Non-Hermitian scattering theory: Resonant tunneling probability amplitude in a quantum dot

Quantum dots play an important role in nanoscaled electronic devices. In order to fully understand the transport properties of quantum dots, phenomena such as tunneling of electrons must be characterized. Phase measurements of an electron traversing a quantum dot via a double-slit interference experiment were carried out by Heiblum and co-workers. In their experiment, the quantum dot ~QD! was inserted into one slit, in a manner that enabled them to control the potential of the electrons trapped in it ~by varying the plunger potential, Vp). The second slit served as a reference. The measured transition probability, utu, oscillated as a function of the plunger potential, Vp . When the scattered electron passed through a resonance state, the transition probability exhibited a Lorentzian-shaped peak, and the phase of the transition-probability amplitude changed by p , as expected by the Breit-Wigner model for resonant tunneling. Surprisingly, the phase does not accumulate but oscillates: although in each resonance the phase changes by p , between resonances it drops sharply by p . In spite of intense work on this problem to the best of our knowledge there is no satisfactory explanation of this phenomenon. Most of the calculations which have been carried out are either for particle space model Hamiltonians or for general mathematical models based on the Friedel sum rule combined with time-reversal symmetry. The conclusion from these studies is that in true one-dimensional ~1D! systems the sharp phase drops by p in the tail of the resonant peak that never occurs. Such a phenomenon is obtained, however, for single electron transport for quasi-1D systems. This result was obtained also in simulation calculations for a real-space model Hamiltonian with a 2D box potential ~single-crossbar cavity!. The discontinuity in the phase evolution of electron transport in the single-crossbar cavity is associated with the interference between two different transmission channels belonging to a localized state in the 2D potential and a continuous state of the transmission channel. An open question we would like to address is whether the discontinuity in the phase evolution would happen also for analytical 2D potential surfaces. Another question we would like to answer is whether the discontinuity in the phase evolution can happen due to another mechanism. As we will show here for analytical 2D potentials a sharp drop ~not a discontinuity! by p happens due to the interference between adjacent resonance