Question

Consider the conductivity equation obtained from the classical electron theory. According to this equation, a bivalent metal, such as zinc, should have a larger conductivity than a monovalent metal, such as copper, because zinc has about twice as many free electrons as copper. Resolve this discrepancy by considering the quantum mechanical equation for conductivity.

Answer 1

Answer & Explanation:

Copper Zinc

Atomic Number 29 30

Electron configuration 3d¹⁰4s¹ 3d¹⁰4s²

Conductivity /100 /28

The conductivity equation from classical theory is given as follows

`\sigma = (N_f re^(2))/(m) ......................................................................1`

The conductivity equation from quantum mechanics is given as

`\sigma = (N(E_f)r \theta _f e^(2) )/( \epsilon ) ......................................................................2`

This quantum mechanical equation reveals that the conductivity depends on the:

Fermi velocity

The relaxation time

and the population density (per unit volume) proportional to the density of states.

Equation (2) is more meaningful than the expression derived from the classical electron theory (eqn 1). Specifically, (Eqn 1) contains the information that not all free electrons Nf are responsible for conduction, i.e., the conductivity in metals depends to a large extent on the population density of the electrons near the Fermi surface. For example, monovalent metals (such as copper, silver, or gold) have partially filled valence bands. Their electron population densities near their Fermi energy are high (fig. 1 attached below), which results in a large conductivity according to (Eqn 2).

Bivalent metals, on the other hand, are distinguished by an overlapping of the upper bands and by a small electron concentration near the bottom of the valence band, As a consequence, the electron population near the Fermi energy is small (fig. 1), which leads to a comparatively low conductivity. Finally, insulators and semiconductors have, under certain conditions, completely filled electron bands, which results in a virtually zero population density near the top of the valence band (Fig. 1). Thus, the conductivity in these materials is extremely small.