No, the potential and electron density is computed fully self-consistently in the entire central region, incl. the electrode extensions. They are only required to match the electrode by geometry, basically for technical reasons, but can have a different electron structure than the corresponding electrode atoms. Of course, in reality you want that this region is very close to bulk-like, but it's not a strict requirement.
Of course, this also points to the fact that in order to have proper convergence and a good physics picture, if you place a gate in the electrode extensions, it should also be present in the electrodes, for consistency. But this is a choice, in terms of the device design. If you want a gate that covers the whole region from source to drain, then you also include it in the electrodes. Otherwise (and typically) you only have a gate in the middle of the scattering region, and enough screening material outside. Compare for instance the device here:
http://quantumwise.com/documents/tutorials/latest/InAsDevice/index.html/chap.nanowiredevice.html#sect2.nanowiredevice.setup.gateA common mistake people make is to make the central region too short...
