Electrical resistivity and conductivity
1
Electrical resistivity and conductivity
Electrical resistivity (also known as resistivity, specific electrical resistance, or volume resistivity) is a measure
of how strongly a material opposes the flow of electric current. A low resistivity indicates a material that readily
allows the movement of electric charge. The SI unit of electrical resistivity is the ohm⋅metre (Ω⋅m). It is commonly
represented by the Greek letter ρ (rho).
Electrical conductivity or specific conductance is the reciprocal quantity, and measures a material's ability to
conduct an electric current. It is commonly represented by the Greek letter σ (sigma), but κ (kappa) (especially in
electrical engineering) or γ gamma are also occasionally used. Its SI unit is siemens per metre (S⋅m−1) and CGSE
unit is reciprocal second (s−1).
Definitions
Scalar form
Electrical resistivity ρ (Greek: rho) is defined by,
where
ρ is the static resistivity of the conductor material (measured in
ohm⋅metres, Ω⋅m),
E is the magnitude of the electric field (in volts per metre,
V⋅m−1),
J is the magnitude of the current density (in amperes per square
metre, A⋅m−2).
in which E and J are inside the conductor.
A piece of resistive material with electrical
contacts on both ends.
Conductiviy is the inverse of resistivity:
which gives an equivalent equation
Many resistors and conductors have a uniform cross section with a uniform flow of electric current and are made of
one material. (See the diagram to the right.) In this case, the above definition of ρ leads to:
where
R is the electrical resistance of a uniform specimen of the material (measured in ohms, Ω)
is the length of the piece of material (measured in metres, m)
A is the cross-sectional area of the specimen (measured in square metres, m²).
The reason resistivity has the dimension units of ohm⋅metres can be seen by transposing the definition to make
resistance the subject (Pouillet's law):
Electrical resistivity and conductivity
The resistance of a given sample will increase with the length, but decrease with greater cross-sectional area.
Resistance is measured in ohms. Length over area has units of 1/distance. To end up with ohms, resistivity must be
in the units of "ohms×distance" (SI ohm⋅metre, US ohm⋅inch).
In a hydraulic analogy, increasing the cross-sectional area of a pipe reduces its resistance to flow, and increasing the
length increases resistance to flow (and pressure drop for a given flow).
Tensor generalization
The equivalent scalar equations
can be generalized to the 3d vector form[1][2]
which is also known as Ohm's law. However this equation is only valid for materials which are homogeneous
(conduction properties are uniform throughout the conductor) and isotropic (properties are independent of
orientation).
For an inhomogeneous material, E, J, ρ and σ vary as a function of position r throughout the conductor. However, if
the material is isotropic, then E and J are collinear. So at all points:
For anisotropic materials, either homogeneous or inhomogeneous, E and J are not always in the same direction, so
the conductivity σ and resistivity ρ become tensors. For homogeneous materials, there is no r dependence, but for the
inhomogeneous materials, E and J are first-order tensor fields (vector fields) and ρ and σ are second-order tensor
fields, all defined at every point in the conductor. The equations are compactly illustrated in component form (using
index notation and the summation convention) [3]:
which can be interpreted
• physically as the conductivity properties (ρ and σ tensor fields) determine how the E-field responds to the J-field
and vice versa,
• mathematically as a vector-matrix equation (order-2 tensors have a matrix representation), i.e. the ρ and σ tensors
re-scale the components of the E and J vector fields.
Analogous to scalar definition above:
where δij is the Kronecker delta (identity matrix), i.e. ρ and σ are still reciprocals of each other. (The sum is over j,
not i).
2
Electrical resistivity and conductivity
3
Causes of resistivity
Band theory simplified
Quantum mechanics states that the energy of an electron in an atom
cannot be any arbitrary value. Rather, there are fixed energy levels which
the electrons can occupy, and values in between these levels are
impossible. When a large number of such allowed energy levels are
spaced close together (in energy-space) i.e. have similar (minutely
differing energies) then we can talk about these energy levels together as
an "energy band".There can be many such energy bands in a material,
depending on the atomic number (number of electrons) and their
distribution (besides external factors like environment modifying the
energy bands).Two such bands important in the discussion of
conductivity of materials are: the valence band and the conduction band
(the latter is generally above the former). Electrons in the conduction
band may move freely throughout the material in the presence of an
electrical field.
Electron energy levels in an insulator
In insulators and semiconductors, the atoms in the substance influence each other so that between the valence band
and the conduction band there exists a forbidden band of energy levels, which the electrons cannot occupy. In order
for a current to flow, a relatively large amount of energy must be furnished to an electron for it to leap across this
forbidden gap and into the conduction band. Thus, even large voltages can yield relatively small currents.
In metals
A metal consists of a lattice of atoms, each with a shell of electrons. This is also known as a positive ionic lattice.
The outer electrons are free to dissociate from their parent atoms and travel through the lattice, creating a 'sea' of
electrons, making the metal a conductor. When an electrical potential difference (a voltage) is applied across the
metal, the electrons drift from one end of the conductor to the other under the influence of the electric field.
Near room temperatures, the thermal motion of ions is the primary source of scattering of electrons (due to
destructive interference of free electron waves on non-correlating potentials of ions), and is thus the prime cause of
metal resistance. Imperfections of lattice also contribute into resistance, although their contribution in pure metals is
negligible.
The larger the cross-sectional area of the conductor, the more electrons are available to carry the current, so the
lower the resistance. The longer the conductor, the more scattering events occur in each electron's path through the
material, so the higher the resistance. Different materials also affect the resistance.[4]
In semiconductors and insulators
In metals, the Fermi level lies in the conduction band (see Band Theory, below) giving rise to free conduction
electrons. However, in semiconductors the position of the Fermi level is within the band gap, approximately
half-way between the conduction band minimum and valence band maximum for intrinsic (undoped)
semiconductors. This means that at 0 kelvins, there are no free conduction electrons and the resistance is infinite.
However, the resistance will continue to decrease as the charge carrier density 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 accepting holes in the valence band. For both types of donor or acceptor atoms,
increasing the dopant density leads to a reduction in the resistance. Highly doped semiconductors hence behave
metallic. At very high temperatures, the contribution of thermally generated carriers will dominate over the
Electrical resistivity and conductivity
4
contribution from dopant atoms and the resistance will decrease exponentially with temperature.
In ionic liquids/electrolytes
In electrolytes, 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 liquids varies tremendously by the concentration
- while distilled water is almost an insulator, salt water is a very efficient electrical conductor. In biological
membranes, currents are carried by ionic salts. Small holes in the membranes, called ion channels, are selective to
specific ions and determine the membrane resistance.
Resistivity of various materials
• A conductor such as a metal has high conductivity and a low resistivity.
• An insulator like glass has low conductivity and a high resistivity.
• The conductivity of a semiconductor is generally intermediate, but varies widely under different conditions, such
as exposure of the material to electric fields or specific frequencies of light, and, most important, with temperature
and composition of the semiconductor material.
The degree of doping in semiconductors makes a large difference in conductivity. To a point, more doping leads to
higher conductivity. The conductivity of a solution of water is highly dependent on its concentration of dissolved
salts, and other chemical species that ionize 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 as specific conductance, relative
to the conductivity of pure water at 25 °C. An EC meter is normally used to measure conductivity in a solution. A
rough summary is as follows:
Material
Resistivity
ρ (Ω•m)
Metals
10−8
Semiconductors
variable
Electrolytes
variable
Insulators
1016
Superconductors 0
This table shows the resistivity, conductivity and temperature coefficient of various materials at 20 °C (68 °F)
Material
ρ (Ω•m) at 20 °C
σ (S/m) at 20 °C
Temperature Reference
[5]
coefficient
(K−1)
Silver
1.59×10−8
6.30×107
0.0038
[6][7]
Copper
1.68×10−8
5.96×107
0.0039
[7]
1.72×10−8
5.80×107
2.44×10−8
4.10×107
0.0034
[6]
Aluminium
2.82×10−8
3.5×107
0.0039
[6]
Calcium
3.36×10−8
2.98×107
0.0041
Tungsten
5.60×10−8
1.79×107
0.0045
Annealed copper
[9]
Gold
[10]
[8]
[6]
Electrical resistivity and conductivity
5
Zinc
5.90×10−8
1.69×107
0.0037
Nickel
6.99×10−8
1.43×107
0.006
Lithium
9.28×10−8
1.08×107
0.006
Iron
1.0×10−7
1.00×107
0.005
[6]
Platinum
1.06×10−7
9.43×106
0.00392
[6]
Tin
1.09×10−7
9.17×106
0.0045
Carbon steel (1010)
1.43×10−7
6.99×106
Lead
2.2×10−7
4.55×106
0.0039
Titanium
4.20×10−7
2.38×106
X
Grain oriented electrical steel 4.60×10−7
2.17×106
Manganin
4.82×10−7
2.07×106
0.000002
[14]
Constantan
4.9×10−7
2.04×106
0.000008
[15]
Stainless steel
6.9×10−7
1.45×106
Mercury
9.8×10−7
1.02×106
0.0009
[14]
Nichrome
1.10×10−6
9.09×105
0.0004
[6]
Carbon (amorphous)
5×10−4 to 8×10−4
1.25 to 2×103
−0.0005
[6][19]
[20]
2.5e×10−6 to 5.0×10−6 ⊥basal
plane
3.0×10−3 //basal plane
2 to 3×105 ⊥basal
plane
3.3×102 //basal plane
[21]
[22]
1×1012
~10−13
[23]
4.6×10−1
2.17
2×10−1
4.8
2×101 to 2×103
5×10−4 to 5×10−2
1.8×105
5.5×10−6
Silicon
6.40×102
1.56×10−3
GaAs
5×10−7 to 10×10−3
5×10−8 to 103
Glass
10×1010 to 10×1014
10−11 to 10−15
?
[6][7]
Hard rubber
1×1013
10−14
?
[6]
Sulfur
1×1015
10−16
?
[6]
Air
1.3×1016 to 3.3×1016
3 to 8×10−15
Paraffin
1×1017
10−18
?
Fused quartz
7.5×1017
1.3×10−18
?
PET
10×1020
10−21
?
[16]
[18]
Carbon (graphite)
Carbon (diamond)
Germanium
[22]
[24]
Sea water
Drinking water
[26]
Deionized water
[22]
[27]
[11]
[12]
[6]
[13]
[17]
−0.048
[6][7]
[25]
[28]
−0.075
[6]
[29]
[30]
[6]
Electrical resistivity and conductivity
Teflon
6
10×1022 to 10×1024
10−25 to 10−23
?
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 at 0 °C.[31]
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 science-popularizing book, One, 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, the free electron model gives a basic description of
electron flow in metals.
Temperature dependence
Linear approximation
The electrical resistivity of most materials changes with temperature. If the temperature T does not vary too much, a
linear approximation is typically used:
where
is called the temperature coefficient of resistivity,
temperature), and
is the resistivity at temperature
is a fixed reference temperature (usually room
. The parameter
is an empirical parameter fitted from
measurement data. 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.[32] When the temperature
varies over a large temperature range, the linear approximation is inadequate and a more detailed analysis and
understanding should be used.
Metals
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 resistivity ρ of a metal is given by the Bloch–Grüneisen formula:
where
is the residual resistivity due to defect scattering, A is a constant that depends on the velocity of
electrons at the Fermi surface, the Debye radius and the number density of electrons in the metal.
is the Debye
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:
1. n=5 implies that the resistance is due to scattering of electrons by phonons (as it is for simple metals)
2. n=3 implies that the resistance is due to s-d electron scattering (as is the case for transition metals)
3. n=2 implies that the resistance is due to electron–electron interaction.
If more than one source of scattering is simultaneously present, Matthiessen's Rule (first formulated by Augustus
Matthiessen in the 1860s) [33][34] says that the total resistance can be approximated by adding up several different
terms, each with the appropriate value of n.
Electrical resistivity and conductivity
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 the residual 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 as
superconductivity.
An investigation of the low-temperature resistivity of metals was the motivation to Heike Kamerlingh Onnes's
experiments that led in 1911 to discovery of superconductivity. For details see History of superconductivity.
Semiconductors
In general, resistivity of intrinsic semiconductors decreases with increasing temperature. The electrons are bumped
to the conduction energy band by thermal energy, where they flow freely and in doing so leave behind holes in the
valence band which also flow freely. The electric resistance of a typical intrinsic (non doped) semiconductor
decreases exponentially with the temperature:
An even better approximation of the temperature dependence of the resistivity of a semiconductor is given by the
Steinhart–Hart equation:
where A, B and C are the so-called Steinhart–Hart coefficients.
This equation is used to calibrate thermistors.
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 it will behave like intrinsic semiconductors as the
carriers from the donors/acceptors become insignificant compared to the thermally generated carriers.[35]
In non-crystalline semiconductors, conduction can occur by charges quantum tunnelling from one localised site to
another. This is known as variable range hopping and has the characteristic form of
,
where n = 2, 3, 4, depending on the dimensionality of the system.
Complex resistivity and conductivity
When analyzing the response of materials to alternating electric fields, in applications such as electrical impedance
tomography,[36] it is necessary to replace resistivity with a complex quantity called impeditivity (in analogy to
electrical impedance). Impeditivity is the sum of a real component, the resistivity, and an imaginary component, the
reactivity (in analogy to reactance). The magnitude of Impeditivity is the square root of sum of squares of
magnitudes of resistivity and reactivity.
Conversely, in such cases the conductivity must be expressed as a complex number (or even as a matrix of complex
numbers, in the case of anisotropic materials) called the admittivity. Admittivity is the sum of a real component
called the conductivity and an imaginary component called the susceptivity.
An alternative description of the response to alternating currents uses a real (but frequency-dependent) conductivity,
along with a real permittivity. The larger the conductivity is, the more quickly the alternating-current signal is
absorbed by the material (i.e., the more opaque the material is). For details, see Mathematical descriptions of opacity.
7
Electrical resistivity and conductivity
8
Resistivity density products
In some applications where the weight of an item is very important resistivity density products are 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 low resistivity density product material (or equivalently a high conductance to density ratio) is desirable.
For example, for long distance overhead power lines— aluminium is frequently used rather than copper because it is
lighter for the same conductance.
Material
Resistivity (nΩ•m) Density (g/cm3)
Resistivity-density
product (nΩ•m•g/cm3)
Sodium
47.7
0.97
46
Lithium
92.8
0.53
49
Calcium
33.6
1.55
52
Potassium
72.0
0.89
64
Beryllium
35.6
1.85
66
Aluminium 26.50
2.70
72
Magnesium 43.90
1.74
76.3
Copper
16.78
8.96
150
Silver
15.87
10.49
166
Gold
22.14
19.30
427
Iron
96.1
7.874
757
Silver, although it is the least resistive metal known, has a high density and does poorly by this measure. Calcium
and the alkali metals make for the best products, but are rarely used for conductors due to their high reactivity with
water and oxygen. Aluminium is far more stable. And the most important attribute, the current price, excludes the
best choice: Beryllium.
Notes and references
Notes
[1] An introduction to Tensor Analysis: For Engineers and Applied Scientists, J.R. Tyldesley, Longman, 1975, ISBN 0-582-44355-5
[2] The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2
[3] Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISBN
978-0-521-86153-3
[4] Suresh V Vettoor Electrical Conduction and Superconductivity (http:/ / www. ias. ac. in/ resonance/ Sept2003/ pdf/ Sept2003p41-48. pdf).
ias.ac.in. September 2003
[5] The numbers in this column increase or decrease the significand portion of the resistivity. For example, at 30 °C (unknown operator:
u'strong' K), the resistivity of silver is 1.65×10−8. This is calculated as Δρ = α ΔT ρo where ρo is the resistivity at 20 °C (in this case) and α is
the temperature coefficient.
[6] Serway, Raymond A. (1998). Principles of Physics (2nd ed ed.). Fort Worth, Texas; London: Saunders College Pub. p. 602.
ISBN 0-03-020457-7.
[7] Griffiths, David (1999) [1981]. "7. Electrodynamics". In Alison Reeves (ed.). Introduction to Electrodynamics (3rd edition ed.). Upper Saddle
River, New Jersey: Prentice Hall. p. 286. ISBN 0-13-805326-X. OCLC 40251748.
[8] Referred to as 100% IACS or International Annealed Copper Standard. The unit for expressing the conductivity of nonmagnetic materials by
testing using the eddy-current method. Generally used for temper and alloy verification of aluminium.
[9] Gold is commonly used in electrical contacts because it does not easily corrode.
[10] Commonly used for high voltage power lines
[11] Physical constants (http:/ / physics. mipt. ru/ S_III/ t). (PDF format; see page 2, table in the right lower corner)]. Retrieved on 2011-12-17.
[12] Matweb (http:/ / www. matweb. com/ search/ DataSheet. aspx?MatGUID=025d4a04c2c640c9b0eaaef28318d761)
[13] JFE steel (http:/ / www. jfe-steel. co. jp/ en/ products/ electrical/ catalog/ f1e-001. pdf)
Electrical resistivity and conductivity
[14] Giancoli, Douglas C. (1995). Physics: Principles with Applications (4th ed ed.). London: Prentice Hall. ISBN 0-13-102153-2.
(see also Table of Resistivity (http:/ / hyperphysics. phy-astr. gsu. edu/ hbase/ Tables/ rstiv. html))
[15] John O'Malley, Schaum's outline of theory and problems of basic circuit analysis, p. 19, McGraw-Hill Professional, 1992 ISBN 0070478244
[16] 18% chromium/ 8% nickel austenitic stainles steel
[17] Glenn Elert (ed.), "Resistivity of steel" (http:/ / hypertextbook. com/ facts/ 2006/ UmranUgur. shtml), The Physics Factbook, retrieved and
archived (http:/ / www. webcitation. org/ 5zURdqzDl) 16 June 2011.
[18] Nickel-iron-chromium alloy commonly used in heating elements.
[19] Y. Pauleau, Péter B. Barna, P. B. Barna, Protective coatings and thin films: synthesis, characterization, and applications, p. 215, Springer,
1997 ISBN 0792343808.
[20] Graphite is strongly anisotropic.
[21] Hugh O. Pierson, Handbook of carbon, graphite, diamond, and fullerenes: properties, processing, and applications, p. 61, William Andrew,
1993 ISBN 0815513399.
[22] The resistivity of semiconductors depends strongly on the presence of impurities in the material.
[23] Lawrence S. Pan, Don R. Kania, Diamond: electronic properties and applications, p. 140, Springer, 1994 ISBN 0792395247.
[24] Corresponds to an average salinity of 35 g/kg at 20 °C.
[25] Physical properties of sea water (http:/ / www. kayelaby. npl. co. uk/ general_physics/ 2_7/ 2_7_9. html). Kayelaby.npl.co.uk. Retrieved on
2011-12-17.
[26] This value range is typical of high quality drinking water and not an indicator of water quality
[27] Conductivity is lowest with monoatomic gases present; changes to 1.2×10−4 upon complete de-gassing, or to 7.5×10−5 upon equilibration to
the atmosphere due to dissolved CO2
[28] Pashley, R. M.; Rzechowicz, M; Pashley, LR; Francis, MJ (2005). "De-Gassed Water is a Better Cleaning Agent". The Journal of Physical
Chemistry B 109 (3): 1231. doi:10.1021/jp045975a. PMID 16851085.
[29] Ohring, Milton (1995). Engineering materials science, Volume 1 (3rd edition ed.). p. 561.
[30] Pawar, S. D.; Murugavel, P.; Lal, D. M. (2009). "Effect of relative humidity and sea level pressure on electrical conductivity of air over
Indian Ocean". Journal of Geophysical Research 114: D02205. Bibcode 2009JGRD..11402205P. doi:10.1029/2007JD009716.
[31] Copper Wire Tables (http:/ / library. bldrdoc. gov/ docs/ nbshb100. pdf). US Dep. of Commerce. National Bureau of Standards Handbook.
February 21, 1966
[32] Ward, MR, Electrical Engineering Science, pp. 36−40, McGraw-Hill, 1971.
[33] A. Matthiessen, Rep. Brit. Ass. 32, 144 (1862)
[34] A. Matthiessen, Progg. Anallen, 122, 47 (1864)
[35] Seymour J, Physical Electronics, chapter 2, Pitman, 1972
[36] Otto H. Schmitt, University of Minnesota Mutual Impedivity Spectrometry and the Feasibility of its Incorporation into Tissue-Diagnostic
Anatomical Reconstruction and Multivariate Time-Coherent Physiological Measurements (http:/ / web. archive. org/ web/ 20101013215436/
http:/ / www. otto-schmitt. org/ OttoPagesFinalForm/ Sounds/ Speeches/ MutualImpedivity. htm). otto-schmitt.org. Retrieved on 2011-12-17.
References
Further reading
• Paul Tipler (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern
Physics (5th ed.). W. H. Freeman. ISBN 0-7167-0810-8.
External links
• New nanomaterial better efficient conductor (http://www.reuters.com/article/environmentNews/
idUSN2041399820080320?rpc=64&pageNumber=1&virtualBrandChannel=10150)
• Resistivity & Mobility Calculator/Graph from BYU cleanroom (http://www.ee.byu.edu/cleanroom/
ResistivityCal.phtml/)
• Bucknell University (http://www.facstaff.bucknell.edu/mastascu/eLessonsHTML/Sensors/TempR.html)
9
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Image:Resistivity geometry.png Source: http://en.wikipedia.org/w/index.php?title=File:Resistivity_geometry.png License: GNU Free Documentation License Contributors: User:Omegatron
Image:Resistance band theory insulator.png Source: http://en.wikipedia.org/w/index.php?title=File:Resistance_band_theory_insulator.png License: Public Domain Contributors: User:Bkell,
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