# Can conductivity (or resistivity) value be assigned to 2D materials?

2D materials, such as graphene monolayer or graphene nanoribbon, don't really have a cross section. By definition, conductivity is $\sigma = \frac{J}{E}$, and $J$, the current density, is calculated per cross section: $J = \frac{I}{A}$.

So how can conductivity (or resistivity, or current density) be reported for 2D materials if no actual cross section can be defined? To my understanding, only conductance (or resistance) can be assigned to 2D materials, as $G = \frac{I}{V}$, which do not depend on cross section.

• Depending on the level of detail you want there are several possibilites. One is to simply define a 2D conductivity. That is, model the current density as $\vec j = (J_x \vec e_x + J_y \vec e_y) \delta(z)$, and then calculate the linear response of the two dimensional vector $\vec J$ to the three dimensional electrical field $\vec E$, which gives at 2x3 matrix for the conductivity. (I guess the $\sigma_{xz}$ and $\sigma_{yz}$ components will be about zero). Commented Dec 10, 2015 at 22:23
• I'd add that, normally, an atomic monolayer will have sufficient symmetry, e.g. mirror symmetry in its own plane, that those two components will be exactly zero. Also, very often people will just use the sheet resistance, with units of ohms, which gives the current per unit length for a given electric field. I think this is generally used much more often than pretending that the film has some exact thickness, the choice of which is necessarily arbitrary. Commented Dec 11, 2015 at 2:37
• Technically, Ohm's law is a tensor form given by: $\mathbf{E} = \overleftrightarrow{\eta} \cdot \mathbf{j}$, where $\overleftrightarrow{\eta}$ is the resistivity tensor. The conductivity only exists in the form you describe if $\overleftrightarrow{\eta}$ can be defined as $\overleftrightarrow{\eta} = \eta \ \mathbb{I}$. In any case, that it is a tensor makes it easier to reduce the dimensions. So yes, you can have a 2D form for the resistivity, similar to how $\mathbf{j} \rightarrow \boldsymbol{\kappa}$ for a 2D current sheet... Commented Dec 11, 2015 at 13:29