Electrical resistivity and conductivity

Resistivity
Common symbols	ρ
SI unit	ohm metre (Ω⋅m)
In SI base units	kg⋅m3⋅s−3⋅A−2
Derivations from other quantities	${\displaystyle \rho =R{\frac {A}{\ell }}}$
Dimension	${\displaystyle {\mathsf {M}}{\mathsf {L}}^{3}{\mathsf {T}}^{-3}{\mathsf {I}}^{-2}}$

Conductivity
Common symbols	σ, κ, γ
SI unit	siemens per metre (S/m)
In SI base units	kg−1⋅m−3⋅s3⋅A2
Derivations from other quantities	${\displaystyle \sigma ={\frac {1}{\rho }}}$
Dimension	${\displaystyle {\mathsf {M}}^{-1}{\mathsf {L}}^{-3}{\mathsf {T}}^{3}{\mathsf {I}}^{2}}$

Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. Its inverse, called electrical conductivity, quantifies how well a material conducts electricity. A low resistivity indicates a material that readily allows electric current. Resistivity is commonly represented by the Greek letter ρ (rho). The SI unit of electrical resistivity is the ohm-meter (Ω⋅m).^[1][2][3] For example, if a 1 m solid cube of material has sheet contacts on two opposite faces, and the resistance between these contacts is 1 Ω, then the resistivity of the material is 1 Ω⋅m.

Electrical conductivity or specific conductance is the reciprocal of electrical resistivity. It represents a material's ability to conduct electric current. It is commonly signified by the Greek letter σ (sigma), but κ (kappa) (especially in electrical engineering) and γ (gamma) are sometimes used. The SI unit of electrical conductivity is siemens per metre (S/m).

Definition[]

Ideal case[]

A piece of resistive material with electrical contacts on both ends.

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. Many resistors and conductors 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 electrical resistivity ρ (Greek: rho) can be calculated by:

${\displaystyle \rho =R{\frac {A}{\ell }},\,\!}$

where

${\displaystyle R}$ is the electrical resistance of a uniform specimen of the material

${\displaystyle \ell }$ is the length of the specimen

${\displaystyle A}$ is the cross-sectional area of the specimen

Both resistance and resistivity describe how difficult it is to make electrical current flow through a material, but unlike resistance, resistivity is an intrinsic property. This means that all pure copper wires (which have not been subjected to distortion of their crystalline structure etc.), irrespective of their shape and size, have the same resistivity, but a long, thin copper wire has a much larger resistance than a thick, short copper wire. Every material has its own characteristic resistivity. For example, rubber has a far larger resistivity than copper.

In a hydraulic 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 not solely 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 get Pouillet's law (named after Claude Pouillet):

${\displaystyle R=\rho {\frac {\ell }{A}}.\,\!}$

The resistance of a given material is proportional to the length, but inversely proportional to the cross-sectional area. Thus resistivity can be expressed using the SI unit "ohm metre" (Ω⋅m) — i.e. ohms divided by metres (for the length) and then multiplied by square metres (for the cross-sectional area).

For example, if A = 1 m², ${\displaystyle \ell }$ = 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:

${\displaystyle \sigma ={\frac {1}{\rho }}.\,\!}$

Conductivity has SI units of siemens per metre (S/m).

General scalar quantities[]

For less ideal cases, such as more complicated geometry, or when the current and electric field vary in different parts of the material, it is necessary to use a more general expression in which the resistivity at a particular point is defined as the ratio of the electric field to the density of the current it creates at that point:

${\displaystyle \rho ={\frac {E}{J}},\,\!}$

where

${\displaystyle \rho }$ is the resistivity of the conductor material,

${\displaystyle E}$ is the magnitude of the electric field,

${\displaystyle J}$ is the magnitude of the current density,

in which ${\displaystyle E}$ and ${\displaystyle J}$ are inside the conductor.

Conductivity is the inverse (reciprocal) of resistivity. Here, it is given by:

${\displaystyle \sigma ={\frac {1}{\rho }}={\frac {J}{E}}.\,\!}$

For example, rubber is a material with large ρ and small σ — 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 small ρ and large σ — because even a small electric field pulls a lot of current through it.

As shown below, this expression simplifies to a single number when the electric field and current density are constant in the material.

showDerivation from general definition of resistivity

There are three equations to be combined here. The first is the resistivity for parallel current and electric field: ${\displaystyle \rho ={\frac {E}{J}},\,\!}$ If the electric field is constant, the electric field is given by the total voltage V across the conductor divided by length ℓ of the conductor: ${\displaystyle E={\frac {V}{\ell }}}$ If the current density is constant, it is equal to the total current divided by the cross sectional area: ${\displaystyle J={\frac {I}{A}}}$ Plugging in the values of E and J into the first expression, we obtain: ${\displaystyle \rho ={\frac {VA}{I\ell }}}$ Finally, we apply Ohm's law, V/I = R. ${\displaystyle \rho =R{\frac {A}{\ell }}}$

Tensor resistivity[]

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 of Ohm'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 in anisotropic cases, where the more simple definitions cannot be applied. If the material is not anisotropic, 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 of graphite 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]

${\displaystyle \mathbf {J} ={\boldsymbol {\sigma }}\mathbf {E} \,\,\rightleftharpoons \,\,\mathbf {E} ={\boldsymbol {\rho }}\mathbf {J} \,\!}$

where the conductivity σ and resistivity ρ are rank-2 tensors, and electric field E and current density J are vectors. These tensors can be represented by 3×3 matrices, the vectors with 3×1 matrices, with matrix multiplication used on the right side of these equations. In matrix form, the resistivity relation is given by:

${\displaystyle {\begin{bmatrix}E_{x}\\E_{y}\\E_{z}\end{bmatrix}}={\begin{bmatrix}\rho _{xx}&\rho _{xy}&\rho _{xz}\\\rho _{yx}&\rho _{yy}&\rho _{yz}\\\rho _{zx}&\rho _{zy}&\rho _{zz}\end{bmatrix}}{\begin{bmatrix}J_{x}\\J_{y}\\J_{z}\end{bmatrix}}}$

where

${\displaystyle \mathbf {E} }$ is the electric field vector, with components (E_x, E_y, E_z).

${\displaystyle {\boldsymbol {\rho }}}$ is the resistivity tensor, in general a three by three matrix.

${\displaystyle \mathbf {J} }$ is the electric current density vector, with components (J_x, J_y, J_z)

Equivalently, resistivity can be given in the more compact Einstein notation:

${\displaystyle \mathbf {E} _{i}={\boldsymbol {\rho }}_{ij}\mathbf {J} _{j}}$

In either case, the resulting expression for each electric field component is:

${\displaystyle E_{x}=\rho _{xx}J_{x}+\rho _{xy}J_{y}+\rho _{xz}J_{z}.}$

${\displaystyle E_{y}=\rho _{yx}J_{x}+\rho _{yy}J_{y}+\rho _{yz}J_{z}.}$

${\displaystyle E_{z}=\rho _{zx}J_{x}+\rho _{zy}J_{y}+\rho _{zz}J_{z}.}$

Since the choice of the coordinate system is free, the usual convention is to simplify the expression by choosing an x-axis parallel to the current direction, so J_y = J_z = 0. This leaves:

${\displaystyle \rho _{xx}={\frac {E_{x}}{J_{x}}},\quad \rho _{yx}={\frac {E_{y}}{J_{x}}},{\text{ and }}\rho _{zx}={\frac {E_{z}}{J_{x}}}.}$

Conductivity is defined similarly:^[7]

${\displaystyle {\begin{bmatrix}J_{x}\\J_{y}\\J_{z}\end{bmatrix}}={\begin{bmatrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{yx}&\sigma _{yy}&\sigma _{yz}\\\sigma _{zx}&\sigma _{zy}&\sigma _{zz}\end{bmatrix}}{\begin{bmatrix}E_{x}\\E_{y}\\E_{z}\end{bmatrix}}}$

or

${\displaystyle \mathbf {J} _{i}={\boldsymbol {\sigma }}_{ij}\mathbf {E} _{j}}$

Both resulting in:

${\displaystyle J_{x}=\sigma _{xx}E_{x}+\sigma _{xy}E_{y}+\sigma _{xz}E_{z}}$

${\displaystyle J_{y}=\sigma _{yx}E_{x}+\sigma _{yy}E_{y}+\sigma _{yz}E_{z}}$

${\displaystyle J_{z}=\sigma _{zx}E_{x}+\sigma _{zy}E_{y}+\sigma _{zz}E_{z}}$

Looking at the two expressions, ${\displaystyle {\boldsymbol {\rho }}}$ and ${\displaystyle {\boldsymbol {\sigma }}}$ are the matrix inverse of each other. However, in the most general case, the individual matrix elements are not necessarily reciprocals of one another; for example, σ_xx may not be equal to 1/ρ_xx. This can be seen in the Hall effect, where ${\displaystyle \rho _{xy}}$ is nonzero. In the Hall effect, due to rotational invariance about the z-axis, ${\displaystyle \rho _{yy}=\rho _{xx}}$ and ${\displaystyle \rho _{yx}=-\rho _{xy}}$, so the relation between resistivity and conductivity simplifies to:^[8]

${\displaystyle \sigma _{xx}={\frac {\rho _{xx}}{\rho _{xx}^{2}+\rho _{xy}^{2}}},\quad \sigma _{xy}={\frac {-\rho _{xy}}{\rho _{xx}^{2}+\rho _{xy}^{2}}}}$

If the electric field is parallel to the applied current, ${\displaystyle \rho _{xy}}$ and ${\displaystyle \rho _{xz}}$ are zero. When they are zero, one number, ${\displaystyle \rho _{xx}}$, is enough to describe the electrical resistivity. It is then written as simply ${\displaystyle \rho }$, and this reduces to the simpler expression.

Conductivity and current carriers[]

Relation between current density and electric current velocity[]

Electric current is the ordered movement of electric charges. These charges are called current carriers. In metals and semiconductors, electrons are the current carriers; in electrolytes and ionized gases, positive and negative ions. In the general case, the current density of one carrier is determined by the formula:^[9]

${\displaystyle {\vec {j}}=qn{\vec {\upsilon }}_{a}}$,

where