Resonant tunneling and Fano resonance in quantum dots with electron-phonon interaction

We theoretically study the resonant tunneling and Fano resonance in quantum dots with electron-phonon (e-ph) interaction. We examine the bias-voltage (V) dependence of the decoherence, using the Keldysh Green function method and perturbation with respect to the e-ph interaction. With optical phonons of energy ω0, only the elastic process takes place when eV< ω0, in which electrons emit and absorb phonons virtually. The process suppresses the resonant amplitude. When eV> ω0, the inelastic process is possible, which is accompanied by real emission of phonons. It results in the dephasing and broadens the resonant width. The bias-voltage dependence of the decoherence is not easy to obtain by the canonical transformation method of the e-ph interaction with neglecting the renormalization of tunnel coupling. With acoustic phonons, the asymmetric shape of the Fano resonance grows like a symmetric one as the bias voltage increases, in qualitative accordance with experimental results.