Electrical Resistivity

How can a metal have infinite conductivity?
In fact, the electrical resistivity of all metals and alloys decreases when they are cooled. So firstly we have to think what causes a conductor to have resistance.

A metal consists of a lattice of atoms , each with a shell of electrons. This can also be known as a positive ionic lattice. A normal metal consist of a regular crystalline lattice of positively charged ions and gas of free, on –interacting conduction electrons that fill the inter ionic space of lattice. If there is one electron per ion, we should have approximately Because of the electrons are the opposite charge from the ions, the total charge is balanced and metal is electrically neutral. When an electrical potential difference is applied across the metal, the electrons drift from one end of the conductor to the other under the influence of the electrical field. Because of the external force accelerate the free electrons a current flow occurs within the metal.

So the current in a conductor is carried by “conduction electrons” which are free to move through the material. An electron is able to pass through a perfect crystal without any loss of momentum in its original direction . However, any fault in the periodicity of the crystal will scatter the electron and introduce some resistance. So there are two effects which can spoil the perfect periodicity of crystal lattice and so introduce resistance.

1. At temperatures above absolute zero the atoms are vibrating and electrons equilibrium positions will be displaced by various amounts. As we increase the temperature, the electrons also get scattered by thermal vibrations of the lattice, called phonons, so the resistivity rises with temperature. This contribution is called phonon resistivity, . When the temperature is lowered, the thermal vibrations of atoms decreases and the conduction electrons are less frequently scattered.

A phonon, is a quantum of energy, relating to a mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid

Figure 2.1- The schematic diagram of phonon resistivity


2. Foreign atoms or other defects (imperfections in periodicity like dislocations) randomly distributed can interrupt the perfect periodicity. As every real metal contains some imperfections and impurities, observes some finite resistivity at very low temperatures (extrapolated T=0) . Therefore the electrons, in addition to being scattered bye the impurities ,and this impurity scattering is more or less independent of temperature. As a result there is certain resistivity called “residual resistivity”, , which remains even at the lowest temperatures. The more impure the metal, the larger will be its residual resistivity. (Figure 2.2)

Figure 2.2- Variation of resistance of metals with temperature

Near room temperatures, the thermal motion of ions is the primary source of scattering of electrons (due to destructive interference of free electron waves on non-correlating potentials of ions), and is thus the prime cause of metal resistance. Imperfections of lattice also contribute into resistance, although their contribution in pure metals is negligible.

The larger the cross-sectional areas of the conductor, the more electrons are available to carry the current, so the lower the resistance. The longer the conductor, the more scattering events occur in each electron's path through the material, so the higher the resistance.


And, some metals ,however, show a very remarkable behavior; when they are cooled their electrical resistance decrease in the usual way, but on reaching a temperature a few degrees above absolute zero they suddenly lose all trace of electrical resistance (Figure 2.3) . They are then said to have passed into the superconducting state. And this transformation to the superconducting state may occur even if the metal is so impure that it would otherwise have had large residual resistivity.

Figure 2.3 The difference of behavior ,non-superconductive metal and superconductor metal about loss of resistance at low temperatures.