_0 = q E_0 @@ -121,7 +121,7 @@ $$\begin{aligned} v_0 = \frac{q \tau}{m (1 - i \omega \tau)} E_0 \end{aligned}$$ -From $v(t)$, we find the resulting average current density $J(t)$ to be as follows: +From $$v(t)$$, we find the resulting average current density $$J(t)$$ to be as follows: $$\begin{aligned} J(t) @@ -129,8 +129,8 @@ $$\begin{aligned} = \sigma E(t) \end{aligned}$$ -Where $\sigma(\omega)$ is the **AC conductivity**, -which depends on the **DC conductivity** $\sigma_0$: +Where $$\sigma(\omega)$$ is the **AC conductivity**, +which depends on the **DC conductivity** $$\sigma_0$$: $$\begin{aligned} \boxed{ @@ -145,7 +145,7 @@ $$\begin{aligned} \end{aligned}$$ We can use these quantities to rewrite -the dielectric function $\varepsilon_r$ from earlier: +the dielectric function $$\varepsilon_r$$ from earlier: $$\begin{aligned} \boxed{ @@ -164,12 +164,12 @@ 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 $m^*$. +by replacing the actual carrier mass $$m$$ +by the effective mass $$m^*$$. Furthermore, semiconductors already have -a high intrinsic permittivity $\varepsilon_{\mathrm{int}}$ +a high intrinsic permittivity $$\varepsilon_{\mathrm{int}}$$ before the dopant is added, -so the diplacement field $D$ is: +so the diplacement field $$D$$ is: $$\begin{aligned} D @@ -177,9 +177,9 @@ $$\begin{aligned} = \varepsilon_{\mathrm{int}} \varepsilon_0 E - \frac{N q^2}{m^* (\omega^2 + i \gamma \omega)} E \end{aligned}$$ -Where $P_{\mathrm{int}}$ is the intrinsic undoped polarization, -and $P_{\mathrm{free}}$ is the contribution of the free carriers. -The dielectric function $\varepsilon_r(\omega)$ is therefore given by: +Where $$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{ @@ -188,8 +188,8 @@ $$\begin{aligned} } \end{aligned}$$ -Where the plasma frequency $\omega_p$ has been redefined as follows -to include $\varepsilon_\mathrm{int}$: +Where the plasma frequency $$\omega_p$$ has been redefined as follows +to include $$\varepsilon_\mathrm{int}$$: $$\begin{aligned} \boxed{ @@ -198,14 +198,14 @@ $$\begin{aligned} } \end{aligned}$$ -The meaning of $\omega_p$ is the same as for metals, -with high absorption for $\omega < \omega_p$. -However, due to the lower carrier density $N$ in a semiconductor, -$\omega_p$ lies in the IR rather than UV. +The meaning of $$\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$$ lies in the IR rather than UV. -However, instead of asymptotically going to $1$ for $\omega > \omega_p$ like a metal, -$\varepsilon_r$ tends to $\varepsilon_\mathrm{int}$ instead, -and crosses $1$ along the way, +However, instead of asymptotically going to $$1$$ for $$\omega > \omega_p$$ like a metal, +$$\varepsilon_r$$ tends to $$\varepsilon_\mathrm{int}$$ instead, +and crosses $$1$$ along the way, at which point the reflectivity is zero. This occurs at: @@ -214,9 +214,9 @@ $$\begin{aligned} = \frac{\varepsilon_{\mathrm{int}}}{\varepsilon_{\mathrm{int}} - 1} \omega_p^2 \end{aligned}$$ -This is used to experimentally determine the effective mass $m^*$ +This is used to experimentally determine the effective mass $$m^*$$ of the doped semiconductor, -by finding which value of $m^*$ gives the measured $\omega$. +by finding which value of $$m^*$$ gives the measured $$\omega$$. |