The reason graphene is great for transistors is with its higher band energy its 5x harder for it to soft set itself. So if we pretend todays 11nm transistors have a 1% chance of electron tunneling, carbon would have a 0.2%.
The problem is that same switch would take 5x as much power to switch. Which means a modern 220w cpu would now need 1100watts of power :x
I should have said electron affinity rather than electronegativity: You're not trying to get electrons from the valence band into the vacuum, just from the conduction band to the vacuum. Diamond actually has a negative electron affinity, which is why the vacuum electronic folks were excited about it.
Those are not the same thing. You can have 5x voltage without changing the power or the energy.
That said, except for very few embedded applications, computers today operate on a voltage that's much bigger than the silicon band gap. I'm not convinced it'd be a problem.
P = CV^2 * f (multiply by activity factor if pedantic; assuming all transistors toggle every cycle here.)
The band gap is not a function of the element alone, but mainly of the crystal structure. Different allotropes of carbon have vastly different band structures.
Diamond has a high band gap, as you mentioned. But Graphene has none, and that is one of the major obstacles of the material.
But you know, it is really fast! Just almost useless transconductance and no off state, but we can fix it later!