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diff --git a/source/know/concept/drude-model/index.md b/source/know/concept/drude-model/index.md index b175e64..c4faf81 100644 --- a/source/know/concept/drude-model/index.md +++ b/source/know/concept/drude-model/index.md @@ -9,124 +9,117 @@ categories: layout: "concept" --- -The **Drude model** classically predicts -the dielectric function and electric conductivity of a gas of free charge carriers, +The **Drude model**, also known as +the **Drude-Lorentz model** due to its analogy +to the *Lorentz oscillator model* +classically predicts the [dielectric function](/know/concept/dielectric-function/) +and electric conductivity of a gas of free charges, 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: +In a metal, the conduction electrons can roam freely. +When an [electromagnetic wave](/know/concept/electromagnetic-wave-equation/) +passes by, its oscillating [electric field](/know/concept/electric-field/) +$$\vb{E}(t) = \vb{E}_0 e^{- i \omega t}$$ exerts a force on those electrons, +so the displacement $$\vb{x}(t)$$ of a particle from its initial position +obeys this equation of motion: $$\begin{aligned} - m \dvn{2}{x}{t} + m \gamma \dv{x}{t} - = q E_0 \exp(- i \omega t) + m \dvn{2}{\vb{x}}{t} + = q \vb{E} - \gamma m \dv{\vb{x}}{t} \end{aligned}$$ -Inserting the ansatz $$x(t) = x_0 \exp(- i \omega t)$$ -and isolating for the displacement $$x_0$$ yields: +Where $$m$$ and $$q < 0$$ are the mass and charge of the electron. +The first term is Newton's third law, +and the last term represents a damping force +slowing down the electrons at rate $$\gamma$$. -$$\begin{aligned} - - x_0 m \omega^2 - i x_0 m \gamma \omega - = q E_0 - \quad \implies \quad - x_0 - = - \frac{q E_0}{m (\omega^2 + i \gamma \omega)} -\end{aligned}$$ - -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: +Inserting the ansatz $$\vb{x}(t) = \vb{x}_0 e^{- i \omega t}$$ +and isolating for the displacement $$\vb{x}$$, we find: $$\begin{aligned} - P(t) - = N p(t) - = N q x(t) - = - \frac{N q^2}{m (\omega^2 + i \gamma \omega)} E(t) + \vb{x}(t) + = \vb{x}_0 e^{- i \omega t} + = - \frac{q \vb{E}}{m (\omega^2 + i \gamma \omega)} \end{aligned}$$ -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: +The polarization density $$\vb{P}(t)$$ is therefore as shown below. +Note that the dipole moment vector $$\vb{p}$$ is defined +as pointing from negative to positive, +whereas the electric field $$\vb{E}$$ goes from positive to negative. +Let $$N$$ be the metal's electron density, then: $$\begin{aligned} - D - = \varepsilon_0 \varepsilon_r E - = \varepsilon_0 E + P - = \varepsilon_0 \bigg( 1 - \frac{N q^2}{\varepsilon_0 m} \frac{1}{\omega^2 + i \gamma \omega} \bigg) E + \vb{P}(t) + = N \vb{p}(t) + = N q \vb{x}(t) + = - \frac{N q^2}{m (\omega^2 + i \gamma \omega)} \vb{E}(t) \end{aligned}$$ -The parenthesized expression is the desired dielectric function $$\varepsilon_r$$, -which depends on $$\omega$$: +The electric displacement field $$\vb{D}$$ is then as follows, +where the parenthesized expression is the dielectric function +$$\varepsilon_r$$ of the material: $$\begin{aligned} - \boxed{ - \varepsilon_r(\omega) - = 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega} - } + \vb{D} + = \varepsilon_0 \vb{E} + \vb{P} + = \varepsilon_0 \bigg( 1 - \frac{N q^2}{\varepsilon_0 m} \frac{1}{\omega^2 + i \gamma \omega} \bigg) \vb{E} + = \varepsilon_0 \varepsilon_r \vb{E} \end{aligned}$$ -Where we have defined the important so-called **plasma frequency** like so: +From this, we define the **plasma frequency** $$\omega_p$$ +at which the conductor "resonates", +leading to so-called **plasma oscillations** of the electron density +(see also [Langmuir waves](/know/concept/langmuir-waves/)): $$\begin{aligned} + \varepsilon_r(\omega) + = 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega} + \qquad\qquad \boxed{ \omega_p \equiv \sqrt{\frac{N q^2}{\varepsilon_0 m}} } \end{aligned}$$ -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: - -$$\begin{aligned} - m \dv{v}{t} + m \gamma v = q E(t) -\end{aligned}$$ +Suppose that $$\gamma = 0$$, +then we can identify three distinct scenarios for $$\varepsilon_r$$ here: -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: +* $$\omega < \omega_p$$, so $$\varepsilon_r < 0$$, + so the refractive index $$\sqrt{\varepsilon_r}$$ is imaginary, + meaning high absorption and high reflectivity + (due to the large complex index difference between media). +* $$\omega = \omega_p$$, so $$\varepsilon = 0$$, + allowing for self-sustained plasma oscillations. +* $$\omega > \omega_p$$, so $$\varepsilon_r > 0$$, + so the index $$\sqrt{\varepsilon}$$ is real and asymptotically goes to $$1$$, + leading to high transparency and low reflectivity from air. -$$\begin{aligned} - \dv{p}{t} - = - \frac{p}{\tau} + q E -\end{aligned}$$ +For most metals $$\omega_p$$ is ultraviolet, +which explains why they typically appear shiny to us. +In reality $$\gamma > 0$$, reducing the reflectivity somewhat when $$\omega < \omega_p$$. -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: +The Drude model also lets us calculate the metal's conductivity. +We already have an expression for $$\vb{x}(t)$$, +which we differentiate to get the velocity $$\vb{v}(t)$$: $$\begin{aligned} - - i m \omega v_0 + \frac{m}{\tau} v_0 = q E_0 - \quad \implies \quad - v_0 = \frac{q \tau}{m (1 - i \omega \tau)} E_0 + \vb{v}(t) + = \dv{\vb{x}}{t} + = - i \omega \vb{x} + = \frac{i \omega q \vb{E}}{m (\omega^2 + i \gamma \omega)} + = \frac{q \vb{E}}{m (\gamma - i \omega)} \end{aligned}$$ -From $$v(t)$$, we find the resulting average current density $$J(t)$$ to be as follows: +Consequently the average current density $$\vb{J}(t)$$ is found to be: $$\begin{aligned} - J(t) - = - N q v(t) - = \sigma E(t) + \vb{J}(t) + = N q \vb{v}(t) + = \sigma \vb{E}(t) \end{aligned}$$ Where $$\sigma(\omega)$$ is the **AC conductivity**, @@ -134,57 +127,76 @@ which depends on the **DC conductivity** $$\sigma_0$$: $$\begin{aligned} \boxed{ - \sigma - = \frac{\sigma_0}{1 - i \omega \tau} + \sigma(\omega) + = \frac{\gamma \sigma_0}{\gamma - i \omega} } - \qquad \quad + \qquad\qquad \boxed{ \sigma_0 - = \frac{N q^2 \tau}{m} + \equiv \frac{N q^2}{\gamma m} } \end{aligned}$$ -We can use these quantities to rewrite -the dielectric function $$\varepsilon_r$$ from earlier: +Recall that $$\gamma$$ measures friction. +Specifically, Drude assumed that the electrons often collide with obstacles, +each time resetting their momentum to zero; +in that case $$\vb{v}$$ should be interpreted as the average "drift" +of many electrons in an ensemble. +The mean time between those collisions is +the **momentum scattering time** $$\tau \equiv 1 / \gamma$$, so: + +$$\begin{aligned} + \sigma(\omega) + = \frac{\sigma_0}{1 - i \omega \tau} + \qquad\qquad + \sigma_0 + = \frac{N q^2 \tau}{m} +\end{aligned}$$ + +After defining all those quantities, +the dielectric function $$\varepsilon_r(\omega)$$ can be written as: $$\begin{aligned} \boxed{ - \varepsilon_r(\omega) - = 1 + \frac{i \sigma(\omega)}{\varepsilon_0 \omega} + \begin{aligned} + \varepsilon_r(\omega) + &= 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega} + \\ + &= 1 + \frac{i \sigma(\omega)}{\varepsilon_0 \omega} + \end{aligned} } \end{aligned}$$ + ## 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 $$m^*$$. +free electrons (n-type doping) or free holes (p-type doping), +which can be treated as free charge carriers moving through the material, +so the Drude model is also relevant in this case. + +We must replace the carriers' true mass $$m$$ with their *effective mass* $$m^*$$ +found from the material's electronic band structure. Furthermore, semiconductors already have -a high intrinsic permittivity $$\varepsilon_{\mathrm{int}}$$ -before the dopant is added, -so the diplacement field $$D$$ is: +a high intrinsic dielectric function $$\varepsilon_{\mathrm{int}}$$ +before being doped, so the displacement field $$\vb{D}$$ becomes: $$\begin{aligned} - D - = \varepsilon_0 E + P_{\mathrm{int}} + P_{\mathrm{free}} - = \varepsilon_{\mathrm{int}} \varepsilon_0 E - \frac{N q^2}{m^* (\omega^2 + i \gamma \omega)} E + \vb{D} + = \varepsilon_0 \vb{E} + \vb{P}_{\mathrm{int}} + \vb{P}_{\mathrm{free}} + = \varepsilon_0 \varepsilon_{\mathrm{int}} \vb{E} - \frac{N q^2}{m^* (\omega^2 + i \gamma \omega)} \vb{E} + = \varepsilon_0 \varepsilon_r \vb{E} \end{aligned}$$ -Where $$P_{\mathrm{int}}$$ is the intrinsic undoped polarization, -and $$P_{\mathrm{free}}$$ is the contribution of the free carriers. +Where $$\vb{P}_{\mathrm{int}}$$ is the intrinsic polarization before doping, +and $$\vb{P}_{\mathrm{free}}$$ is the expression we calculated above for metals. The dielectric function $$\varepsilon_r(\omega)$$ is therefore given by: $$\begin{aligned} \boxed{ \varepsilon_r(\omega) - = \varepsilon_{\mathrm{int}} \Big( 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega} \Big) + = \varepsilon_{\mathrm{int}} \bigg( 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega} \bigg) } \end{aligned}$$ @@ -194,29 +206,28 @@ to include $$\varepsilon_\mathrm{int}$$: $$\begin{aligned} \boxed{ \omega_p - = \sqrt{\frac{N q^2}{\varepsilon_{\mathrm{int}} \varepsilon_0 m^*}} + \equiv \sqrt{\frac{N q^2}{\varepsilon_0 \varepsilon_{\mathrm{int}} m^*}} } \end{aligned}$$ 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. +but the free carrier density $$N$$ is typically lower in this case, +so $$\omega_p$$ is usually infrared rather than ultraviolet. -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: +Furthermore, instead of $$\varepsilon_r \to 1$$ +for $$\omega \to \infty$$ like a metal, +now $$\varepsilon_r \to \varepsilon_\mathrm{int}$$. +Along the way, there is a point where $$\varepsilon_r = 1$$ +and the reflectivity becomes 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 $$\omega$$. +If $$N$$ and $$\varepsilon_\mathrm{int}$$ are known, +this can be used to experimentally determine $$m^*$$ +by finding which value of $$\omega_p$$ would lead to the measured zero-reflectivity point. |