For mobility calculation, you may do a classical molecular dynamics (MD) simulation for 'DeviceConfiguration', see
http://docs.quantumwise.com/tutorials/md_basics.html, combined with conductance calculations. In your case, Device Configuration is a silicon nanowire with a given length attached to the left and right electrodes, which can be designed as semi-infinite silicon nanowires to avoid a contact resistance issue as it would be the case for metal electrodes.
The actual purpose of the classical MD simulation is to use its trajectory to calculate the conductance for a set of frames along the trajectory. Each frame corresponds to a certain atom arrangement in the finite-length nanowire, which is a central region in the Device Configuration described in the previous paragraph.
The mobility or conductivity of an infinite silicon nanowire can then be related to the finite-length nanowire conductance averaged over the trajectory frames selected. Assuming that the nanowire length in the device central region is more than the electron mean free path, the conductivity can be defined as the trajectory-averaged conductance multiplied by the nanowire length. If the nanowire diameter is large enough to neglect a surface effect, the conductance can be divided by the nanowire diameter to get a bulk-like conductivity. To reach the actual bulk limit, the nanowire diameter should exceed the mean free path indeed.
The mobility can be related to the trajectory-averaged conductivity as discussed in the theory section of the tutorial on 'Mobility',
http://docs.quantumwise.com/tutorials/mobility.html.
The theoretical grounds for the described procedure are based on the fact that electron dynamics is much faster than that of ions, i.e., the electrons go through the nanowire faster than ions change their positions, so that the electrons "see" a static lattice of ions at the time scale of electron dynamics. The described approach does not require explicit calculation of the electron-phonon coupling. Note that It accounts for the elastic electron-phonon scattering only. This is perhaps good enough for silicon since it is not a polar semiconductor, i.e., it does not have polar optical phonon modes like in III-V semiconductors.
To avoid direct MD simulations, one may also follow the procedure described in
http://journals.aps.org/prb/abstract/10.1103/PhysRevB.91.220405 and applied to metals, but this approach is limited to harmonic approximation, and would also require an additional effort to be interfaced with ATK, unlike the MD simulator that is already a part of ATK.
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