Electrical resistivity and conductivity

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 resistance of the conductor is directly proportional to its length and inversely proportional to its cross-sectional area, where the electrical resistivity ρ (Greek: rho) is the constant of proportionality. This is written as:

$${\displaystyle R\propto {\frac {\ell }{A}}}$$

$${\displaystyle {\begin{aligned}R&=\rho {\frac {\ell }{A}}\\[3pt]{}\Leftrightarrow \rho &=R{\frac {A}{\ell }},\end{aligned}}}$$

where

The resistivity can be expressed using the SI unit ohm metre (Ω⋅m) — i.e. ohms multiplied by square metres (for the cross-sectional area) then divided by metres (for the length).

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 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 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 element is proportional to the length, but inversely proportional to 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).

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

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.

Derivation 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: $${\displaystyle \rho =R{\frac {A}{\ell }}.}$$

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

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:

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

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:

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

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.

Electric current is the ordered movement of electric charges.[2]

Filling of the electronic states in various types of materials at equilibrium. Here, height is energy while width is the density of available states for a certain energy in the material listed. The shade follows the Fermi–Dirac distribution (black: all states filled, white: no state filled). In metals and semimetals the Fermi level E_F lies inside at least one band.

In insulators and semiconductors the Fermi level is inside a band gap; however, in semiconductors the bands are near enough to the Fermi level to be thermally populated with electrons or holes.

According to elementary quantum 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[lower-alpha 1] and their distribution within the crystal.[lower-alpha 2]

The material's electrons seek to minimize the total energy in the material by settling into low energy states; however, the Pauli 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 the Fermi 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 the Fermi 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.

A metal consists of a lattice of atoms, 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 (a voltage) is applied across the metal, the resulting electric field causes electrons to drift towards the positive terminal. The actual drift 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 large current density.[10] The mechanism is similar to transfer of momentum of balls in a Newton'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 waves interfere, 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, the Fermi level lies in the conduction band (see Band Theory, above) giving rise to free conduction electrons. However, in semiconductors 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 the charge 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.

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 solutions (electrolytes) varies tremendously with concentration – while distilled water is almost an insulator, salt water is a reasonable electrical conductor. Conduction in ionic liquids is also controlled by the movement of ions, but here we are talking about molten salts rather than solvated ions. In biological membranes, currents are carried by ionic salts. Small holes in cell membranes, called ion 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 ${\displaystyle \alpha }$, which is the ratio of the concentration of ions ${\displaystyle N}$ to the concentration of molecules of the dissolved substance ${\displaystyle N_{0}}$:

$${\displaystyle N=\alpha N_{0}~.}$$

The specific electrical conductivity (${\displaystyle \sigma }$) of a solution is equal to:

$${\displaystyle \sigma =q\left(b^{+}+b^{-}\right)\alpha N_{0}~,}$$

where ${\displaystyle q}$: module of the ion charge, ${\displaystyle b^{+}}$ and ${\displaystyle b^{-}}$: mobility of positively and negatively charged ions, ${\displaystyle N_{0}}$: concentration of molecules of the dissolved substance, ${\displaystyle \alpha }$: the coefficient of dissociation.

Original data from the 1911 experiment by Heike Kamerlingh Onnes showing the resistance of a mercury wire as a function of temperature. The abrupt drop in resistance is the superconducting transition.

The electrical resistivity of a metallic conductor decreases gradually as temperature is lowered. In normal (that is, non-superconducting) conductors, such as copper or silver, this decrease is limited by impurities and other defects. Even near absolute 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 of superconducting wire can persist indefinitely with no power source.[16]

In a class of superconductors known as type II superconductors, including all known high-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 of magnetic 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.

Lightning is an example of plasma present at Earth's surface. Typically, lightning discharges 30,000 amperes at up to 100 million volts, and emits light, radio waves, and X-rays.[17] Plasma temperatures in lightning might approach 30,000 kelvin (29,727 °C) (53,540 °F), or five times hotter than the temperature at the sun surface, and electron densities may exceed 10²⁴ m⁻³.

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 the plasma potential, or space potential. If an electrode is inserted into a plasma, its potential generally lies considerably below the plasma potential, due to what is termed a Debye sheath. The good electrical conductivity of plasmas makes their electric fields very small. This results in the important concept of quasineutrality, which says the density of negative charges is approximately equal to the density of positive charges over large volumes of the plasma (n_e = ⟨Z⟩ > n_i), but on the scale of the Debye length there can be charge imbalance. In the special case that double 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 net charge density. A common example is to assume that the electrons satisfy the Boltzmann relation:

$${\displaystyle n_{\text{e}}\propto \exp \left(e\Phi /k_{\text{B}}T_{\text{e}}\right).}$$

Differentiating this relation provides a means to calculate the electric field from the density:

$${\displaystyle \mathbf {E} =-{\frac {k_{\text{B}}T_{\text{e}}}{e}}{\frac {\nabla n_{\text{e}}}{n_{\text{e}}}}.}$$

(∇ is the vector gradient operator; see nabla symbol and gradient 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 repulsive electrostatic force dissipates it.

In astrophysical plasmas, Debye screening prevents electric fields from directly affecting the plasma over large distances, i.e., greater than the Debye 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 of Debye lengths. The dynamics of plasmas interacting with external and self-generated magnetic fields are studied in the academic discipline of magnetohydrodynamics.

Plasma is often called the fourth state of matter after solid, liquids and gases.[18][19] It is distinct from these and other lower-energy states 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:

The electrical resistivity of most materials changes with temperature. If the temperature T does not vary too much, a linear approximation is typically used:

$${\displaystyle \rho (T)=\rho _{0}[1+\alpha (T-T_{0})],}$$

where ${\displaystyle \alpha }$ is called the temperature coefficient of resistivity, ${\displaystyle T_{0}}$ is a fixed reference temperature (usually room temperature), and ${\displaystyle \rho _{0}}$ is the resistivity at temperature ${\displaystyle T_{0}}$. The parameter ${\displaystyle \alpha }$ is an empirical parameter fitted from measurement data, equal to 1/${\displaystyle \kappa }$. Because the linear approximation is only an approximation, ${\displaystyle \alpha }$ is different for different reference temperatures. For this reason it is usual to specify the temperature that ${\displaystyle \alpha }$ was measured at with a suffix, such as ${\displaystyle \alpha _{15}}$, and the relationship only holds in a range of temperatures around the reference.[53] When the temperature varies over a large temperature range, the linear 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 resistivity ρ of a metal can be approximated through the Bloch–Grüneisen formula:[54]

$${\displaystyle \rho (T)=\rho (0)+A\left({\frac {T}{\Theta _{R}}}\right)^{n}\int _{0}^{\Theta _{R}/T}{\frac {x^{n}}{(e^{x}-1)(1-e^{-x})}}\,dx,}$$

where ${\displaystyle \rho (0)}$ 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. ${\displaystyle \Theta _{R}}$ 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.

The Bloch–Grüneisen formula is an approximation obtained assuming that the studied metal has spherical Fermi surface inscribed within the first Brillouin zone and a Debye phonon spectrum.[55]

If more than one source of scattering is simultaneously present, Matthiessen's rule (first formulated by Augustus Matthiessen in the 1860s)[56][57] 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 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.

The Wiedemann–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:

$${\displaystyle {\kappa \over \sigma }={\pi ^{2} \over 3}\left({\frac {k}{e}}\right)^{2}T,}$$

where ${\displaystyle \kappa }$ is the thermal conductivity, ${\displaystyle k}$ is the Boltzmann constant, ${\displaystyle e}$ is the electron charge, ${\displaystyle T}$ is temperature, and ${\displaystyle \sigma }$ is the electric 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 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 temperature following an Arrhenius model:

$${\displaystyle \rho =\rho _{0}e^{\frac {E_{A}}{k_{B}T}}.}$$

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

$${\displaystyle {\frac {1}{T}}=A+B\ln \rho +C(\ln \rho )^{3},}$$

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

This equation is used to calibrate thermistors.

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.[58]

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

$${\displaystyle \rho =A\exp \left(T^{-1/n}\right),}$$

where n = 2, 3, 4, depending on the dimensionality of the system.