# Electric conductivity

Physical size
Surname electric conductivity
Formula symbol ${\ displaystyle \ sigma}$, ,${\ displaystyle \ gamma}$${\ displaystyle \ kappa}$
Size and
unit system
unit dimension
SI S · m -1 = ( Ω · m ) -1 M -1 · L -3 · T 3 · I 2
Gauss ( cgs ) s −1 T −1
esE ( cgs ) s −1 T −1
emE ( cgs ) cm -2 · s L -2 · T

The electrical conductivity , also as conductivity or EC value (from the English electrical conductivity hereinafter) is a physical quantity that indicates how much the capability of a substance is, the electric current conducting.

The symbol for electrical conductivity is ( Greek sigma ), also (gamma), in electrochemistry (kappa). The derived SI unit of electrical conductivity is S / m ( Siemens per meter). The reciprocal of the electrical conductivity is called the specific resistance . ${\ displaystyle \ sigma}$ ${\ displaystyle \ gamma}$${\ displaystyle \ kappa}$

The electrical conductivity is defined as the proportionality constant between the current density and the electrical field strength : ${\ displaystyle {\ vec {J}}}$ ${\ displaystyle {\ vec {E}}}$

${\ displaystyle {\ vec {J}} = \ sigma \; {\ vec {E}}}$

In the special case of constant electrical conductivity, this definition equation corresponds to Ohm's law .

## Conductivity as a tensor

In the special case of an isotropic (direction-independent) and linear (field size-independent) medium, the electrical conductivity is a scalar (one-dimensional value). Only in this simple but common case is the current conduction proportional and in the same direction as the electric field causing the current density.

In an anisotropic and linear material, the electrical conductivity is a 2nd level tensor ( dyad ), i.e. a multi-dimensional value. Examples of materials with such properties are materials with structures such as graphite , crystals and high-temperature superconductors .

## Connections and units

It should be noted that the above equation - it is one of the three fundamental material equations  - can not be derived from Maxwell's equations . Maxwell's equations with the laws of continuity and the material equations represent the foundation of the non-relativistic electrodynamic field theory.

The conductance as the reciprocal of the resistance is a property of a body. Conductivity as the reciprocal of the specific resistance is a property of a material. and are linked to one another via a factor that results from the geometric structure of the body. ${\ displaystyle G}$${\ displaystyle \ sigma}$${\ displaystyle G}$${\ displaystyle \ sigma}$

Note: The basic standards such as DIN 1304, DIN EN 80000-6, IEC 60050 or IEV use the term “conductivity” or “electrical conductivity”, but there is no addition “specific” in connection with conductivity. The dependence on the respective material is already in the definition of the term.

The derived SI unit of electrical conductivity is S / m ( Siemens per meter). S / cm, m / (Ω · mm 2 ) and S · m / mm 2 are also common , with the relationships 1 S / cm = 100 S / m and 1 m / (Ω · mm 2 ) = 1 S · m / mm 2  = 10 6  S / m apply.

Another particularly common in the US unit IACS , for English International Annealed Copper Standard . Here the conductivity is expressed in relation to the conductivity in pure annealed copper : 100% IACS = 58 · 10 6  S / m.

## Electrical conductivity of various substances

Electrical conductivity of selected materials at 20 to 25  ° C. The data depends partly on the degree
of purity
material classification σ in S / m source
Graph Non-metal 100e6th
silver metal 61e6th
copper metal 58e6th
gold metal 45e6th
aluminum metal 37e6th
tungsten metal 19the6th
iron metal 10e6th
Steel C35 metal 8th.6the6th
Graphite (parallel to layers) Non-metal 3e6th
Graphite (across layers) Non-metal 3e2
Stainless steel WNr. 1.4301 metal 1.4the6th
mercury metal 1.0e6th
manganese metal 0.69e6th
Germanium (foreign content <10 −9 ) semiconductor 2e0
Silicon (foreign content <10 −12 ) semiconductor 0.5e-3
Conductive polymers polymer 10 −11 to 10 5
Polytetrafluoroethylene ("Teflon") polymer <10 −16
Sea water electrolyte 5e0
tap water electrolyte 5… 50e-3
Ultrapure water electrolyte 5e-6th

The electrical conductivity is preferably obtained without changing the substance due to the transport of electrons. Such substances are divided into

Below a material- dependent transition temperature , the electrical resistance drops to zero and the conductivity becomes infinite.
Typical (at 25 ° C):> 10 6  S / m.
Manganese has the lowest electrical conductivity of all pure metals, silver has the highest, which conducts 88 times better.
In the case of semiconductors, the conductivity depends to a great extent on the degree of purity, and more on the temperature and pressure than in the case of metals. Intrinsic conductivity can be specified as a fairly reproducible material property. The conductivity of semiconductors lies between that of conductors and non-conductors. This classification dates back to times when the possibility of changing their conductivity extremely through the targeted storage of foreign atoms ( doping ) was not yet known (factor 10 6 ). For this purpose, our own semiconductor technology has been developed.
Typically <10 −8  S / m.
The conductivity of good insulators is approx. 10 −16  S / m.

In addition, there is an ionic conduction associated with material transport in electrolytes .

## Cause of conductivity

The conductivity of a substance or a mixture of substances depends on the availability and density of mobile charge carriers. These can be loosely bound electrons such as in metals, but also ions or delocalized electrons in organic molecules, as they are often described by mesomeric boundary structures. Substances with many freely moving charge carriers are therefore conductive.

In reality, every material has a certain, albeit sometimes very low, conductivity. Even all non-conductors and electrical insulating materials or insulators cannot completely prevent the flow of current. However, the currents are so small that they can often be neglected.

All non-conductors or insulators can conduct (higher or high) electrical currents when a sufficiently high voltage is applied or through strong heating, but the structure of the non-conductor is mostly destroyed (it disintegrates or melts), especially if it was a solid.

For example, diamond and glass become conductive when red heat (approx. 1000 K).

### Examples

#### Metals

Metals are electronic conductors . Their electrons in the conduction band are mobile and transport the electric current very well.

#### Ionic conduction

Ultrapure water has a certain conductivity ( ionic conductivity , approx. 1:10 13 times less than metals, but still approx. 1000 times more conductive than an insulating material). If salts , acids or bases are added to the water , which release free-moving ions in an aqueous solution , the conductivity increases (even tap water has a conductivity around 4 powers of ten ).

Fires in low-voltage systems up to 1000 V can be extinguished largely without problems with water; In high-voltage systems (e.g. switchgears ), fires should not be extinguished with water, in order not to expose the fire-fighting personnel to the risk of electric shock . According to DIN VDE 0132, wet extinguishers (extinguishing agent water) can be used in low-voltage systems from a distance of at least 1 m (spray jet) or 3 m (full jet).

#### Doping (electrons, holes)

With doping , the conductivity of semiconductors can be strongly influenced (by many powers of ten). If the ( extremely pure ) basic material is mixed with electron donors (elements with more external electrons than the basic material), one speaks of n-doping (negatively charged, quasi-free charge carriers in excess compared to the positively charged), with the addition of electron acceptors (elements with fewer Electrons as the basic material) against it from p-doping (positively charged quasi-free charge carriers in excess compared to the negatively charged). The p-doping creates electron defects , also known as holes or defect electrons, which also enable the conduction of electrical current, as do the surplus electrons in the case of n-doped semiconductors. The conductivity arises from the fact that the holes or electrons are mobile - although not as mobile as the electrons in metals.

Semiconductor components such as diodes and transistors are based on the effects at the boundary points of differently doped areas , in which the conductivity depends, for example, on the amount and direction of the electric field strength.

A model for illustrating or explaining the conductivity of a crystal is given by the ribbon model .

Since the thermal conductivity in metallic solids is mainly determined by the electrons, electrical and thermal conductivity are linked by the Wiedemann-Franz law .

## Cause of the electrical resistance

In 1900 Paul Drude formulated a model named after him , according to which the electrical resistance is caused by the collision of the conduction electrons with the atomic cores of the metal, which are assumed to be rigid . After that is the conductivity

${\ displaystyle \ sigma = {\ frac {ne ^ {2} \ tau} {m}}}$.

Here is the concentration of free electrons, the charge, the mass of an electron and the mean flight time of the electron between two collisions ( relaxation time ). This model illustrates the electrical conductivity quite well, but predicts some experimental results incorrectly, since the assumption of the free electron gas is too imprecise: electrons are fermions , that is, each energy state in reciprocal k-space can only be occupied by two electrons, so that even at absolute zero energy levels up to the Fermi energy are occupied and form the Fermi sphere . The temperature-dependent probability of whether an energy level is occupied by electrons is determined by the Fermi-Dirac distribution${\ displaystyle n}$${\ displaystyle e}$${\ displaystyle m}$${\ displaystyle \ tau}$ ${\ displaystyle E (k) = E (p) (\ approx E (v))}$ ${\ displaystyle E _ {\ text {F}}}$${\ displaystyle E (k)}$

${\ displaystyle f_ {0} (k, T) = {\ frac {1} {\ mathrm {e} ^ {\ frac {E (k) -E _ {\ text {F}}} {k _ {\ text { B}} T}} + 1}}}$

specified. Since the Fermi energy with a few electron volts is much larger than the thermal energy with a few dozen milli-electron volts, only electrons close to the Fermi energy are excited and contribute to the electrical conductivity. In the non-equilibrium state, the time dependence of the distribution is described by the Boltzmann equation . With this improvement, the Sommerfeld theory , the same conductivity follows as according to Drude, but with two decisive changes: ${\ displaystyle E _ {\ text {F}}}$${\ displaystyle k _ {\ text {B}} T}$

• The relaxation time is the relaxation time of the electrons at the Fermi edge , i.e. that of the electrons with the energy .${\ displaystyle \ tau}$${\ displaystyle E _ {\ text {F}}}$
• The mass of the electrons in the crystal apparently has a different, effective mass , which is direction-dependent and therefore also a tensor quantity.${\ displaystyle m}$ ${\ displaystyle m ^ {*}}$

The reciprocal of the relaxation time, the scatter rate (number of scatterings per time), is the sum of the individual scatter rates of the electrons on vibrations of the atomic cores (the phonons ), on other electrons, on lattice defects (foreign atoms, defects, etc.) in the crystal or else the walls of the crystal. This results in a generalization of Matthiessen's rule :

${\ displaystyle {\ frac {1} {\ tau}} = {\ frac {1} {\ tau _ {\ text {Phonon}}}} + {\ frac {1} {\ tau _ {\ text {electron }}}} + {\ frac {1} {\ tau _ {\ text {defects}}}} + \ dotsb \ propto \ rho = {\ frac {1} {\ sigma}}}$

The individual relaxation times lead to the different temperature dependencies of the conductivity in the metal. So is z. B. the scattering at faults temperature-independent and leads to the residual resistance , whereas the electron-phonon scattering at room temperature is proportional to the temperature.

If one takes into account the mobility of the charge carriers in a general solid , the result is: ${\ displaystyle \ mu = e \ tau / m}$

${\ displaystyle \ sigma = en \ mu \,}$

where expresses the charge carrier density (number per volume). ${\ displaystyle n}$

If you expand this expression further, you get:

${\ displaystyle \, \ sigma = e (n \ mu _ {n} + p \ mu _ {p})}$

Thereby the electron density and its mobility as well as the defect electron density and its mobility . ${\ displaystyle n}$${\ displaystyle \ mu _ {n}}$${\ displaystyle p}$${\ displaystyle \ mu _ {p}}$

## Measurement

The electrical conductivity cannot be measured directly, but is usually determined by means of transport measurements from current strength , voltage drop and sample geometry analogous to the specific resistance . Different methods can be used depending on the sample geometry.

In liquids z. B. For simple measurements electrodes of known area and distance are used and the voltage and current are measured, see conductivity meter . The formula for this is: ${\ displaystyle A}$${\ displaystyle l}$${\ displaystyle U}$${\ displaystyle I}$

${\ displaystyle \ sigma = {\ frac {I \ cdot l} {U \ cdot A}}}$

In the case of a good conductor with a known cross-section, preferably extended in one dimension (such as a wire), the conductivity is determined by means of four-wire measurement , the current through the conductor and the voltage drop between two measuring contacts located at a distance . The current is fed in beyond these measuring contacts in order to avoid measuring errors. ${\ displaystyle A}$${\ displaystyle I}$${\ displaystyle U}$${\ displaystyle l}$

A method for measuring the specific sheet resistance of a large, homogeneous layer is the four-point method and is mainly used in the semiconductor industry . If, on the other hand, the layer is small and has any shape, the conductivity can be determined using the Van der Pauw measurement method .

The first conductivity measuring devices , also known as conductometers, go back to the work of Jean-Jacques Rousseau and the historical measuring device diagometer .

## Temperature dependence

The electrical conductivity depends on the temperature. The course of this temperature dependency depends on the structure and type of material or on the dominant mechanisms for the transport of electrical charges.

The temperature curve is often linear only within small temperature changes or even shows sudden changes (for example during phase transitions such as melting or when the transition temperature is reached in superconductors ).

In metals, the conductivity decreases with increasing temperature due to increasing lattice vibrations which hinder the flow of electrons. They have a positive temperature coefficient of electrical resistance. An electric incandescent lamp has a much higher conductivity when de-energized than when it is in operation. At the moment of switching on, a high inrush current flows (up to ten times greater than the operating current). If the filament is heated, the current drops to the nominal value. A rule of thumb is that for every degree of temperature increase, the resistance increases by 0.5% of its value. Incandescent lamps can therefore be used to limit the current or as a thermal fuse, e.g. B. to protect tweeters in loudspeaker boxes. Small incandescent lamps were also used to control the gain or amplitude in Wien bridge sine wave generators .

In semiconductors, mobility also decreases due to the lattice vibrations, but the charge carrier density can also change. In the area of ​​the impurity reserve and intrinsic conduction , it increases disproportionately (more precisely: exponentially) when electrons are excited into the conduction band . In contrast, the charge carrier density remains approximately constant in the area of ​​the impurity conduction . The conductivity can therefore increase or decrease slightly with the temperature and thus also depends on the doping.

A practical application of temperature dependency in semiconductors is temperature measurement with the aid of a current-carrying diode - its forward voltage decreases strictly linearly with increasing temperature. For temperature measurement and for inrush current limitation are NTC thermistors used, the conductivity with temperature greatly rises. With PTC thermistors , the resistance increases when heated; they are used, for example, as thermal or self-resetting fuse .

In superconductors , the resistance drops to zero below the critical temperature, i.e. it disappears. When the transition temperature is exceeded, the resistance reappears just as suddenly, which can lead to destruction by quenching of current-carrying coils made of superconductors, i.e. massive overheating of the affected area.

In gases, solutions and electrolytes , the resistance is strongly temperature-dependent, since there the mobility and the number of ions increase sharply with increasing temperature (with weak electrolytes, the degree of dissociation is strongly temperature-dependent). As a rule, the charge carrier mobility increases with temperature and the conductivity increases.

## literature

• Neil W. Ashcroft, N. David Mermin: Solid State Physics . Saunders College Publishing, New York 1976, ISBN 0-03-083993-9 .

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6. Typically approx. 56 · 10 6  S / m applies to copper cables (not pure copper), see specific resistance .
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