Drude Theory of Metals

Table of Contents

Metals

Metals occupy a special place in solid state physics. They share striking properties:

• High electrical and thermal conductivity
• Ductility, malleability, and metallic luster

Although most solids are nonmetallic, metals have played a central role in theory from the late 19th century onward. The metallic state is considered a fundamental state of matter — over two-thirds of elements are metals.

To understand nonmetals, it’s often necessary to understand metals first (e.g. copper vs. salt).

Over the past 100+ years, physicists have built models to describe metals both qualitatively and quantitatively.

This chapter introduces the Drude model (ca. 1900):

• Simple, intuitive picture of metallic conduction
• Useful for rough estimates, still in use today
• Fails to explain some key experiments and deeper phenomena

Basic Assumptions of the Drude Model

• After the discovery of the electron (1897), Drude applied kinetic theory of gases to metals, modeling them as a gas of electrons.
• In simplest kinetic theory:
  • Electrons are like gas molecules → rigid spheres moving in straight lines until they collide.
  • Collisions are instantaneous and no forces act between them except during collisions.

Electrons in a Metal

• A metal contains both:
  • Light, mobile valence electrons
  • Heavy, immobile positive ions
• When atoms condense into a metal:
  • Valence electrons become delocalized → form an electron gas
  • Core electrons + nucleus = immobile ion background

Electron Density

Drude treated conduction electrons as a gas of particles of mass $m$. Their density:

Where:

• $Z$: conduction electrons per atom
• $\rho_m$: mass density in g/cm³
• $A$: atomic mass

Also used: Wigner-Seitz radius $r_s$ — radius of a sphere per conduction electron:

Key Assumptions of Drude Theory

1. Independent electron approximation:

   • Ignore electron-electron and electron-ion interactions during collisions.
   • Each electron moves freely between collisions.

2. Collisions are instantaneous:

   • Modeled as abrupt velocity changes (like bouncing off hard walls).
   • No gradual scattering — just sudden redirection.

3. Collision rate:

   • Probability of collision per time = $1/\tau$
   • After each collision, an electron emerges with a random velocity, unrelated to its prior motion.

4. Local thermal equilibrium:

   • Electrons regain thermal distribution after each collision.
   • Hotter regions → faster electrons emerge.

Despite its simplicity, the Drude model gives a surprisingly good first approximation to electron behavior in metals, especially for:

• Estimating conductivity
• Understanding basic scattering processes
• Building intuition for metallic transport

Table: Free Electron Densities

DC Electrical Conductivity of a Metal

Ohm’s Law:

Drude theory aims to estimate $R$ from microscopic principles. To eliminate geometry dependence, we define resistivity $\rho$ via:

Current Density

Let current $I$ flow through a wire of length $L$, cross-sectional area $A$:

Using $V = \rho j L \Rightarrow R = \rho \frac{L}{A}$

Drude model: electrons of charge $-e$, number density $n$, average velocity $\\mathbf{v}$:

Step-by-Step Conductivity Derivation

Starting from the current density equation:

$\\mathbf{j} = -ne \\mathbf{v}$

The average drift velocity in an electric field is:

$\mathbf{v}_{\text{avg}} = -\frac{e \tau}{m} \mathbf{E}$

Substituting the drift velocity into the current equation:

$\mathbf{j} = -ne \left(-\frac{e \tau}{m} \mathbf{E}\right)$

Simplifying the expression:

$\mathbf{j} = \frac{ne^2 \tau}{m} \mathbf{E}$

This gives us the famous Drude conductivity:

$\sigma = \frac{ne^2 \tau}{m}$

So from $\mathbf{j} = -ne \mathbf{v}_{\text{avg}}$:

with:

Estimating Relaxation Time

Rearranging:

This allows $\tau$ to be estimated from experimental resistivity.

Resistivity vs Temperature

Table 1.2: Selected values of $\rho$ (μΩ⋅cm)

• Resistivity rises roughly linearly with $T$ at room temp.
• Drops steeply as $T \to 0$

• At room temperature, $\tau \sim 10^{-14} \, \text{sec}$
• A more intuitive idea: estimate the mean free path, $\ell = \bar{v} \tau$, the average distance an electron travels between collisions

Drude Relaxation Times (Approximate, $10^{-14}$ sec)

• Relaxation time decreases with increasing temperature
• At low $T$, scattering is dominated by impurities/defects
• At high $T$, phonon scattering dominates

Time-Dependent View: Momentum and Collisions

• Average electron velocity: $\mathbf{v} = \mathbf{p}(t)/m$
• So current density is:

To model $\mathbf{p}(t)$, assume:

• Electrons collide randomly with a probability $dt/\tau$
• Between collisions, motion evolves under external fields (electric, magnetic)

Evolution of Electron Momentum

• Total momentum per electron satisfies:

where $\mathbf{f}(t)$ is the force per electron due to external fields.

Taking the limit as $dt \to 0$:

This is the Drude momentum equation — Newton’s second law with a damping term due to collisions.

Hall Effect and Magnetoresistance

In 1879, E. H. Hall tested whether magnetic forces act on:

• The whole wire (current), or
• Just the moving electrons

He reasoned: if a magnetic field deflects moving charges, they should shift sideways in the conductor → causing a transverse voltage. Hall was able to measure this.

Lorentz Force and Setup

Electric field $\mathbf{E}_x$ drives current density $j_x$ in the $x$-direction.
Magnetic field $\mathbf{H}$ points in the $z$-direction.

Lorentz force on electrons:

• Electrons deflect in $-y$ direction
• Accumulate on one side → build up transverse electric field $\mathbf{E}_y$
• This transverse field cancels Lorentz force in steady state

Measured Quantities

Two main observables:

1. Magnetoresistance (longitudinal):

   $$\rho(H) = \frac{E_x}{j_x}$$

2. Hall Coefficient:

   $$R_H = \frac{E_y}{j_x H}$$

• Negative $R_H$ implies negative charge carriers (electrons)
• Surprisingly, some metals show positive $R_H$

Hall Effect Analysis

Starting with the Lorentz force on moving electrons:

$\mathbf{F} = -\frac{e}{c} \mathbf{v} \times \mathbf{H}$

Newton’s second law with all forces (electric field, magnetic field, and collisions):

$\frac{d\mathbf{p}}{dt} = -e\mathbf{E} - \frac{e}{c} \mathbf{v} \times \mathbf{H} - \frac{\mathbf{p}}{\tau}$

In steady state, breaking into x and y components:

$0 = -eE_x - \omega_c p_y - \frac{p_x}{\tau}$

$0 = -eE_y + \omega_c p_x - \frac{p_y}{\tau}$

This leads to the Hall coefficient:

$R_H = \frac{E_y}{j_x H} = -\frac{1}{nec}$

Total force per electron:

Steady-state: $\frac{d\mathbf{p}}{dt} = 0$

Break into components:

Where cyclotron frequency:

Current Components

From $\mathbf{j} = -ne\mathbf{v} = -\frac{ne}{m}\mathbf{p}$, we get:

• $\sigma_0$ is the Drude conductivity with no magnetic field
• The angle $\phi$ between $\mathbf{E}$ and $\mathbf{j}$ is the Hall angle: $$\tan \phi = \omega_c \tau$$

Table: Hall Coefficients at High Fields

• For alkali metals, Drude’s prediction works well: one electron per atom.

• For multivalent metals (Be, Mg, In, Al), observed values deviate from simple theory — sometimes even suggesting positive charge carriers.

• $R_H$ depends only on carrier type and density

• Surprisingly, in real materials:

  • $R_H$ often deviates from the Drude prediction
  • Depends on magnetic field strength and sample preparation

• Quantum theory is required to explain deviations

Drude model predicts:

Cyclotron Frequency and Magnetic Effects

The cyclotron frequency:

With:

• $\omega_c \tau$ is a dimensionless measure of magnetic field strength
• If $\omega_c \tau \ll 1$: field barely perturbs orbits
• If $\omega_c \tau \gtrsim 1$: strong deflection → major magnetic effects

AC Electrical Conductivity of a Metal

Let the electric field oscillate with frequency $\omega$:

Momentum equation:

Assume steady-state solution:

So that,

Current density:

Define frequency-dependent (AC) conductivity:

Electromagnetic Wave Propagation

We derive wave equations from Maxwell’s laws:

Assume $e^{-i\omega t}$ time dependence and substitute $\mathbf{j} = \sigma(\omega) \mathbf{E}$:

Wave equation with complex dielectric:

High-Frequency Limit and Plasma Frequency

If $\omega \tau \gg 1$, then:

Define plasma frequency:

No propagation if $\omega < \omega_p$ (since $\epsilon < 0$).

Estimate:

Transparency Threshold

Frequency and wavelength at which metals become transparent:

Table 1.5: Transparency Threshold (approximate)

Plasma Oscillations (Plasmons)

Charge oscillations with time dependence $e^{-i\omega t}$.

From continuity and Gauss’s law:

Using $\mathbf{j}(\omega) = \sigma(\omega) \mathbf{E}(\omega)$:

Same condition as for electromagnetic wave propagation: $\epsilon(\omega) = 0$.

Simple Plasma Oscillation Model

Assume slab of $N$ electrons displaced by $d$:

• Surface charge: $\sigma = \pm nde$
• Electric field: $E = 4\pi nde$
• Equation of motion: $$Nm \ddot{d} = -Ne(4\pi nde) = -4\pi ne^2 N d$$

→ Harmonic oscillator at $\omega_p$.

Plasmons observable via electron energy loss spectroscopy.

Thermal Conductivity of a Metal

Wiedemann–Franz Law

One of the Drude model’s key successes was explaining the Wiedemann–Franz law:

This states that the ratio of thermal conductivity $\kappa$ to electrical conductivity $\sigma$ is proportional to $T$. The constant of proportionality is the Lorenz number:

This relationship holds remarkably well across many metals.

Experimental Values

Units:

• $\kappa$: W/cm·K
• $\kappa/\sigma T$: W·Ω/K²

Heat Conduction in Metals

Drude assumed:

• Heat is carried by conduction electrons, not ions
• Electron collisions randomize direction
• Temperature gradient leads to a net thermal current

Thermal current density (Fourier’s Law):

Microscopic Picture

Electrons from hotter side carry more energy than those from colder side. Assume:

• Electrons come equally from left ($x - vt$) and right ($x + vt$)
• Energy per electron: $\varepsilon(T)$

Thermal current density:

For small gradients:

In 3D:

Wiedemann–Franz from Drude

Using:

• $c_v = \frac{3}{2}n k_B$
• $v^2 = \frac{3k_B T}{m}$

We get:

This is half the experimental value. Drude’s estimate was off due to errors in both $c_v$ and $v^2$, which accidentally canceled.

Thermoelectric Field (Seebeck Effect)

A temp gradient also produces an electric field:

For no net current ($v_Q + v_E = 0$), use:

• Drift from temp gradient:

  $$v_Q = -\tau \frac{d}{dx} \left( \frac{v^2}{2} \right) = -\tau v \frac{dT}{dx} \text{(1D)}$$ $$v_Q = -\frac{\tau}{6} \frac{dv^2}{dT} \nabla T \text{(3D)}$$

• Drift from electric field:

  $$v_E = -\frac{eE\tau}{m}$$

To cancel, require: