Categories: Electromagnetism, Optics, Physics.

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 has an oscillating 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.

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.
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