Mathematical Model of Phonon-Dependent Electron Tunneling Transport in Nanosystems

A mathematical model is proposed that describes the tunneling of electrons in nanosystems, taking into account the interaction with a phonon subsystem, which acts as a dissipative environment influencing the tunneling process. This model allows for the detailed analysis of quantum transport phenomena in nanostructures by incorporating the effects of energy exchange and decoherence caused by phonon interactions. The implementation of the mathematical model is performed through the construction of a corresponding difference scheme, which ensures numerical stability and accuracy during simulation. The developed approach demonstrates a good agreement with theoretical expectations and is suitable for modeling electron tunneling in nanosystems within relevant temporal and spatial regimes. The model also enables a quantitative assessment of how the phonon system affects electron transport by calculating the transparency coefficients of the nanosystem under various conditions. Numerical simulations have been conducted, showing how different modes of phonon system influence the efficiency and transparency coefficient of electron tunneling, offering insights into the control of quantum processes in nanostructured materials.