Electrical properties of metals - delocalised electron field , ohm's law , resistivity vs temp for metals and insulators

Why do metals conduct electricity so well while wood does not? The answer lies in the microscopic chaos of the "Electron Sea." This post explores the delocalized electron model, the physical origins of Ohm's Law, and the contrasting ways temperature affects metals versus insulators.

Figure 1: The "Sea of Electrons" model showing positive ion cores submerged in a cloud of delocalized valence electrons.

Electrical Properties: Metals vs. Insulators

1. The Delocalized Electron Field

The defining characteristic of a metal is the presence of delocalized electrons (often called free electrons).

• Ion Cores: The atoms in a metal lattice lose their valence electrons, becoming positive ions fixed in a periodic arrangement.
• Electron Gas: These valence electrons are not bound to any single atom. Instead, they move freely throughout the crystal, behaving like a gas.

Key Concept: Conductivity occurs because these delocalized electrons can drift easily under the influence of an external Electric Field (\( \vec{E} \)).

2. Ohm's Law in Metals

We established in previous posts that Ohm's Law in vector form is \( \vec{J} = \sigma \vec{E} \). For metals, the conductivity (\( \sigma \)) depends on two factors:
1. The density of free electrons (\( n \)).
2. The relaxation time (\( \tau \)), which is the average time an electron travels before crashing into a lattice ion.

\[ \sigma = \frac{n e^2 \tau}{m} \]

Resistivity (\( \rho \)) is the inverse of conductivity: \[ \rho = \frac{1}{\sigma} = \frac{m}{n e^2 \tau} \]

3. Resistivity vs. Temperature (\( \rho \) vs \( T \))

The effect of heat on electrical flow distinguishes metals from non-metals.

[INSERT IMAGE OF RESISTIVITY VS TEMPERATURE GRAPH HERE]
Figure 2: Comparison of Resistivity trends. Metals show a linear increase, while Semiconductors/Insulators show an exponential decrease with temperature.

A. Metals (Positive Temperature Coefficient)

Behavior: As temperature increases, resistivity increases.
Why?
In metals, the electron density (\( n \)) is constant. However, as temperature rises, the positive ions in the lattice vibrate more roughly (phonon scattering). This increases the frequency of collisions, effectively reducing the relaxation time (\( \tau \)).

Formula: \[ \rho(T) = \rho_0 [1 + \alpha(T - T_0)] \] Where \( \alpha \) is the temperature coefficient of resistivity.

B. Insulators & Semiconductors (Negative Coefficient)

Behavior: As temperature increases, resistivity decreases.
Why?
In these materials, the primary factor is the number of charge carriers (\( n \)). At low temperatures, electrons are stuck in the valence band. As heat is applied, electrons gain enough energy to jump the band gap (\( E_g \)) into the conduction band. This exponential increase in \( n \) outweighs the increase in collisions.

Formula: \[ \rho(T) \approx A e^{ \frac{-E_g}{2k_B T} } \]