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In $2d$ systems, the relation between the conductance $G$ and conductivity $\sigma$ is given by $$G = \sigma W/L$$ (page 1, Electronic Transport in Mesoscopic systems, Supriyo Datta) where $W$ is the width and $L$ is the length of the sample.

In usual $3d$ systems, the relation between them is $G = \sigma A/L$, where $A$ is the cross section area. In $2d$ systems, one of the dimension is so small that the electrons are confined in that direction. However, if we compare the two equations, it looks like the width of the sample in the confined direction appears to be unity (so that $A = W$), which it cannot be.

The other possible explanation is, $\sigma$ is defined differently, as it must have different dimensions. For $3d$ systems, we can write $\sigma = n e^2 \tau/m$. What will be the form of $\sigma$ in terms of the scattering time $\tau$ in a $2d$ system?

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This formula for $\sigma$ is also true for $2d$ systems. $n$ is the density of the charge carriers. In $3d$ systems it has dimensions of inverse volume while in $2d$ systems it has dimensions of inverse area. So dimensions-wise $[n_{3d} A] = [n_{2d} W]=[1/L]$. It means that indeed $\sigma$ has different dimensions in $2d$ and $3d$ systems, but that's ok.

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