resistivity

[ree-zis-tiv-i-tee]

Electrical resistance of a conductor of unit cross-sectional area and unit length. The resistivity of a conductor depends on its composition and its temperature. As a characteristic property of each material, resistivity is useful in comparing various materials on the basis of their ability to conduct electric current. As temperature increases, the resistivity of a metallic conductor usually increases and that of a semiconductor usually decreases.

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Encyclopedia Britannica, 2008. Encyclopedia Britannica Online.

Resistivity

Electrical resistivity (also known as specific electrical resistance) 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 electrical charge. The SI unit of electrical resistivity is the ohm meter.

Definitions

The electrical resistivity ρ (rho) of a material is given by

{rho={R left. frac{A}{ell} right.}}

where

ρ is the static resistivity (measured in ohm metres, Ω-m);

R is the electrical resistance of a uniform specimen of the material (measured in ohms, Ω);

ell

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).

Electrical resistivity can also be defined as

rho={E over J}

where

E is the magnitude of the electric field (measured in volts per metre, V/m);

J is the magnitude of the current density (measured in amperes per square metre, A/m²).

Finally, electrical resistivity is also defined as the inverse of the conductivity σ (sigma), of the material, or

rho = {1oversigma}.

Table of resistivities

This table shows the resistivity and temperature coefficient of various materials at 20 °C (68 °F)

Material	Resistivity (Ω-m) at 20 °C	Coefficient*
Silver	1.59×10⁻⁸	.0038
Copper	1.72×10⁻⁸	.0039
Gold	2.44×10⁻⁸	.0034
Aluminium	2.82×10⁻⁸	.0039
Calcium	3.3x10^-8
Tungsten	5.60×10⁻⁸	.0045
Nickel	6.99×10⁻⁸	?
Iron	1.0×10⁻⁷	.005
Tin	1.09×10⁻⁷	.0045
Platinum	1.1×10⁻⁷	.00392
Lead	2.2×10⁻⁷	.0039
Manganin	4.82×10⁻⁷	.000002
Constantan	4.9×10⁻⁷	0.00001
Mercury	9.8×10⁻⁷	.0009
Nichrome	1.10×10⁻⁶	.0004
Carbon	3.5×10⁻⁵	-.0005
Germanium	4.6×10⁻¹	-.048
Silicon	6.40×10²	-.075
Glass	10¹⁰ to 10¹⁴	?
Hard rubber	approx. 10¹³	?
Sulfur	10¹⁵	?
Paraffin	10¹⁷	?
Quartz (fused)	7.5×10¹⁷	?
PET	10²⁰	?
Teflon	10²² to 10²⁴	?

*The numbers in this column increase or decrease the significand portion of the resistivity. For example, at 30°C (303.15 K), the resistivity of silver is 1.65×10⁻⁸. This is calculated as Δρ = α ΔT ρ_o where ρ_o is the resistivity at 20°C and α is the temperature coefficient

Temperature dependence

In general, electrical resistivity of metals increases with temperature, while the resistivity of semiconductors decreases with increasing temperature. In both cases, 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:

rho(T)=rho(0)+Aleft(frac{T}{Theta_R}right)^nint_0^{frac{Theta_R}{T}}frac{x^n}{(e^x-1)(1-e^{-x})}dx

where $rho(0)$ 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. $Theta_R$ 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:

n=5 implies that the resistance is due to scattering of electrons by phonons (as it is for simple metals)
n=3 implies that the resistance is due to s-d electron scattering (as is the case for transition metals)
n=2 implies that the resistance is due to electron-electron interaction.

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 even better approximation of the temperature dependence of the resistivity of a semiconductor is given by the Steinhart–Hart equation:

1/T = A + B ln(rho) + C (ln(rho))^3 ,

where A, B and C are the so-called Steinhart–Hart coefficients.

This equation is used to calibrate thermistors.

In non-crystalline semi-conductors, 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 $rho = Ae^{T^{-1/n}}$ , where n=2,3,4 depending on the dimensionality of the system.

Complex resistivity

When analyzing the response of materials to alternating electric fields, as is done in certain types of tomography, 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)

Resistivity density products

In some applications where the weight of an item is very important resistivity density products are more important than absolute low resistance- 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.

This fact is used for long distance overhead powerline transmission- aluminium is used rather than copper because it is lighter for the same conductance. Calcium, with a resistivity density product lower than aluminium, is rarely if ever used due to its highly reactive nature.

Material	Resistivity (nΩ·m)	Density (g/cm^3)	Resistivity - density product (nΩ·m·g/cm^3)
Calcium	33.6	1.55	52
Aluminium	26.50	2.70	72
Copper	16.78	8.96	150
Silver	15.87	10.49	166

Sources

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.