Resonant tunneling of electrons interacting with phonons

The effect of the electron-phonon interaction on resonant tunneling of electrons through a two-barrier nanostructure is investigated in the framework of a consistent quantum-mechanical model. The wave function is determined by solving the Schrödinger equation with correct boundary conditions in the semiclassical approximation in the electron-phonon interaction. The current calculated with the help of the wave function is averaged over the phonon subsystem with the help of the Bloch theorem. The analytic expressions derived for static and varying currents in a resonance tunnel diode taking into account the electron-phonon interaction formally coincide with the Mössbauer effect probability. In the adiabatic limit and for a strong electron-phonon interaction, the static current decreases in proportion to η, while the varying low-frequency current is proportional to η². The shape of the resonance curve becomes Gaussian with a width of τ _ph ^?1 . The fundamental result is that the properties inherent in coherent tunneling are preserved even in the limit η?1 (which is often regarded as incoherent). The most striking effect (analogous to the Mössbauer effect) is the conservation of a narrow Lorentzian resonance curve in the limit η?1, ω_ph?Γ. This means that even for η?1, the resonance current is due to coherent electrons (experiencing interference), but their fraction decreases in view of the electron-phonon interaction. It is concluded that the application of the rate equations and other approximate methods disregarding interference may lead to incorrect results. The expressions for the high-frequency and nonlinear responses are also derived. The quantum-mechanical regime is found to be less sensitive to the effect of phonons than the classical regime.