## Electrical Conductivity

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# Electrical Conductivity

History

Materials

Metals

Semiconductors

Non-ohmic conductors

Resources

Conductivity is the ability of a material medium to permit the passage of charged particles or thermal energy. Thermal conductivity is the ability of a material to transmit heat energy, and electrical conductivity is its ability to transmit current (the movement of charged particles, most often electrons). Together, these are the most significant examples of a broader classification of phenomena known as transport processes. In metals, electrical conductivity and thermal conductivity are related since both involve aspects of electronmotion.

## History

The early studies of electrical conduction in metals were done in the eighteenth and early nineteenth centuries. Benjamin Franklin (17061790) in his experiments with lightning (leading to his invention of the lightning rod), reasoned that the charge would travel along the metallic rod. Alessandro Volta (17451827) derived the concept of electrical potential from his studies of static electricity, and then discovered the principle of the battery in his experiments with dissimilar metals in common contact with moisture. Once batteries were available for contact with metals, electric currents were produced and studied. Georg Simon Ohm (17871854) found the direct proportion relating current and potential difference, which became a measure of the ability of various metals to conduct electricity. Extensive theoretical studies of currents were carried out by Andre´ Marie Ampe`re (17751836).

To honor these scientists, the syste`me internationale (SI) units use their names. The unit of potential difference is the volt, and potential difference is more commonly called voltage. The unit of electrical resistance is the ohm, and the unit of current is the ampere. The relation among these functions is known as Ohms law.

Franklin is remembered for an unlucky mistake. He postulated that there was only one type of electricity, not two as others thought, in the phenomena known in his day. He arbitrarily called one form of static electric charge positive and attributed the opposite charge to the absence of the positive. All subsequent studies continued the convention he established. Late in the nineteenth century, when advancements in both electrical and vacuum technology led to the discovery of cathode rays, streams of particles issuing from a negative electrode in an evacuated tube, Sir Joseph John Thomson (18561940) identified these particles as common to all metals used as cathodes and negatively charged. The historical concept of a positive current issuing from an anode is mathematically self-consistent and leads to no analytical errors, so the convention is maintained but understood to be a convenience.

## Materials

Electrical conduction can take place in a variety of substances. The most familiar conducting substances are metals, in many of which the outermost electrons of the atoms can move easily in the interatomic spaces. Other conducting materials include semiconductors, electrolytes, and ionized gases, which are discussed later in this article.

## Metals

Metals are primarily elements characterized by atoms in which the outermost orbital shell has few electrons. The highest conductivity occurs in metals with only one electron occupying a state in the outermost 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 ionizedthat is, left with a net positive chargeand 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:

I = q/t

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:

I=GV

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 Ohms 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 Ohms law is:

I = V/R

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:

E = (V/L)

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 Earths 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:

R = ρ(L/A)

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

G = ζ(A/L)

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

This analysis may be extended by substitution of equivalent expressions:

G = I/V

ζ(A/L) = I/EL

ζ = I/AE

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

J = I/A

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

ζ = J/E

This equation is called the field form of Ohms 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:

n = N/U

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:

q = Ne = nUe

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:

J = nevd

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:

ζ = J/E = nevd/E = neu

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

F = ma

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:

F = eE

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:

ρ = ρ0[1 + α(T T0)]

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.

## Semiconductors

Semiconductors are materials in which the conductivity is much lower than for metals, and widely variable through control of their composition. These substances are now known to be poor insulators rather than poor conductors, in terms of their atomic structure. Though some semiconducting substances had been identified and studied by the latter half of the nineteenth century, their properties could not be explained on the basis of classical physics. It was not until the mid-twentieth century, when modern quantum-mechanical principles were applied to the analysis of both metals and semiconductors, that theoretical calculations of conductivity values agreed with the results of experimental measurements.

In a good insulator, electrons cannot move because nearly all allowed orbital states are occupied. Energy must then be supplied to remove an electron from an outermost bound position to a higher allowed state. This leaves a vacancy into which another bound electron can hop under the influence of an electric field. Thus, both the energized electron and its vacancy become mobile. The vacancy acts like a positive charge, called a hole, and drifts in the direction opposite to electrons. Electrons and holes are more generally termed charge carriers.

In good insulators, the activation energy of charge carriers is high, and their availability requires a correspondingly high temperature. In poor insulators, that is, semiconductors, activation occurs at temperatures moderately above 80.6°F (27°C). Each substance has a characteristic value.

There are many more compounds than elements that can be classed as semiconductors. The elements are a few of those in column IV of the periodic table, which have covalent bonds: carbon (C), germanium (Ge), and silicon (Si). For carbon, only the graphite form is semiconducting; diamond is an excellent insulator. The next element down in this column, tin (Sn), undergoes a transition from semiconductor to metal at 59°F (15°C), below room temperature, indicative of an unusefully low activation energy. Other elements that exhibit semiconductor behavior are found in the lower portion of column VI, specifically selenium (Se) and tellurium (Te).

There are two principal groups of compounds with semiconducting properties, named for the periodic table columns of their constituents: III-V, including gallium arsenide (GaAs) and indium antimonide (InSb), among others; and II-VI, including zinc sulfide (ZnS), selenides, tellurides, and some oxides. In many respects these compounds mimic the behavior of column IV elements. Their chemical bonds are mixed covalent and ionic. There are also some organic semi-conducting compounds, but their analysis is beyond the scope of this article.

A semiconductor is called intrinsic if its conductivity is the result of equal contributions from its own electrons and holes. The equation must then be expanded:

σ = nee μe + nhe μh

In an intrinsic semiconductor, ne =nh, and e has the same numerical value for an electron (-) and the hole left behind (+). The mobilities are usually different. These terms add because the opposite charges move in opposite directions, resulting in a pair of like signs in each product.

For application in devices, semiconductors are rarely used in their pure or intrinsic composition. Under carefully controlled conditions, impurities are introduced which contribute either an excess or a deficit of electrons. Excess electrons neutralize holes so that only electrons are available for conduction. The resulting material is called n-type, n for negative carrier. An example of n-type material is Si with Sb, a column IV element with a column V impurity known as a donor. In n-type material, donor atoms remain fixed and positively ionized. When a column III impurity is infused into a column IV element, electrons are bound and holes made available. That material is called p-type, p for positive carrier. Column III impurities are known as acceptors; in the material acceptor atoms remain fixed and negatively ionized. An example of p-type material is Si with Ga. Both n-type and p-type semiconductors are referred to as extrinsic.

Thermal kinetic energy is not the only mechanism for the release of charge carriers in semiconductors. Photons with energy equal to the activation energy can be absorbed by a bound electron, which, in an intrinsic semiconductor, adds both itself and a hole as mobile carriers. These photons may be in the visible range or in the near infrared, depending on EG . In extrinsic semiconductors, photons of much lower energies can contribute to the pool of the prevailing carrier type, provided the material is cooled to cryogenic temperatures in order to reduce the population of thermally activated carriers. This behavior is known as photoconductivity.

Each separate variety of semiconductor is ohmic, with the conductivity constant at constant temperature. However, as the temperature is increased, the conductivity increases very rapidly. The concentration of available carriers varies in accordance with an exponential function:

n α exp[(EG/kT)]

where EG is the gap or activation energy, k is Boltzmanns constant (1.38× 1023 joules/kelvin), T is absolute (kelvin) temperature, and the product kT is the thermal energy corresponding to temperature T. The increase in available charge carriers overrides any decrease in mobility, and this leads to a negative value for a. Indeed, a decrease in resistance with increasing temperature is a reliable indication that a substance is a semiconductor, not a metal. Graphite is an example of a conductor that appears metallic in many ways except for a negative α. The converse, a positive α, is not as distinct a test for metallic conductivity.

The Fermi level, EF, can be shown differently for intrinsic, n-type, and p-type semiconductors. However, for materials physically connected, EF must be the same for thermal equilibrium. This is a consequence of the laws of thermodynamics and energy conservation. Thus, the behavior of various junctions, in which the interior energy levels shift to accommodate the alignment of the Fermi level, is extremely important for the semiconductor devices.

## Non-ohmic conductors

Non-ohmic conduction is marked by nonlinear graphs of current vs. voltage. It occurs in semiconductor junctions, electrolytic solutions, some ionic solids not in solution, ionized gases, and vacuum tubes. Respective examples include semiconductor p-n diodes, battery acid or alkaline solutions, alkali halide crystals, the ionized mercury vapor in a fluorescent lamp, and cathode ray tubes.

Ionic conductivities are much lower than electronic, because the masses and diameters of ions make them much less mobile. While ions can drift slowly in a gas or liquid, their motions through the interstices of a solid lattice are much more restricted. Yet, with their thermal kinetic energy, ions will diffuse through a lattice, and in the presence of an electric field, will wander toward the appropriate electrode. In most instances, both ionic and electronic conduction will occur, depending on impurities. Thus, for studies of ionic conductivity, the material must be a very pure solid.

In gases, the gas atoms must be ionized by an electric field sufficient to supply the ionization energy of the gas in the tube. For stable currents, the ratio of field to gas pressure, E/P, is a major parameter. Electrons falling back into bound states produce the characteristic spectrum of the gas, qualitatively associated with color, e.g., red for neon, yellow-orange for sodium vapor, or blue-white for mercury vapor.

The basic definition of a plasma in physics includes all material conductors, ohmic and nonohmic. A plasma is a medium in which approximately equal numbers of opposite charges are present, so that the medium is neutral or nearly so. In a metal the negative electrons are separated from an equal number of positive ion cores. In a semiconductor there may be holes and electrons (intrinsic), holes and ionized acceptors (p-type), or electrons and ionized donors (n-type). In an electrolytic solution and in an ionic solid there are positive and negative ions. An ionized gas contains electrons and positive ions. A small distinction among these may be made as to whether the medium has one or two mobile carriers.

In contemporary usage, the term plasma usually refers to extremely hot gases such as those used in the Tokamak for nuclear fusion experiments. High-energy plasmas are discussed in the article on fusion as a means of generating electric power.

### KEY TERMS

Lattice The structure of atoms in a solid. In a conducting material, ion cores make up the lattice.

Potential difference In a conductor carrying an electric current, it is the difference of potential energy per unit charge.

The remaining non-ohmic conduction category is the vacuum tube, in which a beam of electrons is emitted from either a heated cathode (thermionic) or a suitably illuminated cathode (photoelectric), and moves through evacuated space to an anode. The beam in its passage is subjected to electrostatic or magnetic fields for control. The evacuated space cannot be classed either as a material with a definable conductivity or as a plasma, since only electrons are present. However, there are relations of current and voltage to be analyzed. These graphs are generally nonlinear or linear over a limited range. But vacuum tubes are not called ohmic even in their linear ranges because there is no material undergoing the lattice behavior previously described as the basis for ohmic resistance.

Electrical conduction in the human body and other animal organisms is primarily ionic, since body fluids contain vital electrolytes subject to electrochemical action in organs. Further information is available in other articles, particularly those on the heart, the brain, and neurons.

## Resources

### BOOKS

Ellse, Mark and Chris Honeywill. Electricity and Thermal Physics. Cheltenham, UK: Nelson Thornes, 2004.

Halliday, David, et al. Fundamentals of Physics. New York: Wiley, 2004.

Frieda A. Stahl

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# Electrical conductivity

Conductivity is the term used to describe the ability of a material medium to permit the passage of particles or energy . Electrical conductivity refers to the movement of charged particles through matter . Thermal conductivity refers to the transmission of heat energy through matter. Together, these are the most significant examples of a broader classification of phenomena known as transport processes. In metals, electrical conductivity and thermal conductivity are related since both involve aspects of electron motion .

## History

The early studies of electrical conduction in metals were done in the eighteenth and early nineteenth centuries. Benjamin Franklin (1706-1790) in his experiments with lightning (leading to his invention of the lightning rod), reasoned that the charge would travel along the metallic rod. Alessandro Volta (1745-1827) derived the concept of electrical potential from his studies of static electricity , and then discovered the principle of the battery in his experiments with dissimilar metals in common contact with moisture. Once batteries were available for contact with metals, electric currents were produced and studied. Georg Simon Ohm (1787-1854) found the direct proportion relating current and potential difference, which became a measure of the ability of various metals to conduct electricity. Extensive theoretical studies of currents were carried out by André Marie Ampère (1775-1836).

To honor these scientists, the système internationale (SI) units use their names. The unit of potential difference is the volt, and potential difference is more commonly called voltage. The unit of electrical resistance is the ohm, and the unit of current is the ampere. The relation among these functions is known as Ohm's law .

Franklin is remembered for an unlucky mistake. He postulated that there was only one type of electricity, not two as others thought, in the phenomena known in his day. He arbitrarily called one form of static electric charge positive and attributed the opposite charge to the absence of the positive. All subsequent studies continued the convention he established. Late in the nineteenth century, when advancements in both electrical and vacuum technology led to the discovery of cathode rays, streams of particles issuing from a negative electrode in an evacuated tube, Sir Joseph John Thomson (1856-1940) identified these particles as common to all metals used as cathodes and negatively charged. The historical concept of a positive current issuing from an anode is mathematically self-consistent and leads to no analytical errors, so the convention is maintained but understood to be a convenience.

## Materials

Electrical conduction can take place in a variety of substances. The most familiar conducting substances are metals, in which the outermost electrons of the atoms can move easily in the interatomic spaces. Other conducting materials include semiconductors, electrolytes, and ionized gases, which are discussed later in this article.

## Metals

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.

## Semiconductors

Semiconductors are materials in which the conductivity is much lower than for metals, and widely variable through control of their composition. These substances are now known to be poor insulators rather than poor conductors, in terms of their atomic structure. Though some semiconducting substances had been identified and studied by the latter half of the nineteenth century, their properties could not be explained on the basis of classical physics . It was not until the mid-twentieth century, when modern quantum-mechanical principles were applied to the analysis of both metals and semiconductors, that theoretical calculations of conductivity values agreed with the results of experimental measurements.

In a good insulator, electrons cannot move because nearly all allowed orbital states are occupied. Energy must then be supplied to remove an electron from an outermost bound position to a higher allowed state. This leaves a vacancy into which another bound electron can hop under the influence of an electric field. Thus, both the energized electron and its vacancy become mobile. The vacancy acts like a positive charge, called a hole, and drifts in the direction opposite to electrons. Electrons and holes are more generally termed charge carriers.

In good insulators the activation energy of charge carriers is high, and their availability requires a correspondingly high temperature. In poor insulators, that is, semiconductors, activation occurs at temperatures moderately above 80.6°F (27°C). Each substance has a characteristic value.

There are many more compounds than elements that can be classed as semiconductors. The elements are a few of those in column IV of the periodic table, which have covalent bonds: carbon (C), germanium (Ge), and silicon (Si). For carbon, only the graphite form is semiconducting; diamond is an excellent insulator. The next element down in this column, tin (Sn), undergoes a transition from semiconductor to metal at 59°F (15°C), below room temperature, indicative of an unusefully low activation energy. Other elements that exhibit semiconductor behavior are found in the lower portion of column VI, specifically selenium (Se) and tellurium (Te).

There are two principal groups of compounds with semiconducting properties, named for the periodic table columns of their constituents: III-V, including gallium arsenide (GaAs) and indium antimonide (InSb), among others; and II-VI, including zinc sulfide (ZnS), selenides, tellurides, and some oxides. In many respects these compounds mimic the behavior of column IV elements. Their chemical bonds are mixed covalent and ionic. There are also some organic semiconducting compounds, but their analysis is beyond the scope of this article.

A semiconductor is called intrinsic if its conductivity is the result of equal contributions from its own electrons and holes. The equation must then be expanded:

In an intrinsic semiconductor, ne = nh, and e has the same numerical value for an electron (-) and the hole left behind (+). The mobilities are usually different. These terms add because the opposite charges move in opposite directions, resulting in a pair of like signs in each product.

For application in devices, semiconductors are rarely used in their pure or intrinsic composition. Under carefully controlled conditions, impurities are introduced which contribute either an excess or a deficit of electrons. Excess electrons neutralize holes so that only electrons are available for conduction. The resulting material is called n-type, n for negative carrier. An example of n-type material is Si with Sb, a column IV element with a column V impurity known as a donor. In n-type material, donor atoms remain fixed and positively ionized. When a column III impurity is infused into a column IV element, electrons are bound and holes made available. That material is called p-type, p for positive carrier. Column III impurities are known as acceptors; in the material acceptor atoms remain fixed and negatively ionized. An example of p-type material is Si with Ga. Both n-type and p-type semiconductors are referred to as extrinsic.

Thermal kinetic energy is not the only mechanism for the release of charge carriers in semiconductors. Photons with energy equal to the activation energy can be absorbed by a bound electron which, in an intrinsic semiconductor, adds both itself and a hole as mobile carriers. These photons may be in the visible range or in the near infrared, depending on Eg. In extrinsic semiconductors, photons of much lower energies can contribute to the pool of the prevailing carrier type, provided the material is cooled to cryogenic temperatures in order to reduce the population of thermally activated carriers. This behavior is known as photoconductivity.

Each separate variety of semiconductor is ohmic, with the conductivity constant at constant temperature. However, as the temperature is increased, the conductivity increases very rapidly. The concentration of available carriers varies in accordance with an exponential function:

where Eg is the gap or activation energy, k is Boltzmann's constant (1.38 × 1023 joules/kelvin), T is absolute (kelvin) temperature, and the product kT is the thermal energy corresponding to temperature T. The increase in available charge carriers overrides any decrease in mobility, and this leads to a negative value for a. Indeed, a decrease in resistance with increasing temperature is a reliable indication that a substance is a semiconductor, not a metal. Graphite is an example of a conductor that appears metallic in many ways except for a negative ALPHA. The converse, a positive ALPHA, is not as distinct a test for metallic conductivity.

The Fermi level, Ef, can be shown differently for intrinsic, n-type, and p-type semiconductors. However, for materials physically connected, Ef must be the same for thermal equilibrium. This is a consequence of the laws of thermodynamics and energy conservation. Thus, the behavior of various junctions, in which the interior energy levels shift to accommodate the alignment of the Fermi level, is extremely important for the semiconductor devices.

## Non-ohmic conductors

Non-ohmic conduction is marked by nonlinear graphs of current vs. voltage. It occurs in semiconductor junctions, electrolytic solutions, some ionic solids not in solution , ionized gases, and vacuum tubes. Respective examples include semiconductor p-n diodes, battery acid or alkaline solutions, alkali halide crystals, the ionized mercury vapor in a fluorescent lamp, and cathode ray tubes.

Ionic conductivities are much lower than electronic, because the masses and diameters of ions make them much less mobile. While ions can drift slowly in a gas or liquid, their motions through the interstices of a solid lattice are much more restricted. Yet, with their thermal kinetic energy, ions will diffuse through a lattice, and in the presence of an electric field, will wander toward the appropriate electrode. In most instances, both ionic and electronic conduction will occur, depending on impurities. Thus, for studies of ionic conductivity, the material must be a very pure solid.

In gases, the gas atoms must be ionized by an electric field sufficient to supply the ionization energy of the gas in the tube. For stable currents, the ratio of field to gas pressure , E/P, is a major parameter. Electrons falling back into bound states produce the characteristic spectrum of the gas, qualitatively associated with color , e.g., red for neon, yellow-orange for sodium vapor, or blue-white for mercury vapor.

The basic definition of a plasma in physics includes all material conductors, ohmic and non-ohmic. A plasma is a medium in which approximately equal numbers of opposite charges are present, so that the medium is neutral or nearly so. In a metal the negative electrons are separated from an equal number of positive ion cores. In a semiconductor there may be holes and electrons (intrinsic), holes and ionized acceptors (p-type), or electrons and ionized donors (n-type). In an electrolytic solution and in an ionic solid there are positive and negative ions. An ionized gas contains electrons and positive ions. A small distinction among these may be made as to whether the medium has one or two mobile carriers.

In contemporary usage, the term plasma usually refers to extremely hot gases such as those used in the Tokamak for nuclear fusion experiments. High-energy plasmas are discussed in the article on fusion as a means of generating electric power.

The remaining non-ohmic conduction category is the vacuum tube , in which a beam of electrons is emitted from either a heated cathode (thermionic) or a suitably illuminated cathode (photoelectric), and moves through evacuated space to an anode. The beam in its passage is subjected to electrostatic or magnetic fields for control. The evacuated space cannot be classed either as a material with a definable conductivity or as a plasma, since only electrons are present. However, there are relations of current and voltage to be analyzed. These graphs are generally nonlinear or linear over a limited range. But vacuum tubes are not called ohmic even in their linear ranges because there is no material undergoing the lattice behavior previously described as the basis for ohmic resistance.

Electrical conduction in the human body and other animal organisms is primarily ionic, since body fluids contain vital electrolytes subject to electrochemical action in organs. Further information is available in other articles, particularly those on the heart , the brain , and neurons.

## Resources

### books

Halliday, David, Robert Resnick, and Kenneth Krane. Physics. 4th ed. New York: John Wiley and Sons, 1992.

Serway, Raymond A. Physics for Scientists and Engineers. 3rd ed. Philadelphia: W. B. Saunders Co.

Frieda A. Stahl

## KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lattice

—The structure of atoms in a solid. In a conducting material, ion cores make up the lattice.

Potential difference

—In a conductor carrying an electric current, it is the difference of potential energy per unit charge.

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# Electrical conductivity

Electrical conductivity is the ability of a material to carry the flow of an electric current (a flow of electrons). Imagine that you attach the two ends of a battery to a bar of iron and a galvanometer. (A galvanometer is an instrument for measuring the flow of electric current.) When this connection is made, the galvanometer shows that electric current is flowing through the iron bar. The iron bar can be said to be a conductor of electric current.

Replacing the iron bar in this system with other materials produces different galvanometer readings. Other metals also conduct an electric current, but to different extents. If a bar of silver or aluminum is used, the galvanometer shows a greater flow of electrical current than with the iron bar. Silver and aluminum are better conductors of electricity than is iron. If a lead bar is inserted, the galvanometer shows a lower reading than with iron. Lead is a poorer conductor of electricity than are silver, aluminum, or iron.

Many materials can be substituted for the original iron bar that will produce a zero reading on the galvanometer. These materials do not permit the flow of electric current at all. They are said to be nonconductors, or insulators. Wood, paper, and most plastics are common examples of insulators.

## Electrical Resistance

Another way of describing the conductivity of a material is through resistance. Resistance can be defined as the extent to which a material prevents the flow of electricity. Silver, aluminum, iron and other metals have a low resistance (and a high conductivity). Wood, paper, and most plastics have a high resistance (and a low conductivity).

The unit of measurement for electrical resistance is called the ohm (abbreviation: ). The ohm was named for German physicist Georg Simon Ohm (17891854), who first expressed the mathematical laws of electrical conductance and resistance in detail. Interestingly enough, the unit of electrical conductance is called the mho (ohm written backwards). This choice of units clearly illustrates the reciprocal (opposite) relationship between electrical resistance and conductivity.

## How conductance takes place

Electrical conductivity occurs because of the ease with which electrons can be removed from atoms. All substances consist of atoms. In turn, all atoms consist of two main parts: a positively charged nucleus and one or more negatively charged electrons. An atom of iron, for example, consists of a nucleus with 26 positive charges and 26 negatively charged electrons.

The electrons in an atom are not all held with equal strength. Electrons close to the nucleus are strongly attracted by the positive charge of the nucleus and are removed from the atom only with great difficulty. Electrons farthest from the nucleus are held only loosely and are removed quite easily.

A block of iron can be thought of as a huge collection of iron atoms. Most of the electrons in these atoms are held tightly by the iron nuclei. But a few electrons are held looselyso loosely that they act as if they don't even belong to atoms at all. Scientists sometimes refer to this condition as a cloud of electrons.

Normally these "free" electrons have no place to go. They just spin around randomly among the iron atoms. That situation changes, however, when a battery (or other source of electric current) is attached to the iron block. Electrons flow out of one end of the battery and into the other. At the electron-rich end of the battery, electrons flow into the piece of iron, pushing iron electrons ahead of them. Since all electrons have the same negative charge, they repel each other. Iron electrons are pushed away from the electron-rich end of the battery towards the electron-poor end. In other words, an electric current flows through the iron.

Insulators have a very different structure. They too consist of atoms (nuclei and electrons), but very few free electrons can be found in insulators. Those electrons tend to be bound tightly to nuclei in chemical bonds. Attaching a battery to an insulator has no effect since there are no free electrons to be pushed through the material.

## Solution conductivity

Electrons are not the only particles capable of carrying an electric current. Ions can do it, too. An ion is an atom or group of atoms with an electric charge. Suppose you dissolve a crystal of table salt (sodium chloride) in water. Salt crystals consist of positive sodium ions and negative chloride ions. In the solid state, these ions are not free to move around. Once they are dissolved in water, however, they become completely mobile. They are free to "swim" about in the water and to respond to an electric current from a battery. That current supplies electrons that cause positive sodium ions to flow in one direction and negative chloride ions to flow in the opposite direction.

A good example of this effect can be seen in the conductivity of water. Pure water consists only of water molecules. The electrons in water molecules are held tightly by hydrogen and oxygen atoms and are not free to move. Attaching a battery to a container of water produces no electric current because pure water is an insulator. But a few grains of table salt added to the water changes things completely. Sodium ions and chloride ions are released from the salt, and the salt water solution becomes conductive.

## Semiconductivity and superconductivity

Some materials cannot be classified as either conductors or insulators. Semiconductors, for example, are materials that conduct an electric current but do so very poorly. Semiconductors were not well understood until the mid-twentieth century, when a series of remarkable discoveries revolutionized the field of electrical conductivity. These discoveries have made possible a virtually limitless variety of electronic devices, ranging from miniature radios and handheld calculators to massive solar power arrays and orbiting telescopes.

Superconductivity is a property that appears only at very low temperatures, usually close to absolute zero (273°C). At such temperatures, certain materials lose all resistance to electric current; they become perfect conductors. Once an electric current is initiated in such materials, it continues to flow without diminishing and can go on essentially forever.

The discovery of superconductivity holds enormous potential for the development of electric appliances. In such appliances, a large fraction of the electrical energy supplied to the device is lost in overcoming electrical resistance within the device. That lost energy shows up as waste heat. If the same appliance were made of a superconducting material, no energy would be lost because there would be no resistance to overcome. The appliance would become, at least in principle, 100 percent efficient.