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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 $\frac{\Omega}{\mu m^2}$?

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Through a bit more accurate research on Wikipedia and in the paper mentioned above I found out where this unit comes from.

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.

For this reason the resistance per unit width is very often used in literature to describe such devices.

$$\tilde{R}_C = \frac{\rho_c}{w} \rightarrow \Big[\frac{\Omega~m^2}{m}\Big] = [\Omega~m]$$

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