Electrical Conductivity

Metals are now known to be primarily elements characterized by atoms in which the outermost orbital shell has very few electrons with corresponding values of energy. The highest conductivity occurs in metals with only one electron occupying a state in that shell. Silver, copper, and gold are examples of high-conductivity metals. Metals are found mainly toward the left side of the periodic table of the elements, and in the transition columns. The electrons contributing to their conductivity are also the electrons that determine their chemical valence in forming compounds. Some metallic conductors are alloys of two or more metal elements, such as steel, brass, bronze, and pewter.

A piece of metal is a block of metallic atoms. In individual atoms the valence electrons are loosely bound to their nuclei. In the block, at room temperature, these electrons have enough kinetic energy to enable them to wander away from their original locations. However, that energy is not sufficient to remove them from the block entirely because of the potential energy of the surface, the outermost layer of atoms. Thus, at their sites, the atoms are ionized—that is, left with a net positive charge—and are referred to as ion cores. Overall, the metal is electrically neutral, since the electrons' and ion cores' charges are equal and opposite. The conduction electrons are bound to the block as a whole rather than to the nuclei.

These electrons move about as a cloud through the spaces separating the ion cores. Their motion is random, bearing some similarities to gas molecules, especially scattering, but the nature of the scattering is different. Electrons do not obey classical gas laws; their motion in detail must be analyzed quantum-mechanically. However, much information about conductivity can be understood classically.

A particular specimen of a metal may have a convenient regular shape such as a cylinder (wire) or a prism (bar). When a battery is connected across the ends of a wire, the electrochemical energy of the battery imparts a potential difference, or voltage between the ends. This electrical potential difference is analogous to a hill in a gravitational system. Charged particles will then move in a direction analogous to downhill. In the metal, the available electrons will move toward the positive terminal, or anode, of the battery. As they reach the anode, the battery injects electrons into the wire in equal numbers, thereby keeping the wire electrically neutral. This circulation of charged particles is termed a current, and the closed path is termed a circuit. The battery acts as the electrical analog of a pump. Departing from the gravitational analogy, in which objects may fall and land, the transport of charged particles requires a closed circuit.

Current is defined in terms of charge transport:

where I is current, q is charge, and t is time. Thus q/t is the rate of charge transport through the wire. In a metal, as long as its temperature remains constant, the current is directly proportional to the voltage. This direct proportion in mathematical terms is referred to as linear, because it can be described in a simple linear algebraic equation:

In this equation, V is voltage and G is a constant of proportionality known as conductance, which is independent of V and remains constant at constant temperature. This equation is one form of Ohm's law, a principle applicable only to materials in which electrical conduction is linear. In turn, such materials are referred to as ohmics.

The more familiar form of Ohm's law is:

where R is 1/G and is termed resistance.

Conceptually, the idea of resistance to the passage of current preceded the idea of charge transport in historical development.

The comparison of electrical potential difference to a hill in gravitational systems leads to the idea of a gradient, or slope. The rate at which the voltage varies along the length of the wire, measured relative to either end, is called the electric field:

The field E is directly proportional to V and inversely proportional to L in a linear or ohmic conductor. This field is the same as the electrostatic field defined in the article on electrostatics. The minus sign is associated with the need for a negative gradient to represent "downhill." The electric field in this description is conceptually analogous to the gravitational field near the earth's surface.

Experimental measurements of current and voltage in metallic wires of different dimensions, with temperature constant, show that resistance increases in direct proportion to length and inverse proportion to cross-sectional area. These variations allow the metal itself to be considered apart from specimen dimensions. Using a proportionality constant for the material property yields the relation:

where ρ is called the resistivity of the metal. Inverting this equation places conduction rather than resistance uppermost:

where σ is the conductivity, the reciprocal (1/ρ) of the resistivity.

This analysis may be extended by substitution of equivalent expressions:

Introducing the concept of current density, or current flowing per unit cross-sectional area:

yields an expression free of all the external measurements required for its actual calculation:

This equation is called the field form of Ohm's law, and is the first of two physical definitions of conductivity, rather than mathematical.

The nature of conductivity in metals may be studied in greater depth by considering the electrons within the bulk metal. This approach is termed microscopic, in contrast to the macroscopic properties of a metal specimen. Under the influence of an internal electric field in the material, the electron cloud will undergo a net drift toward the battery anode. This drift is very slow in comparison with the random thermal motions of the individual electrons. The cloud may be characterized by the concentration of electrons, defined as total number per unit volume:

where n is the concentration, N the total number, and U the volume of metal (U is used here for volume instead of V, which as an algebraic symbol is reserved for voltage). The total drifting charge is then:

where e is the charge of each electron.

N is too large to enumerate; however, if as a first approximation each atom is regarded as contributing one valence electron to the cloud, the number of atoms can be estimated from the volume of a specimen, the density of the metal, and the atomic mass. The value of n calculated this way is not quite accurate even for a univalent metal, but agrees in order of magnitude. (The corrections are quantum-mechanical in nature; metals of higher valence and alloys require more complicated quantum-based corrections.) The average drift velocity of the cloud is the ratio of wire length to the average time required for an electron to traverse that length. Algebraic substitutions similar to those previously shown will show that the current density is proportional to the drift velocity:

The drift velocity is superimposed on the thermal motion of the electrons. That combination of motions, in which the electrons bounce their way through the metal, leads to the microscopic description of electrical resistance, which incorporates the idea of a limit to forward motion. The limit is expressed in the term mobility:

so that mobility, the ratio of drift velocity to electric field, is finite and characteristic of the particular metal.

Combining these last two equations produces the second physical definition of conductivity:

The motion of electrons among vibrating ion cores may be analyzed by means of Newton's second law, which states that a net force exerted on a mass produces an acceleration:

Acceleration in turn produces an increasing velocity. If there were no opposition to the motion of an electron in the space between the ion cores, the connection of a battery across the ends of a wire would produce a current increasing with time, in proportion to such an increasing velocity. Experiment shows that the current is steady, so that there is no net acceleration.

Yet the battery produces an electric field in the wire, which in turn produces an electric force on each electron:

Thus, there must be an equal and opposite force associated with the behavior of the ion cores. The analogy here is the action of air molecules against an object falling in the atmosphere, such as a raindrop. This fluid friction generates a force proportional to the velocity, which reaches a terminal value when the frictional force becomes equal to the weight. This steady state, for which the net force is zero, corresponds to the drift velocity of electrons in a conductor. Just as the raindrop quickly reaches a steady speed of fall, electrons in a metal far more quickly reach a steady drift velocity manifested in a constant current.

Thus far, this discussion has required that temperature be held constant. For metals, experimental measurements show that conductivity decreases as temperature increases. Examination suggests that, for a metal with n and e fixed, it is a decrease in mobility that accounts for that decrease in conductivity. For moderate increases in temperature, the experimental variation is found to fit a linear relation:

Here the subscript "0" refers to initial values and a is called the temperature coefficient of resistivity. This coefficient is found to vary over large temperature changes.

To study the relationship between temperature and electron mobility in a metal, the behavior of the ion cores must be considered. The ion cores are arranged in a three-dimensional crystal lattice. In most common metals the structure is cubic, and the transport functions are not strongly dependent on direction. The metal may then be treated as isotropic, that is, independent of direction, and all the foregoing equations apply as written. For anisotropic materials, the orientational dependence of transport in the crystals leads to families of equations with sets of directional coefficients replacing the simple constants used here.

Temperature is associated with the vibrational kinetic energy of the ion cores in motion about their equilibrium positions. They may be likened to masses interconnected by springs in three dimensions, with their bonds acting as the springs. Electrons attempting to move among them will be randomly deflected, or scattered, by these lattice vibrations, which are quantized. The vibrational quanta are termed phonons, in an analogy to photons. Advanced conductivity theory is based on analyses of the scattering of electrons by phonons.

With the increase in vibrational energy as temperature is increased, the scattering is increased so that the drift motion is subjected to more disruption. Maintenance of a given current would thus require a higher field at a higher temperature.

If the ion cores of a specific metal were identical and stationary in their exact equilibrium lattice sites, the electron cloud could drift among them without opposition, that is, without resistance. Thus, three factors in resistance can be identified: (a) lattice vibrations, (b) ion core displacement from lattice sites, and (c) chemical impurities, which are wrong ion cores. The factors (a) and (b) are temperature-dependent, and foreign atoms contribute their thermal motions as well as their wrongness. Additionally, sites where ions are missing, or vacancies, also are wrong and contribute to scattering. Displacements, vacancies, and impurities are classed as lattice defects.

A direct extension of thermal behavior downward toward the absolute zero of temperature suggests that resistance should fall to zero monotonically. This does not occur because lattice defects remain wrong and vibrational energy does not drop to zero-quantum mechanics accounts for the residual zero-point energy. However, in many metals and many other substances at temperatures approaching zero, a wholly new phenomenon is observed, the sudden drop of resistivity to zero. This is termed superconductivity.