Quantum Well Transistors

My understanding of a quantum well transistor is that it is created by sandwiching materials with higher bandgaps with smaller ones, so that it creates a potential well. So for instance, electrons will be forced to be confined within the well, as they would travel along the Quantum Well channel.
So my question is, I understand how the concept of quantum wells work, but when they are applied to a transistor, how exactly are the charge carriers being transported when the supply voltage, for instance Vds is increased? Is it that when there is a potential difference across the transistor, the electrons, or carriers, in the QW (usually with higher mobility) would be forced to travel along the route with least resistance. I'm just trying to understand that what forces the injected electrons from the metal contacts on the transistors to move towards the channel...is it because as I already stated, they are more prone to travel through a lower energy bandgap?

I hope someone can help clarify or give me some direction, because I have looked online and have failed to find some decent resources. Thanks.

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