# Why is contact resistance measured in $\Omega~\mu m$?

In many papers the contact resistance of a metal in contact with a semiconductor is given in units of $$\Omega~\mu m$$, for example in the paper by Li et al. (Appl. Phys. Lett. 102 (2013), p. 183110):

$$R_C$$ for contacts formed to epitaxial graphene on SiC have been reported to be less than 100 $$\Omega~\mu m$$ and with specific contact resistivity ($$\rho_c$$) of order $$10^{-7} \Omega~cm^2$$.

I'm not sure where this unit comes from. From the transfer line method one can determine the contact resistance as a value in $$\Omega$$. Normalizing it to the contact area however should bring the unit to $$\frac{\Omega}{\mu m^2}$$, i.e. it should be divided by the area rather than be multiplied by a length.

How does one arrive at the unit $$\Omega~\mu m$$ for the contact resistance? What assumptions regarding the geometry of the contact enter into this? Why is the correct unit not just $$\frac{\Omega}{\mu m^2}$$?

The contact resistivity is defined as the slope of the $$j$$-$$V$$ curve at 0V, where $$j$$ is the current density. $$\rho_c = \frac{\partial V}{\partial j}|_{V=0}$$ This gives it units of $$\frac{V}{\frac{I}{m^2}} = \frac{V m^2}{I} = \Omega~m^2$$
The contact resistance (in $$\Omega$$) is simply the contact resistivity divided by the effective contact area $$A = w \cdot L_T \cdot \coth\big(\frac{L_C}{L_T}\big)$$ $$R_C = \frac{\rho_c}{A} = \frac{\rho_c}{w \cdot L_T \cdot \coth\big(\frac{L_C}{L_T}\big)}$$
In the specific case of a metal-graphene contact, the current flows into the metal contact only in a very thin strip at the edge. This shortens the transfer length $$L_T$$, which is a measure for the effective contact length. In this particular system this length is quite short (~100 nm) compared to the physical size of the contact (~ 100 $$\mu m$$), and can therefore be ignored. This leaves only $$w$$ in the denominator.
$$\tilde{R}_C = \frac{\rho_c}{w} \rightarrow \Big[\frac{\Omega~m^2}{m}\Big] = [\Omega~m]$$