Sheet resistivity (${\mathbf{\rho }}_s={\mathbf{\sigma }}_s^{-1}$)

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.