Electrical resistivity (${\rho }_{\mathrm{el}}=1/\sigma$)

Electrical resistivity is the inverse of scalar electrical conductivity in an isotropic Ohmic medium. In its rest frame, $\mathbf{E}={\rho }_{\mathrm{el}}\mathbf{j}$, so the electric field does work per unit volume $\mathbf{j}\cdot \mathbf{E}={\rho }_{\mathrm{el}}|\mathbf{j}{|}^2$. In Gaussian units, finite conductivity gives magnetic diffusivity $\eta =c^2/(4\pi \sigma )$; neglecting this diffusion is a distinct approximation from neglecting electron inertia in Hall magnetohydrodynamics.

A phenomenological label for metallic responses that depart from ordinary Fermi-liquid expectations, commonly including a broad approximately temperature-linear resistivity regime in some cuprates. The observed range depends on doping, field and material. A linear law alone does not uniquely identify the scattering mechanism, and the ordinary metal's high-temperature phonon-linear resistivity is not by itself evidence of the same regime.

The resistivity tensor is the matrix relating electric field to current density in linear local transport. It is the matrix inverse of the conductivity tensor, not the componentwise reciprocal. A Hall response produces antisymmetric off-diagonal components. In a multicarrier system, the conductivities add because each species sees the same electric field; invert only after summing the species' conductivity tensors.

The residual resistivity is the low-temperature electrical resistivity remaining in a normal metal when thermally excited scattering becomes negligible. Static disorder, impurities and boundaries can relax carrier momentum without requiring a thermal population of phonons. Its value depends on sample purity and structure; a superconducting transition is a different phenomenon.

Matthiessen's rule approximates the total electrical resistivity of a metal by the sum of separate scattering contributions. For a common parabolic energy band, approximately independent scattering mechanisms give $1/\tau ={\sum }_{j}1/{\tau }_j$ for the respective momentum relaxation times, and hence ${\rho }_{\mathrm{el}}=m/(n{e}^2\tau )={\sum }_{j}m/(n{e}^2{\tau }_j)$. Correlated scattering, strongly angle-dependent processes or a complicated changing energy band structure can invalidate the simple addition.

The sheet resistivity tensor relates tangential electric field to surface current density: ${\mathbf{E}}_{\parallel }={\mathbf{\rho }}_s\mathbf{K}$. It is the matrix inverse of the sheet conductivity tensor, when that inverse exists. Its SI unit is the ohm, rather than the ohm times metre used for bulk electrical resistivity. With a transverse response, individual tensor components are not generally reciprocals of the corresponding sheet conductivity components.

For a spin-degenerate parabolic two-dimensional band, the Drude model gives $\tau =m^{\ast }/(n{e}^2{\rho }_{\square })$ and $v_F=\hslash k_F/m^{\ast }$. Their product is the displayed mass-independent mean free path. A Fermi-surface transport interpretation requires a sufficiently degenerate electron gas; a quantum lifetime need not equal the transport relaxation time.

The in-plane dissipative response along an imposed sheet current. In an isotropic Hall system, matrix inversion gives ${\rho }_{xx}={\sigma }_{xx}/({\sigma }_{xx}^2+{\sigma }_{xy}^2)$. It is a sheet quantity in ohms, while a homogeneous strip's measured longitudinal resistance additionally includes its length-to-width ratio.