Graphene Vs Copper?

I was wondering about the amazing conductive properties of graphene, lets assume a large copper bar that is 10 kg, current of 1kA and probably more can flow in it, what about graphene? It certainly can since its a better conductor, but it would be much lighter? In the milligrams possibly?

Also, can graphene replace the heavy busbars that carry MW of power? Making it much lighter?

• Currently Aluminum is the next best conductor to copper, when it comes to bang for the buck; but aluminum is only 61% as efficient at conductivity as copper is, making it less desirable. The weight of aluminum and its low cost are big plusses when compared to copper. What do you suppose you'd have to do to augment aluminum with graphene in order to increase it's conductivity, while keeping its low-weight and flexibility. If you could do that, you could revolutionize the power transmission industry, saving us all a ton of money. – user36022 Oct 13 '16 at 22:38

The trick with graphene is that a lot of its amazing properties only work when you have continuous perfect sheets of it, and making graphene like this is currently beyond us, for large scales anyways. It is true that graphene has very high electron mobility $\approx10^{5}~\mathrm{cm^2/Vs}$ at room temperature, which works out to on the order of $10~\mathrm{n\Omega\cdot m}$ in bulk (which assumes you can make a perfect multilayer structure that maintains the properties of a single sheet). An impressive figure, but only $\approx40\%$ better than copper.
Of course, copper is about 6 times more dense than graphene, so if mass is the main concern, graphene would be a pretty good improvement. Still, we're not talking about replacing $10~\mathrm{kg}$ of copper with a few milligrams. Given how much cheaper and easier it is to make metal wires, we're not at the point of using graphene for this kind of bulk application. There are also some more mundane problems to sort out, e.g. graphene is brittle like a ceramic, which might cause mechanical issues.
Stanene is strictly theoretical at this point as was the theoretical hardest substance C3N4 for many years ($$\ce{C3N4}$$ is extremely expensive and produced in very small quantities in research laboratories - a lot like antimatter). Graphene is 1.4 times as conductive as copper by volume, but if you factor in density, it is 5.8 times more conductive (if a given sample of copper can carry $$\pu{1kA}$$ with a $$\pu{1V}$$ drop in voltage, the same weight, and length, of graphene, could carry $$\pu{5.8 kA}$$ with with a $$\pu{1V}$$ drop in voltage - in theory). Of course, fabrication methods can degrade this. For electrical power transmission, which depends a lot on weight, $$\ce{Al}$$ is 2 times as conductive as copper BY WEIGHT and is, therefore, the current choice for power transmission. The fact that it costs less is a bonus. A graphene/$$\ce{Al}$$ alloy might give a conductor with the fracture resistance near $$\ce{Al}$$ but better conductivity. This is an active area of research currently.