Drude Model: Dielectric Function of Metals & Doped Semiconductors

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Hello! Welcome to our new module, "Optical Properties of Solids." In our previous discussions on electron-phonon interactions, we saw how the electronic system can profoundly influence the lattice, leading to phenomena like the Kohn anomaly. Now, we will explore the direct interaction of electrons with an external electromagnetic field—light. Your experimental background in photoluminescence and other optical spectroscopies will be particularly relevant here.

Today's lesson addresses the first and most fundamental model of a conductor's optical response. Our learning outcome is to derive the frequency-dependent dielectric function for metals and doped semiconductors using the Drude model. The dielectric function, $ϵ (ω)$ , is the master key that unlocks a material's optical properties, telling us why metals are shiny and why they can become transparent to high-energy light.

We'll be focusing on the collective response of free carriers (conduction electrons), which is known as the intraband response. This lays the groundwork for our next lesson, where we'll examine transitions between different energy bands, or the interband response.

1. The Classical Model of a Free Electron Gas

The Drude model, proposed around 1900, is a remarkably successful classical model that treats the conduction electrons in a metal as a gas of free particles. Let's start by understanding its core assumptions and how it differs from the Lorentz model used for insulators (dielectrics).

Lecture 2 (EM21) -- Lorentz and Drude models

This video from EMPossible provides a great conceptual introduction by contrasting the Lorentz model for bound electrons in dielectrics with the Drude model for free electrons in metals. This distinction is crucial.

Please watch the segments from 00:35 to 05:42 and from 40:55 to 42:36. Focus on: The analogy of a mass on a spring for a bound electron (Lorentz model) and the concept of a restoring force. How the Drude model is simply the Lorentz model with the restoring force set to zero, because electrons in a metal are not bound to any particular atom.

As the video explains, the key difference is the restoring force. In an insulator, an electron displaced by an electric field is pulled back by its parent nucleus, like a mass on a spring. In a metal, a conduction electron is free to move throughout the crystal, so there is no restoring force.

This allows us to write a simple equation of motion for an electron in an AC electric field $E (t) = E_{0} e^{- iω t}$ . The forces acting on the electron are:

Driving Force: From the electric field, $F_{d r i v e} = - e E$ .
Damping Force: A phenomenological term representing momentum loss through collisions (with phonons, impurities, etc.), $F_{d am p} = - \frac{m _{e}}{τ} v$ . Here, $v$ is the electron's velocity, $m_{e}$ is the electron mass, and $τ$ is the relaxation time, which represents the average time between scattering events.

Applying Newton's second law, $m_{e} \frac{d v}{d t} = F_{d r i v e} + F_{d am p}$ , we get:

m_{e} \frac{d v}{d t} + \frac{m _{e}}{τ} v = - e E (t)

This is the fundamental equation of motion in the Drude model.

2. From Electron Motion to the Dielectric Function

Now, let's solve this equation to find the optical response of the material. The goal is to connect the microscopic motion of a single electron to the macroscopic dielectric function $ϵ (ω)$ .

This process involves a few clear steps:

Solve the equation of motion for the electron velocity $v (ω)$ .
Calculate the current density $J (ω) = - n e v (ω)$ , where $n$ is the density of free electrons.
From Ohm's Law, $J (ω) = σ (ω) E (ω)$ , find the AC conductivity $σ (ω)$ .
Finally, relate the AC conductivity to the dielectric function $ϵ (ω)$ .

The following video provides an excellent walkthrough of this entire derivation.

Optical Properties of Nanomaterials 07: Drude Model of the dielectric function

This video from the Vogel lab derives the Drude dielectric function step-by-step, starting from the equation of motion we just established. It clearly explains all the physics involved.

Please watch from 14:34 to 24:46. Follow the derivation that starts with the equation of motion and arrives at the final expression for the frequency-dependent dielectric function, \epsilon(\omega).

Let's summarize the key results from the derivation.
By assuming a harmonic solution $v (t) = v_{0} e^{- iω t}$ , we find the frequency-dependent velocity:

v (ω) = - \frac{e τ / m _{e}}{1 - iω τ} E (ω)

The AC conductivity $σ (ω)$ is then:

σ (ω) = \frac{n e ^{2} τ / m _{e}}{1 - iω τ} = \frac{σ _{0}}{1 - iω τ}

where $σ_{0} = n e^{2} τ / m_{e}$ is the familiar DC conductivity.

The crucial link to the dielectric function comes from Maxwell's equations. In a conductive medium, the total current can be incorporated into an effective dielectric function. Ampere's Law in the frequency domain is $\nabla \times H = J - iω D$ . With $D = ϵ_{0} ϵ_{bk g d} E$ and $J = σ E$ , we have $\nabla \times H = (σ - iω ϵ_{0} ϵ_{bk g d}) E$ . We can define a total dielectric function $ϵ (ω)$ such that this is written as $\nabla \times H = - iω ϵ_{0} ϵ (ω) E$ . Comparing these gives the vital relationship:

ϵ (ω) = ϵ_{bk g d} + \frac{iσ ( ω )}{ω ϵ _{0}}

Here, $ϵ_{bk g d}$ is the background dielectric constant from bound electrons (interband transitions), which is approximately 1 for simple metals but significant for semiconductors.

Substituting our expression for $σ (ω)$ gives the Drude dielectric function:

ϵ (ω) = ϵ_{bk g d} - \frac{n e ^{2}}{m _{e} ϵ _{0}} \frac{1}{ω ^{2} + iω / τ}

3. Plasma Frequency and Optical Properties

The expression for $ϵ (ω)$ can be simplified by introducing a characteristic frequency of the electron gas.

The Plasma Frequency, $ω_{p}$

Let's define the plasma frequency $ω_{p}$ as:

ω_{p}^{2} = \frac{n e ^{2}}{m _{e} ϵ _{0}}

The Drude dielectric function becomes:

ϵ (ω) = ϵ_{bk g d} - \frac{ω _{p}^{2}}{ω ( ω + i / τ )}

Physically, the plasma frequency represents the natural frequency of collective oscillations of the entire electron gas if it is displaced relative to the fixed positive ion cores. It is a fundamental property of any conductor.

Optical Properties of Nanomaterials 07: Drude Model of the dielectric function

The same video now gives a clear physical interpretation of the plasma frequency and uses it to explain the characteristic optical properties of metals.

Please watch from 24:46 to 50:35. Pay close attention to: The physical meaning of \omega_p as a turning point where electrons can no longer follow the oscillating field. The behavior of \epsilon(\omega) in the simplified case of no damping (1/ au o 0). How the sign of the real part of \epsilon(\omega) determines whether the material is reflective or transparent.

Let's analyze the behavior of $ϵ (ω)$ by separating it into real and imaginary parts, $ϵ (ω) = ϵ_{1} (ω) + i ϵ_{2} (ω)$ :

ϵ_{1} (ω) = ϵ_{bk g d} - \frac{ω _{p}^{2} τ ^{2}}{1 + ω ^{2} τ ^{2}}

ϵ_{2} (ω) = \frac{ω _{p}^{2} τ}{ω ( 1 + ω ^{2} τ ^{2} )}

The optical properties are primarily determined by $ϵ_{1} (ω)$ :

Below the plasma frequency ( $ω < ω_{p}$ ): For typical metals where $ω_{p} τ ≫ 1$ , $ϵ_{1} (ω)$ is large and negative. A negative dielectric constant implies an imaginary refractive index, meaning the electromagnetic wave cannot propagate inside the material. Instead, it is evanescent and decays rapidly, leading to high reflectivity. This is why metals are shiny.
Above the plasma frequency ( $ω > ω_{p}$ ): $ϵ_{1} (ω)$ becomes positive. The refractive index is real, and the material becomes transparent to light. For most metals, $ω_{p}$ lies in the ultraviolet range, which is why they are opaque in the visible spectrum but become transparent to X-rays.

The imaginary part, $ϵ_{2} (ω)$ , is related to absorption. The damping term $1/ τ$ ensures that there is always some absorption due to electron scattering.

Test your understanding!

You are studying two metals, A and B. Metal A has a free electron density of $n_{A} = 5 \times 1 0^{28} m^{- 3}$ and Metal B has a density of $n_{B} = 2 \times 1 0^{29} m^{- 3}$ . Assuming all other parameters are similar, which metal would you expect to become transparent at a higher frequency of light? Why?

Show answer

Metal B would become transparent at a higher frequency. The transition from reflective to transparent behavior occurs around the plasma frequency, $ω_{p}$ . Since $ω_{p}^{2} = n e^{2} / m_{e} ϵ_{0}$ , the plasma frequency is proportional to the square root of the electron density ( $ω_{p} \propto n$ ). Metal B has a higher electron density $n_{B} > n_{A}$ , so it will have a higher plasma frequency $ω_{p, B} > ω_{p, A}$ . Therefore, it will remain reflective up to higher frequencies.

4. Application to Doped Semiconductors

The Drude model is not limited to metals. It works equally well for doped semiconductors, which contain a finite density of free carriers (electrons in the conduction band or holes in the valence band).

Optical Properties of Solids Robert A. Taylor

These lecture notes by Prof. Robert Taylor at Oxford provide a concise, formal treatment of the Drude model and its application to doped semiconductors.

Please read the section 'V Optical response of a free electron gas' (starts on page 10). Focus on the main derivation and then skim to the subsection 'Example: Reflection from doped semiconductors' (page 13), paying attention to the graph for InSb.

When applying the Drude model to a doped semiconductor, we must consider three key modifications:

Carrier Density $n$ : The free carrier density is determined by the doping level and is typically many orders of magnitude lower than in a metal (e.g., $1 0^{16}$ - $1 0^{18}$ cm $^{- 3}$ vs. $1 0^{22}$ cm $^{- 3}$ ).
Effective Mass $m^{*}$ : The inertia of the charge carriers is described by the band-structure effective mass $m^{*}$ , not the free electron mass $m_{e}$ .
Background Dielectric $ϵ_{bk g d}$ : The host semiconductor crystal (e.g., Si, GaAs) has a significant background dielectric constant ( $ϵ_{bk g d} \approx 10 - 12$ ) due to the polarizability of its valence electrons.

The plasma frequency for a doped semiconductor is therefore:

ω_{p}^{2} = \frac{n e ^{2}}{m ^{*} ϵ _{0} ϵ _{bk g d}}

Because $n$ is much smaller and $ϵ_{bk g d}$ is larger than 1, the plasma frequency in semiconductors is much lower than in metals, typically falling in the infrared (IR) or Terahertz (THz) range. This means that doped semiconductors are often reflective in the far-infrared and transparent in the near-infrared and visible regions, a phenomenon known as free-carrier reflectivity. The sharp drop in reflectivity seen in the InSb data is the plasma edge for that material.

Conclusion

Today, we have derived one of the cornerstones of solid-state optics. The Drude model, despite its classical simplicity, provides a powerful and intuitive framework for understanding the optical response of free carriers.

Key Takeaways:

The Drude model treats conduction electrons as a free gas subject to a driving electric field and a damping force from scattering.
The frequency-dependent dielectric function $ϵ (ω)$ is derived by relating electron velocity to current density, then to AC conductivity $σ (ω)$ , and finally to $ϵ (ω)$ .
The plasma frequency $ω_{p}$ is a critical parameter determined by the carrier density, marking the transition from metallic (reflective) to dielectric (transparent) behavior.
For frequencies $ω < ω_{p}$ , the real part of $ϵ (ω)$ is negative, leading to high reflectivity.
For frequencies $ω > ω_{p}$ , the real part of $ϵ (ω)$ is positive, and the material becomes transparent (if absorption is weak).
The model applies to doped semiconductors by adjusting for carrier density ( $n$ ), effective mass ( $m^{*}$ ), and the background dielectric constant ( $ϵ_{bk g d}$ ), placing their plasma edge in the IR/THz range.

Preview of the Next Lesson

The Drude model successfully explains the optical properties governed by free carriers (intraband processes). However, it cannot explain the colors of metals like gold and copper, nor the fundamental absorption edge of semiconductors. These phenomena arise from electrons jumping between different bands. In our next lesson, we will use Fermi's golden rule to calculate the optical absorption coefficient for a semiconductor with a direct bandgap, which is the quantum mechanical description of these interband transitions.