Nanoelectronic Modeling Lecture 10: Open 1D Systems - Transmission through & over 1 Barrier
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Tunneling and interference are critical in the understanding of quantum mechanical systems. The 1D time independent Schrödinger equation can be easily solved analytically in a scattering matrix approach for a system of a single potential barrier. The solution is obtained by matching wavefunction values and derivatives at the two interfaces in the spatial domain. Classical particles would be completely reflected if their kinetic energy does not exceed the potential barrier and would be completely transmitted if their kinetic energy suffices to overcome the potential barrier. Quantum mechanical waves, however can tunnel through barriers even though their kinetic energy does not exceed the potential barrier. The rate or probability of tunneling depends exponentially on the barrier height and thickness. Even if the kinetic energy of the quantum mechanical wave exceeds the potential barrier energy, it “feels” the perturbation in the potential landscape and waves are not necessarily completely transmitted, but can be reflected depending on their energy. The two subsequent interfaces at the beginning and the end of the barrier set up an interference between partial waves reflected at these potential variations. Quasi-bound states or resonance states are formed above the barrier which allow resonant transmission of unit value at resonant energies. For other energies the waves suffer from partial reflection.
This simple example shows the extended nature of wavefunctions, the non-local effects of local potential variations, the formation of resonant states through interference, and quantum mechanical tunneling in its simplest form.
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Università di Pisa, Pisa, Italy