| Resistivity | |
|---|---|
Common symbols | ρ |
| SI unit | ohm metre (Ω⋅m) |
Other units | s (Gaussian/ESU) |
| InSI base units | kg⋅m3⋅s−3⋅A−2 |
Derivations from other quantities | |
| Dimension | |
| Conductivity | |
|---|---|
Common symbols | σ, κ, γ |
| SI unit | siemens per metre (S/m) |
Other units | (Gaussian/ESU) |
Derivations from other quantities | |
| Dimension | |
Electrical resistivity (also calledvolume resistivity orspecific electrical resistance) is a fundamentalspecific property of a material that measures itselectrical resistance or how strongly it resistselectric current. A low resistivity indicates a material that readily allows electric current. Resistivity is commonly represented by theGreek letterρ (rho). TheSI unit of electrical resistivity is theohm-metre (Ω⋅m).[1][2][3] For example, if a1 m3 solid cube of material has sheet contacts on two opposite faces, and theresistance between these contacts is1 Ω, then the resistivity of the material is1 Ω⋅m.
Electrical conductivity (orspecific conductance) is the reciprocal of electrical resistivity. It represents a material's ability to conduct electric current. It is commonly signified by the Greek letterσ (sigma), butκ (kappa) (especially in electrical engineering)[citation needed] andγ (gamma)[citation needed] are sometimes used. The SI unit of electrical conductivity issiemens permetre (S/m). Resistivity and conductivity areintensive properties of materials, giving the opposition of a standard cube of material to current.Electrical resistance and conductance are correspondingextensive properties that give the opposition of a specific object to electric current.
In an ideal case, cross-section and physical composition of the examined material are uniform across the sample, and the electric field and current density are both parallel and constant everywhere. Manyresistors andconductors do in fact have a uniform cross section with a uniform flow of electric current, and are made of a single material, so that this is a good model. (See the adjacent diagram.) When this is the case, the resistance of the conductor is directly proportional to its length and inversely proportional to its cross-sectional area, where the electrical resistivityρ (Greek:rho) is the constant of proportionality. This is written as:
where
The resistivity can be expressed using theSI unitohm metre (Ω⋅m)—i.e. ohms multiplied by square metres (for the cross-sectional area) then divided by metres (for the length).
Bothresistance andresistivity describe how difficult it is to make electrical current flow through a material, but unlike resistance, resistivity is anintrinsic property and does not depend on geometric properties of a material. This means that all pure copper (Cu) wires (which have not been subjected to distortion of their crystalline structure etc.), irrespective of their shape and size, have the sameresistivity, but a long, thin copper wire has a much largerresistance than a thick, short copper wire. Every material has its own characteristic resistivity. For example, rubber has a far larger resistivity than copper.
In ahydraulic analogy, passing current through a high-resistivity material is like pushing water through a pipe full of sand - while passing current through a low-resistivity material is like pushing water through an empty pipe. If the pipes are the same size and shape, the pipe full of sand has higher resistance to flow. Resistance, however, is notsolely determined by the presence or absence of sand. It also depends on the length and width of the pipe: short or wide pipes have lower resistance than narrow or long pipes.
The above equation can be transposed to getPouillet's law (named afterClaude Pouillet):
The resistance of a given element is proportional to the length, but inversely proportional to the cross-sectional area. For example, ifA =1 m2, =1 m (forming a cube with perfectly conductive contacts on opposite faces), then the resistance of this element in ohms is numerically equal to the resistivity of the material it is made of in Ω⋅m.
Conductivity,σ, is the inverse of resistivity:
Conductivity has SI units ofsiemens per metre (S/m).
Conductivity,, is directly proportional to
Where: = electron concentration, = hole concentration, = electron mobility, = hole mobility.
If the geometry is more complicated, or if the resistivity varies from point to point within the material, the current and electric field will be functions of position. Then it is necessary to use a more general expression in which the resistivity at a particular point is defined as the ratio of theelectric field to thedensity of the current it creates at that point:
where
The current density is parallel to the electric field by necessity.
Conductivity is the inverse (reciprocal) of resistivity. Here, it is given by:
For example, rubber is a material with largeρ and smallσ — because even a very large electric field in rubber makes almost no current flow through it. On the other hand, copper is a material with smallρ and largeσ — because even a small electric field pulls a lot of current through it.
This expression simplifies to the formula given above under "ideal case" when the resistivity is constant in the material and the geometry has a uniform cross-section. In this case, the electric field and current density are constant and parallel.
| Derivation of the constant case from the general case |
|---|
| We will combine three equations. Assume the geometry has a uniform cross-section and the resistivity is constant in the material. Then the electric field and current density are constant and parallel, and by the general definition of resistivity, we obtain Since the electric field is constant, it is given by the total voltageV across the conductor divided by the lengthℓ of the conductor: Since the current density is constant, it is equal to the total current divided by the cross sectional area: Plugging in the values ofE andJ into the first expression, we obtain: Finally, we apply Ohm's law,V/I =R: |
When the resistivity of a material has a directional component, the most general definition of resistivity must be used. It starts from the tensor-vector form ofOhm's law, which relates the electric field inside a material to the electric current flow. This equation is completely general, meaning it is valid in all cases, including those mentioned above. However, this definition is the most complicated, so it is only directly used inanisotropic cases, where the more simple definitions cannot be applied. If the material is isotropic, it is safe to ignore the tensor-vector definition, and use a simpler expression instead.
Here,anisotropic means that the material has different properties in different directions. For example, a crystal ofgraphite consists microscopically of a stack of sheets, and current flows very easily through each sheet, but much less easily from one sheet to the adjacent one.[4] In such cases, the current does not flow in exactly the same direction as the electric field. Thus, the appropriate equations are generalized to the three-dimensional tensor form:[5][6]
where the conductivityσ and resistivityρ are rank-2tensors, and electric fieldE and current densityJ are vectors. These tensors can be represented by 3×3 matrices, the vectors with 3×1 matrices, withmatrix multiplication used on the right side of these equations. In matrix form, the resistivity relation is given by:
where
Equivalently, resistivity can be given in the more compactEinstein notation:
In either case, the resulting expression for each electric field component is:
Since the choice of the coordinate system is free, the usual convention is to simplify the expression by choosing anx-axis parallel to the current direction, soJy =Jz = 0. This leaves:
Conductivity is defined similarly:[7]
or
both resulting in:
Looking at the two expressions, and are thematrix inverse of each other. However, in the most general case, the individual matrix elements are not necessarily reciprocals of one another; for example,σxx may not be equal to1/ρxx. This can be seen in theHall effect, where is nonzero. In the Hall effect, due to rotational invariance about thez-axis, and, so the relation between resistivity and conductivity simplifies to:[8]
If the electric field is parallel to the applied current, and are zero. When they are zero, one number,, is enough to describe the electrical resistivity. It is then written as simply, and this reduces to the simpler expression.
Electric current is the ordered movement ofelectrically-charged particles.[2] Specifically, the relation between current density and charged-particle velocity is governed by the equation where is thecurrent density, is thecharge of the carrier, is thedensity of the particles, and theirdrift velocity, a time-averaged measure of their long-term motion.
According to elementaryquantum mechanics, an electron in an atom or crystal can only have certain precise energy levels; energies between these levels are impossible. When a large number of such allowed levels have close-spaced energy values—i.e. have energies that differ only minutely—those close energy levels in combination are called an "energy band". There can be many such energy bands in a material, depending on the atomic number of the constituent atoms[a] and their distribution within the crystal.[b]
The material's electrons seek to minimize the total energy in the material by settling into low energy states; however, thePauli exclusion principle means that only one can exist in each such state. So the electrons "fill up" the band structure starting from the bottom. The characteristic energy level up to which the electrons have filled is called theFermi level. The position of the Fermi level with respect to the band structure is very important for electrical conduction: Only electrons in energy levels near or above theFermi level are free to move within the broader material structure, since the electrons can easily jump among the partially occupied states in that region. In contrast, the low energy states are completely filled with a fixed limit on the number of electrons at all times, and the high energy states are empty of electrons at all times.
Electric current consists of a flow of electrons. In metals there are many electron energy levels near the Fermi level, so there are many electrons available to move. This is what causes the high electronic conductivity of metals.
An important part of band theory is that there may be forbidden bands of energy: energy intervals that contain no energy levels. In insulators and semiconductors, the number of electrons is just the right amount to fill a certain integer number of low energy bands, exactly to the boundary. In this case, the Fermi level falls within a band gap. Since there are no available states near the Fermi level, and the electrons are not freely movable, the electronic conductivity is very low.
Ametal consists of alattice ofatoms, each with an outer shell of electrons that freely dissociate from their parent atoms and travel through the lattice. This is also known as a positive ionic lattice.[9] This 'sea' of dissociable electrons allows the metal to conduct electric current. When an electrical potential difference (avoltage) is applied across the metal, the resulting electric field causes electrons to drift towards the positive terminal. The actualdrift velocity of electrons is typically small, on the order of magnitude of metres per hour. However, due to the sheer number of moving electrons, even a slow drift velocity results in a largecurrent density.[10] The mechanism is similar to transfer of momentum of balls in aNewton's cradle[11] but the rapid propagation of an electric energy along a wire is not due to the mechanical forces, but the propagation of an energy-carrying electromagnetic field guided by the wire.
Most metals have electrical resistance. In simpler models (non quantum mechanical models) this can be explained by replacing electrons and the crystal lattice by a wave-like structure. When the electron wave travels through the lattice, the wavesinterfere, which causes resistance. The more regular the lattice is, the less disturbance happens and thus the less resistance. The amount of resistance is thus mainly caused by two factors. First, it is caused by the temperature and thus amount of vibration of the crystal lattice. Higher temperatures cause bigger vibrations, which act as irregularities in the lattice. Second, the purity of the metal is relevant as a mixture of different ions is also an irregularity.[12][13] The small decrease in conductivity on melting of pure metals is due to the loss of long range crystalline order. The short range order remains and strong correlation between positions of ions results in coherence between waves diffracted by adjacent ions.[14]
In metals, theFermi level lies in theconduction band (see Band Theory, above) giving rise to free conduction electrons. However, insemiconductors the position of the Fermi level is within the band gap, about halfway between the conduction band minimum (the bottom of the first band of unfilled electron energy levels) and the valence band maximum (the top of the band below the conduction band, of filled electron energy levels). That applies for intrinsic (undoped) semiconductors. This means that at absolute zero temperature, there would be no free conduction electrons, and the resistance is infinite. However, the resistance decreases as thecharge carrier density (i.e., without introducing further complications, the density of electrons) in the conduction band increases. In extrinsic (doped) semiconductors,dopant atoms increase the majority charge carrier concentration by donating electrons to the conduction band or producing holes in the valence band. (A "hole" is a position where an electron is missing; such holes can behave in a similar way to electrons.) For both types of donor or acceptor atoms, increasing dopant density reduces resistance. Hence, highly doped semiconductors behave metallically. At very high temperatures, the contribution of thermally generated carriers dominates over the contribution from dopant atoms, and the resistance decreases exponentially with temperature.
Inelectrolytes, electrical conduction happens not by band electrons or holes, but by full atomic species (ions) traveling, each carrying an electrical charge. The resistivity of ionic solutions (electrolytes) varies tremendously with concentration – while distilled water is almost an insulator,salt water is a reasonable electrical conductor. Conduction inionic liquids is also controlled by the movement of ions, but here we are talking about molten salts rather than solvated ions. Inbiological membranes, currents are carried by ionic salts. Small holes in cell membranes, calledion channels, are selective to specific ions and determine the membrane resistance.
The concentration of ions in a liquid (e.g., in an aqueous solution) depends on the degree of dissociation of the dissolved substance, characterized by a dissociation coefficient, which is the ratio of the concentration of ions to the concentration of molecules of the dissolved substance:
The specific electrical conductivity () of a solution is equal to:
where: module of the ion charge, and: mobility of positively and negatively charged ions,: concentration of molecules of the dissolved substance,: the coefficient of dissociation.
The electrical resistivity of a metallic conductor decreases gradually as temperature is lowered. In normal (that is, non-superconducting) conductors, such ascopper orsilver, this decrease is limited by impurities and other defects. Even nearabsolute zero, a real sample of a normal conductor shows some resistance. In a superconductor, the resistance drops abruptly to zero when the material is cooled below its critical temperature. In a normal conductor, the current is driven by a voltage gradient, whereas in a superconductor, there is no voltage gradient and the current is instead related to the phase gradient of the superconducting order parameter.[15] A consequence of this is that an electric current flowing in a loop ofsuperconducting wire can persist indefinitely with no power source.[16]
In a class of superconductors known astype II superconductors, including all knownhigh-temperature superconductors, an extremely low but nonzero resistivity appears at temperatures not too far below the nominal superconducting transition when an electric current is applied in conjunction with a strong magnetic field, which may be caused by the electric current. This is due to the motion ofmagnetic vortices in the electronic superfluid, which dissipates some of the energy carried by the current. The resistance due to this effect is tiny compared with that of non-superconducting materials, but must be taken into account in sensitive experiments. However, as the temperature decreases far enough below the nominal superconducting transition, these vortices can become frozen so that the resistance of the material becomes truly zero.
Plasmas are very good conductors and electric potentials play an important role.
The potential as it exists on average in the space between charged particles, independent of the question of how it can be measured, is called theplasma potential, orspace potential. If an electrode is inserted into a plasma, its potential generally lies considerably below the plasma potential, due to what is termed aDebye sheath. The good electrical conductivity of plasmas makes their electric fields very small. This results in the important concept ofquasineutrality, which says the density of negative charges is approximately equal to the density of positive charges over large volumes of the plasma (ne = ⟨Z⟩ > ni), but on the scale of theDebye length there can be charge imbalance. In the special case thatdouble layers are formed, the charge separation can extend some tens of Debye lengths.
The magnitude of the potentials and electric fields must be determined by means other than simply finding the netcharge density. A common example is to assume that the electrons satisfy theBoltzmann relation:
Differentiating this relation provides a means to calculate the electric field from the density:
(∇ is the vector gradient operator; seenabla symbol andgradient for more information.)
It is possible to produce a plasma that is not quasineutral. An electron beam, for example, has only negative charges. The density of a non-neutral plasma must generally be very low, or it must be very small. Otherwise, the repulsiveelectrostatic force dissipates it.
Inastrophysical plasmas,Debye screening prevents electric fields from directly affecting the plasma over large distances, i.e., greater than theDebye length. However, the existence of charged particles causes the plasma to generate, and be affected by,magnetic fields. This can and does cause extremely complex behavior, such as the generation of plasma double layers, an object that separates charge over a few tens ofDebye lengths. The dynamics of plasmas interacting with external and self-generated magnetic fields are studied in the academic discipline ofmagnetohydrodynamics.
Plasma is often called thefourthstate of matter after solid, liquids and gases.[18][19] It is distinct from these and other lower-energystates of matter. Although it is closely related to the gas phase in that it also has no definite form or volume, it differs in a number of ways, including the following:
| Property | Gas | Plasma |
|---|---|---|
| Electrical conductivity | Very low: air is an excellent insulator until it breaks down into plasma at electric field strengths above 30 kilovolts per centimetre.[20] | Usually very high: for many purposes, the conductivity of a plasma may be treated as infinite. |
| Independently acting species | One: all gas particles behave in a similar way, influenced bygravity and bycollisions with one another. | Two or three:electrons,ions,protons andneutrons can be distinguished by the sign and value of theircharge so that they behave independently in many circumstances, with different bulk velocities and temperatures, allowing phenomena such as new types ofwaves andinstabilities. |
| Velocity distribution | Maxwellian: collisions usually lead to a Maxwellian velocity distribution of all gas particles, with very few relatively fast particles. | Often non-Maxwellian: collisional interactions are often weak in hot plasmas and external forcing can drive the plasma far from local equilibrium and lead to a significant population of unusually fast particles. |
| Interactions | Binary: two-particle collisions are the rule, three-body collisions extremely rare. | Collective: waves, or organized motion of plasma, are very important because the particles can interact at long ranges through the electric and magnetic forces. |
The degree ofsemiconductors doping makes a large difference in conductivity. To a point, more doping leads to higher conductivity. The conductivity of awater/aqueoussolution is highly dependent on itsconcentration of dissolvedsalts and other chemical species thationize in the solution. Electrical conductivity of water samples is used as an indicator of how salt-free, ion-free, or impurity-free the sample is; the purer the water, the lower the conductivity (the higher the resistivity). Conductivity measurements in water are often reported asspecific conductance, relative to the conductivity of pure water at25 °C. AnEC meter is normally used to measure conductivity in a solution. A rough summary is as follows:
| Material | Resistivity,ρ (Ω·m) |
|---|---|
| Superconductors | 0 |
| Metals | 10−8 |
| Semiconductors | Variable |
| Electrolytes | Variable |
| Insulators | 1016 |
| Superinsulators | ∞ |
This table shows the resistivity (ρ), conductivity andtemperature coefficient of various materials at 20 °C (68 °F; 293 K).
| Material | Resistivity,ρ, at20 °C (Ω·m) | Conductivity,σ, at20 °C (S/m) | Temperature coefficient[c] (K−1) | Reference |
|---|---|---|---|---|
| Silver[d] | 1.59×10−8 | 63.0×106 | 3.80×10−3 | [21][22] |
| Copper[e] | 1.68×10−8 | 59.6×106 | 4.04×10−3 | [23][24] |
| Annealedcopper[f] | 1.72×10−8 | 58.0×106 | 3.93×10−3 | [25] |
| Gold[g] | 2.44×10−8 | 41.1×106 | 3.40×10−3 | [21] |
| Aluminium[h] | 2.65×10−8 | 37.7×106 | 3.90×10−3 | [21] |
| Brass (5% Zn) | 3.00×10−8 | 33.4×106 | [26] | |
| Calcium | 3.36×10−8 | 29.8×106 | 4.10×10−3 | |
| Rhodium | 4.33×10−8 | 23.1×106 | ||
| Tungsten | 5.60×10−8 | 17.9×106 | 4.50×10−3 | [21] |
| Zinc | 5.90×10−8 | 16.9×106 | 3.70×10−3 | [27] |
| Brass (30% Zn) | 5.99×10−8 | 16.7×106 | [28] | |
| Cobalt[i] | 6.24×10−8 | 16.0×106 | 7.00×10−3[30] [unreliable source?] | |
| Nickel | 6.99×10−8 | 14.3×106 | 6.00×10−3 | |
| Ruthenium[i] | 7.10×10−8 | 14.1×106 | ||
| Lithium | 9.28×10−8 | 10.8×106 | 6.00×10−3 | |
| Iron | 9.70×10−8 | 10.3×106 | 5.00×10−3 | [21] |
| Platinum | 10.6×10−8 | 9.43×106 | 3.92×10−3 | [21] |
| Tin | 10.9×10−8 | 9.17×106 | 4.50×10−3 | |
| Phosphor Bronze (0.2% P / 5% Sn) | 11.2×10−8 | 8.94×106 | [31] | |
| Gallium | 14.0×10−8 | 7.10×106 | 4.00×10−3 | |
| Niobium | 14.0×10−8 | 7.00×106 | [32] | |
| Carbon steel (1010) | 14.3×10−8 | 6.99×106 | [33] | |
| Lead | 22.0×10−8 | 4.55×106 | 3.90×10−3 | [21] |
| Galinstan | 28.9×10−8 | 3.46×106 | [34] | |
| Titanium | 42.0×10−8 | 2.38×106 | 3.80×10−3 | |
| Grain orientedelectrical steel | 46.0×10−8 | 2.17×106 | [35] | |
| Manganin | 48.2×10−8 | 2.07×106 | 0.002×10−3 | [36] |
| Constantan | 49.0×10−8 | 2.04×106 | 0.008×10−3 | [37] |
| Stainless steel[j] | 69.0×10−8 | 1.45×106 | 0.94×10−3 | [38] |
| Mercury | 98.0×10−8 | 1.02×106 | 0.90×10−3 | [36] |
| Bismuth | 129×10−8 | 7.75×105 | ||
| Manganese | 144×10−8 | 6.94×105 | ||
| Plutonium[39] (0 °C) | 146×10−8 | 6.85×105 | ||
| Nichrome[k] | 110×10−8 | 6.70×105 [citation needed] | 0.40×10−3 | [21] |
| Carbon (graphite) parallel tobasal plane[l] | 250×10−8 to500×10−8 | 2×105 to3×105 [citation needed] | [4] | |
| Carbon (amorphous) | 0.5×10−3 to0.8×10−3 | 1.25×103 to2.00×103 | −0.50×10−3 | [21][40] |
| Carbon (graphite) perpendicular to basal plane | 3.0×10−3 | 3.3×102 | [4] | |
| GaAs | 10−3 to108 [clarification needed] | 10−8 to103 [dubious –discuss] | [41] | |
| Germanium[m] | 4.6×10−1 | 2.17 | −48.0×10−3 | [21][22] |
| Sea water[n] | 2.1×10−1 | 4.8 | [42] | |
| Swimming pool water[o] | 3.3×10−1 to4.0×10−1 | 0.25 to0.30 | [43] | |
| Drinking water[p] | 2×101 to2×103 | 5×10−4 to5×10−2 | [citation needed] | |
| Bone | 1.66×102 | 6×10−3 | [44] | |
| Silicon[m] | 2.3×103 | 4.35×10−4 | −75.0×10−3 | [45][21] |
| Wood (damp) | 103 to104 | 10−4 to10−3 | [46] | |
| Deionized water[q] | 1.8×105 | 4.2×10−5 | [47] | |
| Ultrapure water | 1.82×105 | 5.49×10−6 | [48][49] | |
| Glass | 1011 to1015 | 10−15 to10−11 | [21][22] | |
| Carbon (diamond) | 1012 | ~10−13 | [50] | |
| Hard rubber | 1013 | 10−14 | [21] | |
| Air | 109 to1015 | ~10−15 to10−9 | [51][52] | |
| Wood (oven dry) | 1014 to1016 | 10−16 to10−14 | [46] | |
| Sulfur | 1015 | 10−16 | [21] | |
| Fused quartz | 7.5×1017 | 1.3×10−18 | [21] | |
| PET | 1021 | 10−21 | ||
| PTFE (teflon) | 1023 to1025 | 10−25 to10−23 |
The effective temperature coefficient varies with temperature and purity level of the material. The 20 °C value is only an approximation when used at other temperatures. For example, the coefficient becomes lower at higher temperatures for copper, and the value 0.00427 is commonly specified at0 °C.[53]
The extremely low resistivity (high conductivity) of silver is characteristic of metals.George Gamow tidily summed up the nature of the metals' dealings with electrons in his popular science bookOne, Two, Three...Infinity (1947):
The metallic substances differ from all other materials by the fact that the outer shells of their atoms are bound rather loosely, and often let one of their electrons go free. Thus the interior of a metal is filled up with a large number of unattached electrons that travel aimlessly around like a crowd of displaced persons. When a metal wire is subjected to electric force applied on its opposite ends, these free electrons rush in the direction of the force, thus forming what we call an electric current.
More technically, thefree electron model gives a basic description of electron flow in metals.
Wood is widely regarded as an extremely good insulator, but its resistivity is sensitively dependent on moisture content, with damp wood being a factor of at least1010 worse insulator than oven-dry.[46] In any case, a sufficiently high voltage – such as that in lightning strikes or some high-tension power lines – can lead to insulation breakdown and electrocution risk even with apparently dry wood.[citation needed]
The electrical resistivity of most materials changes with temperature. If the temperatureT does not vary too much, alinear approximation is typically used:
where is called thetemperature coefficient of resistivity, is a fixed reference temperature (usually room temperature), and is the resistivity at temperature. The parameter is an empirical parameter fitted from measurement data, equal to 1/[clarify]. Because the linear approximation is only an approximation, is different for different reference temperatures. For this reason it is usual to specify the temperature that was measured at with a suffix, such as, and the relationship only holds in a range of temperatures around the reference.[54] When the temperature varies over a large temperature range, thelinear approximation is inadequate and a more detailed analysis and understanding should be used.
In general, electrical resistivity of metals increases with temperature. Electron–phonon interactions can play a key role. At high temperatures, the resistance of a metal increases linearly with temperature. As the temperature of a metal is reduced, the temperature dependence of resistivity follows a power law function of temperature. Mathematically the temperature dependence of the resistivityρ of a metal can be approximated through the Bloch–Grüneisen formula:[55]
where is the residual resistivity due to defect scattering, A is a constant that depends on the velocity of electrons at theFermi surface, theDebye radius and thenumber density of electrons in the metal. is theDebye temperature as obtained from resistivity measurements and matches very closely with the values of Debye temperature obtained from specific heat measurements. n is an integer that depends upon the nature of interaction:
The Bloch–Grüneisen formula is an approximation obtained assuming that the studied metal has spherical Fermi surface inscribed within the firstBrillouin zone and aDebye phonon spectrum.[56]
If more than one source of scattering is simultaneously present, Matthiessen's rule (first formulated byAugustus Matthiessen in the 1860s)[57][58] states that the total resistance can be approximated by adding up several different terms, each with the appropriate value of n.
As the temperature of the metal is sufficiently reduced (so as to 'freeze' all the phonons), the resistivity usually reaches a constant value, known as theresidual resistivity. This value depends not only on the type of metal, but on its purity and thermal history. The value of the residual resistivity of a metal is decided by its impurity concentration. Some materials lose all electrical resistivity at sufficiently low temperatures, due to an effect known assuperconductivity.
An investigation of the low-temperature resistivity of metals was the motivation toHeike Kamerlingh Onnes's experiments that led in 1911 to discovery ofsuperconductivity. For details seeHistory of superconductivity.
TheWiedemann–Franz law states that for materials where heat and charge transport is dominated by electrons, the ratio of thermal to electrical conductivity is proportional to the temperature:
where is thethermal conductivity, is theBoltzmann constant, is the electron charge, is temperature, and is theelectric conductivity. The ratio on the rhs is called the Lorenz number.
In general,intrinsic semiconductor resistivity decreases with increasing temperature. The electrons are bumped to theconduction energy band by thermal energy, where they flow freely, and in doing so leave behindholes in thevalence band, which also flow freely. The electric resistance of a typicalintrinsic (non doped)semiconductor decreasesexponentially with temperature following anArrhenius model:
An even better approximation of the temperature dependence of the resistivity of a semiconductor is given by theSteinhart–Hart equation:
whereA,B andC are the so-calledSteinhart–Hart coefficients.
This equation is used to calibratethermistors.
Extrinsic (doped) semiconductors have a far more complicated temperature profile. As temperature increases starting from absolute zero they first decrease steeply in resistance as the carriers leave the donors or acceptors. After most of the donors or acceptors have lost their carriers, the resistance starts to increase again slightly due to the reducing mobility of carriers (much as in a metal). At higher temperatures, they behave like intrinsic semiconductors as the carriers from the donors/acceptors become insignificant compared to the thermally generated carriers.[59]
In non-crystalline semiconductors, conduction can occur by chargesquantum tunnelling from one localised site to another. This is known asvariable range hopping and has the characteristic form of
wheren = 2, 3, 4, depending on the dimensionality of the system.
Kondo insulators are materials where the resistivity follows the formula
where,, and are constant parameters, the residual resistivity, theFermi liquid contribution, a lattice vibrations term and theKondo effect.
When analyzing the response of materials to alternating electric fields (dielectric spectroscopy),[60] in applications such aselectrical impedance tomography,[61] it is convenient to replace resistivity with acomplex quantity calledimpedivity (in analogy toelectrical impedance). Impedivity is the sum of a real component, the resistivity, and an imaginary component, thereactivity (in analogy toreactance). The magnitude of impedivity is the square root of sum of squares of magnitudes of resistivity and reactivity.
Conversely, in such cases the conductivity must be expressed as acomplex number (or even as a matrix of complex numbers, in the case ofanisotropic materials) called theadmittivity. Admittivity is the sum of a real component called the conductivity and an imaginary component called thesusceptivity.
An alternative description of the response to alternating currents uses a real (but frequency-dependent) conductivity, along with a realpermittivity. The larger the conductivity is, the more quickly the alternating-current signal is absorbed by the material (i.e., the moreopaque the material is). For details, seeMathematical descriptions of opacity.
Even if the material's resistivity is known, calculating the resistance of something made from it may, in some cases, be much more complicated than the formula above. One example isspreading resistance profiling, where the material is inhomogeneous (different resistivity in different places), and the exact paths of current flow are not obvious.
In cases like this, the formulas
must be replaced with
whereE andJ are nowvector fields. This equation, along with thecontinuity equation forJ and thePoisson's equation forE, form a set ofpartial differential equations. In special cases, an exact or approximate solution to these equations can be worked out by hand, but for very accurate answers in complex cases, computer methods likefinite element analysis may be required.
In some applications where the weight of an item is very important, the product of resistivity anddensity is more important than absolute low resistivity – it is often possible to make the conductor thicker to make up for a higher resistivity; and then a material with a low resistivity–density product (or equivalently a high conductivity/density ratio) is desirable. For example, for long-distanceoverhead power lines, aluminium is frequently used rather than copper (Cu) because it is lighter for the same conductance.
Silver, although it is the least resistive metal known, has a high density and performs similarly to copper by this measure, but is much more expensive. Calcium and the alkali metals have the best resistivity-density products, but are rarely used for conductors due to their high reactivity with water and oxygen, and lack of physical strength. Aluminium is far more stable. Toxicity excludes the choice of beryllium;[62] pure beryllium is also brittle. Thus, aluminium is usually the metal of choice when the weight or cost of a conductor is the driving consideration.
| Material | Resistivity | Density | Resistivity × density | |||
|---|---|---|---|---|---|---|
| (nΩ·m) | Relative toCu | (g/cm3) | Relative toCu | (g·mΩ/m2) | Relative toCu | |
| Sodium | 47.7 | 2.843 | 0.97 | 0.108 | 46 | 0.31 |
| Lithium | 92.8 | 5.53 | 0.53 | 0.059 | 49 | 0.33 |
| Calcium | 33.6 | 2.002 | 1.55 | 0.173 | 52 | 0.35 |
| Potassium | 72.0 | 4.291 | 0.89 | 0.099 | 64 | 0.43 |
| Beryllium | 35.6 | 2.122 | 1.85 | 0.206 | 66 | 0.44 |
| Aluminium | 26.50 | 1.579 | 2.70 | 0.301 | 72 | 0.48 |
| Magnesium | 43.90 | 2.616 | 1.74 | 0.194 | 76 | 0.51 |
| Copper | 16.78 | 1 | 8.96 | 1 | 150 | 1 |
| Silver | 15.87 | 0.946 | 10.49 | 1.171 | 166 | 1.11 |
| Gold | 22.14 | 1.319 | 19.30 | 2.154 | 427 | 2.84 |
| Iron | 96.1 | 5.727 | 7.874 | 0.879 | 757 | 5.03 |
In a 1774 letter to Dutch-born British scientistJan Ingenhousz,Benjamin Franklin relates an experiment by another British scientist,John Walsh, that purportedly showed this astonishing fact: Although rarified air conducts electricity better than common air, a vacuum does not conduct electricity at all.[63]
Mr. Walsh ... has just made a curious Discovery in Electricity. You know we find that in rarify’d Air it would pass more freely, and leap thro’ greater Spaces than in dense Air; and thence it was concluded that in a perfect Vacuum it would pass any distance without the least Obstruction. But having made a perfect Vacuum by means of boil’d Mercury in a long Torricellian bent Tube, its Ends immers’d in Cups full of Mercury, he finds that the Vacuum will not conduct at all, but resists the Passage of the Electric Fluid absolutely.
However, to this statement a note (based on modern knowledge) was added by the editors—at the American Philosophical Society and Yale University—of the webpage hosting the letter:[63]
We can only assume that something was wrong with Walsh’s findings. ... Although the conductivity of a gas, as it approaches a vacuum, increases up to a point and then decreases, that point is far beyond what the technique described might have been expected to reach. Boiling replaced the air with mercury vapor, which as it cooled created a vacuum that could scarcely have been complete enough to decrease, let alone eliminate, the vapor’s conductivity.
Barlow's law was published in 1825.Ohm's law was published in 1827.Both were empirical fits to measured data.
The first scientific explanation of why electrons behave in this way is theDrude model, first proposed in 1900.
When electrons are conducted through a metal, they interact with imperfections in the lattice and scatter. […] Thermal energy produces scattering by causing atoms to vibrate. This is the source of resistance of metals.
{{cite book}}: CS1 maint: others (link)