The nonequilibrium Greens function (NEGF) method is often used to predict transport in atomistically resolved nanodevices and yields an immense numerical load when inelastic scattering on phonons is included. To ease this load, this work extends the atomistic mode space approach of Ref. [1] to include inelastic scattering on optical and acoustic phonons in silicon nanowires. This work also includes the exact calculation of the real part of retarded scattering self-energies in the reduced basis representation using the Kramers-Kronig relations. The inclusion of the Kramers-Kronig relation for the real part of the retarded scattering self-energy increases the impact of scattering. Virtually perfect agreement with results of the original representation is achieved with matrix rank reductions of more than 97%. Time-to-solution improvements of more than 200$times$ and peak memory reductions of more than 7$times$ are shown. This allows for the solution of electron transport scattered on phonons in atomically resolved nanowires with cross-sections larger than 5 nm $times$ 5 nm.
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