Categories: Electromagnetism, Optics, Physics.

Drude model

The Drude model, also known as the Drude-Lorentz model due to its analogy to the Lorentz oscillator model classically predicts the dielectric function and electric conductivity of a gas of free charges, as found in metals and doped semiconductors.

Metals

In a metal, the conduction electrons can roam freely. When an electromagnetic wave passes by, its oscillating electric field E(t)=E0eiωt\vb{E}(t) = \vb{E}_0 e^{- i \omega t} exerts a force on those electrons, so the displacement x(t)\vb{x}(t) of a particle from its initial position obeys this equation of motion:

md2xdt2=qEγmdxdt\begin{aligned} m \dvn{2}{\vb{x}}{t} = q \vb{E} - \gamma m \dv{\vb{x}}{t} \end{aligned}

Where mm and q<0q < 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 γ\gamma.

Inserting the ansatz x(t)=x0eiωt\vb{x}(t) = \vb{x}_0 e^{- i \omega t} and isolating for the displacement x\vb{x}, we find:

x(t)=x0eiωt=qEm(ω2+iγω)\begin{aligned} \vb{x}(t) = \vb{x}_0 e^{- i \omega t} = - \frac{q \vb{E}}{m (\omega^2 + i \gamma \omega)} \end{aligned}

The polarization density P(t)\vb{P}(t) is therefore as shown below. Note that the dipole moment vector p\vb{p} is defined as pointing from negative to positive, whereas the electric field E\vb{E} goes from positive to negative. Let NN be the metal’s electron density, then:

P(t)=Np(t)=Nqx(t)=Nq2m(ω2+iγω)E(t)\begin{aligned} \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 electric displacement field D(t)\vb{D}(t) is then as follows, where the parenthesized expression is the dielectric function εr\varepsilon_r of the material:

D=ε0E+P=ε0(1Nq2ε0m1ω2+iγω)E=ε0εrE\begin{aligned} \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}

From this, we define the plasma frequency ωp\omega_p at which the conductor “resonates”, leading to so-called plasma oscillations of the electron density (see also Langmuir waves):

εr(ω)=1ωp2ω2+iγωωpNq2ε0m\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}

Suppose that γ=0\gamma = 0, then we can identify three distinct scenarios for εr\varepsilon_r here:

For most metals ωp\omega_p is ultraviolet, which explains why they typically appear shiny to us. In reality γ>0\gamma > 0, reducing the reflectivity somewhat when ω<ωp\omega < \omega_p.

The Drude model also lets us calculate the metal’s conductivity. We already have an expression for x(t)\vb{x}(t), which we differentiate to get the velocity v(t)\vb{v}(t):

v(t)=dxdt=iωx=iωqEm(ω2+iγω)=qEm(γiω)\begin{aligned} \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}

Consequently the average current density J(t)\vb{J}(t) is found to be:

J(t)=Nqv(t)=σE(t)\begin{aligned} \vb{J}(t) = N q \vb{v}(t) = \sigma \vb{E}(t) \end{aligned}

Where σ(ω)\sigma(\omega) is the AC conductivity, which depends on the DC conductivity σ0\sigma_0:

σ(ω)=γσ0γiωσ0Nq2γm\begin{aligned} \boxed{ \sigma(\omega) = \frac{\gamma \sigma_0}{\gamma - i \omega} } \qquad\qquad \boxed{ \sigma_0 \equiv \frac{N q^2}{\gamma m} } \end{aligned}

Recall that γ\gamma measures friction. Specifically, Drude assumed that the electrons often collide with obstacles, each time resetting their momentum to zero; in that case v\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 τ1/γ\tau \equiv 1 / \gamma, so:

σ(ω)=σ01iωτσ0=Nq2τm\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 εr(ω)\varepsilon_r(\omega) can be written as:

εr(ω)=1ωp2ω2+iγω=1+iσ(ω)ε0ω\begin{aligned} \boxed{ \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 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 mm with their effective mass mm^* found from the material’s electronic band structure. Furthermore, semiconductors already have a high intrinsic dielectric function εint\varepsilon_{\mathrm{int}} before being doped, so the displacement field D(t)\vb{D}(t) becomes:

D=ε0E+Pint+Pfree=ε0εintENq2m(ω2+iγω)E=ε0εrE\begin{aligned} \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 Pint\vb{P}_{\mathrm{int}} is the intrinsic polarization before doping, and Pfree\vb{P}_{\mathrm{free}} is the expression we calculated above for metals. The dielectric function εr(ω)\varepsilon_r(\omega) is therefore given by:

εr(ω)=εint(1ωp2ω2+iγω)\begin{aligned} \boxed{ \varepsilon_r(\omega) = \varepsilon_{\mathrm{int}} \bigg( 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega} \bigg) } \end{aligned}

Where the plasma frequency ωp\omega_p has been redefined as follows to include εint\varepsilon_\mathrm{int}:

ωpNq2ε0εintm\begin{aligned} \boxed{ \omega_p \equiv \sqrt{\frac{N q^2}{\varepsilon_0 \varepsilon_{\mathrm{int}} m^*}} } \end{aligned}

The meaning of ωp\omega_p is the same as for metals, but the free carrier density NN is typically lower in this case, so ωp\omega_p is usually infrared rather than ultraviolet.

Furthermore, instead of εr1\varepsilon_r \to 1 for ω\omega \to \infty like a metal, now εrεint\varepsilon_r \to \varepsilon_\mathrm{int}. Along the way, there is a point where εr=1\varepsilon_r = 1 and the reflectivity becomes zero. This occurs at:

ω2=εintεint1ωp2\begin{aligned} \omega^2 = \frac{\varepsilon_{\mathrm{int}}}{\varepsilon_{\mathrm{int}} - 1} \omega_p^2 \end{aligned}

If NN and εint\varepsilon_\mathrm{int} are known, this can be used to experimentally determine mm^* by finding which value of ωp\omega_p would lead to the measured zero-reflectivity point.

References

  1. M. Fox, Optical properties of solids, 2nd edition, Oxford.
  2. S.H. Simon, The Oxford solid state basics, Oxford.