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 Keldysh Green function method and perturbation with respect to the e-ph interaction. With optical phonons of energy $\omega_0$, only the elastic process takes place when $eV<\omega_0$, in which electrons emit and absorb phonons virtually. The process suppresses the resonant amplitude. When $eV>\omega_0$, 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 cannot be obtained by the canonical transformation method to consider the e-ph interaction if its effect on the tunnel coupling is neglected. 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.