Electrical resistivity and conductivity
Electrical resistivity is a fundamental specific property of a material that measures its electrical resistance or how strongly it resists electric current. A low resistivity indicates a material that readily allows electric current. Resistivity is commonly represented by the Greek letter . The SI unit of electrical resistivity is the ohm-metre. For example, if a solid cube of material has sheet contacts on two opposite faces, and the resistance between these contacts is, then the resistivity of the material is.
Electrical conductivity is the reciprocal of electrical resistivity. It represents a material's ability to conduct electric current. It is commonly signified by the Greek letter , but and are sometimes used. The SI unit of electrical conductivity is siemens per metre. Resistivity and conductivity are intensive properties of materials, giving the opposition of a standard cube of material to current. Electrical resistance and conductance are corresponding extensive properties that give the opposition of a specific object to electric current.
Definition
Ideal case
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. 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 is the constant of proportionality. This is written as:where
The resistivity can be expressed using the SI unit ohm metre —i.e. ohms multiplied by square metres then divided by metres.
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 wires, irrespective of their shape and size, have the same, but a long, thin copper wire has a much larger 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 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 :
The resistance of a given element is proportional to the length, but inversely proportional to the cross-sectional area. For example, if =, = , 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,, is the inverse of resistivity:
Conductivity has SI units of siemens per metre.
Conductivity,, is directly proportional to
Where: = electron charge, = electron concentration, = hole concentration, = electron mobility, = hole mobility.
General scalar quantities
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 the electric field to the density of the current it creates at that point:where
The current density is parallel to the electric field by necessity.
Conductivity is the inverse of resistivity. Here, it is given by:
For example, rubber is a material with large and small — 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 and large — 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.
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 voltage across the conductor divided by the length 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 of and into the first expression, we obtain:
Finally, we apply Ohm's law, :
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 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 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. 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:
where the conductivity and resistivity are rank-2 tensors, and electric field and current density 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:
where
Equivalently, resistivity can be given in the more compact Einstein 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 an -axis parallel to the current direction, so. This leaves:
Conductivity is defined similarly:
or
both resulting in:
Looking at the two expressions, and 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, may not be equal to. This can be seen in the Hall effect, where is nonzero. In the Hall effect, due to rotational invariance about the -axis, and, so the relation between resistivity and conductivity simplifies to:
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.
Causes of conductivity
Electric current is the ordered movement of electrically-charged particles. Specifically, the relation between current density and charged-particle velocity is governed by the equation where is the current density, is the charge of the carrier, is the density of the particles, and their drift velocity, a time-averaged measure of their long-term motion.Band theory simplified
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 and their distribution within the crystal.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.