Tunneling Of Electrons Through Parabolic Quantum Well Structures With Double Barriers

The electron tunneling through parabolic quantum wells with double barriers consisting of Al_xGa_1-xAs material are investigated by taking the influence of magnetic field and electron-phonon interaction into account. The relative problems of tunneling dynamics are discussed.The situation with a transverse magnetic field is more complicated, as the transverse-momentum components keeps no longer conservation in the tunneling process. The transmission coefficients of electrons tunneling through several parabolic quantum wells under a uniform transverse magnetic field have been calculated by using a transfer matrix method under consideration of effective potential. It is seen that the influence of the magnetic field is more significant for the parabolic quantum well structures with larger dimension. Furthermore, it is found that the resonant current peaks shift towards the higher bias position. This indicates that the quasi-bound energy level is raised by the magnetic field. Meanwhile, the J-V characteristics of a parabolic quantum well are discussed for the cyclotron center located at different positions. It is also found that the situation,where the cyclotron center is located at the parabolic well center, is helpful to explain the experimental phenomena. The peak current densities decrease and shift to higher bias with increasing magnetic fields. In order to discuss the tunneling dynamics for parabolic well structures, the time-dependent Schrodinger equation has been numerically solved with using of the transfer matrix method and wave packet theory. The result shows that the probability density of an electron in the well reaches its maximum after a build-up time, and then decays exponentially with time. Furthermore, the dependency of the build-up time on the structure parameters is weaker, but the tunneling lifetime decreases rapidly with increase of the well width and barrier thickness. It can be seen from our results that the build-up time may be longer than the tunneling lifetime (decay constant) for parabolic quantum wells with narrower width. Therefore it is not sufficient that the decay constant is used to describe all of the temporal characteristics of the electron tunneling. By comparing the results of parabolic wells with that for square wells, it is also found that the tunneling lifetime for a parabolic well structure is longer than that for a square well in case of wider well and thinner barriers. It can be explained that the level width of the transmission peak in a parabolic well is smaller than that in a square well structure due to the special shape of parabolic well structure.The tunneling assisted by confined longitudinal optical (LO) phonons in double barrier parabolic wells is discussed based on the Fermi gold rule. It is found that the resonant tunneling currents through the systems peak at higher applied voltage and get lower valley current as well as higher ratio of peak to valley because of the stronger confinement of electron states. The theoretical results indicate that the confined LO-phonon peaks are higher than that for square wells. The additionalcurrent peaks assisted by both of the LO-phonons in the well and in the emitter barrier can be easily observed simultaneously at a larger range of the well widths for parabolic quantum well structures.The influence of applied magnetic fields along the direction of superlattice grown on the phonon assisted tunneling is analyzed both for parabolic and square well structures. The results of numerical computation show that there is a highercurrent ratio of peak to valley for phonon assisted tunneling through parabolic quantum well structures with wider well width. A reasonable explanation is that the thick parabolic part of the collector barrier reduces the valley currents and resonant widths. Our results predict that the confined LO phonon peaks in the parabolic quantum well system can be easily distinguished. Additionally, it is also found from our results that the magnetic field shifts the current peaks of tunneling assisted by phonons towards higher bias, as well as sharpens and heightens these peaks.