Material System - Conductors & Insulating Materials Complete Study Notes

Conductors

The electrons in the outermost orbitals of the atoms of a solid determine its electrical properties. The free electron model is introduced, starting with a brief description of the wide range of resistivity of materials. The justification for Ohm’s law is given, in terms of the free electron motion and the collisions processes in crystals. Materials for conductive and resistive functions are discussed.

Conductors are metals and alloys. Gold, silver and copper are among the best conductors of electricity. Therefore, their electrical resistivities are the lowest, as shown in table below. They are followed by aluminium whose resistivity is 60% higher than that of copper. Transition metals such as iron and nickel are not as good conductors as shown below. Still poorer conductors are the semimetals of the fifth column, e.g., antimony and bismuth. Graphite, with one of its bonding electrons resonating between the (sp²) bonds, also fall in this category of semimetals. The electrical resistivity of conductor’s ranges from 10^-9 to 10^-3ohm m. The electrical conductivity, being the reciprocal of resistivity, ranges from 10⁹ ohm^-1 m^-01 to 10³ ohm^-1 m^-1.

Constants

Units

 When the resistivity is in the range 10^-3-10³ ohm m, we have the second category of materials known as semiconductors:- they form the base for a number of solid state devices. Here, the resistivity is a strong function of small concentrations of impurities. Doped germanium, with an impurity content of a few tens per million, can have a resistivity about two orders of magnitude lower than that of pure germanium as shown in above table. Pure silicon has a higher resistivity than pure germanium.

The third category of materials are insulators:- Common electrical insulating materials such as polyethylene, Bakelite, Lucite, mica, PVC, rubber and porcelain fall in this category. The resistivity range for this category extends from 10⁴ to beyond 10¹⁷ ohm-m. Here, a difference in resistivity of some twelve orders of magnitude is noticeable between a silica glass and soda-lime-silica (window glass). This striking difference is a result of the ionic conductivity of window glass. The relatively loosely bound sodium and calcium cations in soda-lime glass diffuse and conduct much more readily, as compared to the tightly bound immobile silicon cations in pure silica. Ionic conduction and ionic diffusivity are closely related phenomena.

The Free Electron Theory

The conducting properties of a solid are not a function of the total number of electrons in the solid, as only the outermost electrons of the atoms can take part in conduction. In the free electron model, the outermost electrons of an atom are not bound to that atom, but are free to move through the whole solid. These electrons have been variously called the free electron cloud, the free electron gas or the Fermi gas.

In the free electron theory, the basic assumption is that the potential field due to the ion cores is uniform throughout the solid. The free electron have the same potential energy everywhere in the solid. Due to the electrostatic attraction between a free electron and the ion cores, this potential energy will be a finite negative value. As we are interested only in energy differences, we can assume this constant potential to be zero. Then the only energy that we have to consider is the kinetic energy. This kinetic energy is substantially lower than that of the bound electrons in an isolated atom, as the field of motion for the free electron is considerably enlarged in the solid as compared to the field around an isolated atom.

Electrons have both particle-like and wave-like characteristics. The de Broglie wavelength of an electron λ is related to its momentum mv as

Where h is Plank’s constant, m is the mass of the free electron and ν is its velocity. The wavelength is inversely related to the magnitude of the wave number vector k:

As the velocity of the free electrons is much smaller than that of light, we can ignore relativistic effects and use the classical relation for kinetic energy E.

Substitute the above values from equations, we obtain

Conduction By Free Electron

The wave number k takes both positive and negative values. For every electron moving with a certain speed in a direction, there is another electron moving with the same speed in the opposite direction. This equal and opposite velocity distribution in a neutral solid can be biased by an externally applied electrical field to yield a net velocity in one direction. With this biasing, the solid conducts electricity.

Figure: Electrons moving towards the positive end of the applied field acquire extra velocity, while those moving in the opposite direction lose some velocity.

The negatively charged electrons are accelerated towards the positive end of the field. The velocity of the fastest electron moving in the direction of the positive end has a larger magnitude than that of the fastest electron moving towards the negative end of the field. Such redistribution is possible, only when empty electron states are available immediately above the Fermi level. This availability is a basic characteristic of conductors, as opposed to semiconductors and insulators.

The force experienced by an electron of charge e in an applied field of gradient  ‘ε’ can be equated to the force as defined in the classical law:

εe = ma

when m is the mass of the electron and a is the acceleration due to the applied field. The electrons that are accelerated towards the positive end of the field do not continue to increase their velocity indefinitely. They collide with obstacles on their way. Depending on the time interval between two successive collisions, the electrons acquire an average increment of velocity called drift velocity, all of which they lose during a collision, as illustrated in Fig.The drift velocity is the extra velocity that electrons acquire over and above their normal velocity in the absence of a field.

Figure: The extra velocity acquired by an electron due to an applied field is lost on collision with an impurity, imperfection or phonon.

If the average collision time is  and  is the drift velocity acquired by the electrons.

The flux J_e due to the flow of electrons is called the current density:

Where n is the number of free electrons of charge e. This is in the form of Ohm’s law. As conductivity σ is by definition the flux per unit potential gradient, we have

The electrical resistivity ρ is the reciprocal of conductivity.

Conductor and Resistor Materials

The resistivity, the temperature coefficient of resistance, the density and the tensile strength of typical conductor and resistor materials are listed in Table below.

Properties of Typical Conductors and Resistors at Room Temperature

 The tensile strength values given here are approximate, as they depend on the prior thermal and mechanical history of the metal.

For use as conductors in applications such as transmission lines and distribution lines, how I²R loss is the primary consideration and the choice would be from amongst the best conductors, keeping in view the cost, fabricability and mechanical strength. Copper and aluminium are the most likely choices. For long distance transmission lines, aluminium is chosen. As a large cross section would reduce the I²R loss, thick cables are preferred. If the elastic modulus of the aluminium cables is improved by reinforcement with steel as in ACSR (aluminium conductor steel reinforced) cables, the distance between successive poles along the transmission line can be substantially increased. More expensive copper is used for distribution lines, busbars and other energy conversion applications, OFHC (oxygen-free high conductivity) copper is often specified., Among the common solutes in copper, Fe, P and As are the most harmful in impairing the electrical conductivity.

For electrical contacts used in switches, brushes and relays, the material must possess high electrical conductivity, high thermal conductivity, high melting point and good oxidation resistance. High thermal conductivity helps to dissipate the heat effectively. High melting point is desirable so that may accidental overheating does not fuse together the contact points. Good oxidation resistance is necessary to keep the contact clean and free of insulating oxides. Copper and silver largely satisfy the above requirements. For low cost, copper is commonly used. For critical contacts such of pure silver is increased by the dispersion of fine particles of CdO. Dislocations moving in silver have to bend around the dispersed CdO particles and a fine dispersion increases the strength. CdO improves the wear resistance of silver. It also decomposes at the melting point of silver, thereby absorbing much of the heat generated by arcing and minimizing the loss of expensive silver by evaporation.

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