Resonant Tunneling in a Symmetrical Rectangular Quadruple‐Barrier Structure

Abstract

The resonant tunneling phenomena in double-barrier structures have been studied both theoretically and experimentally by many researchers in these twenty years /1 to 7 1 . However, few theoretical studies have been made for the quadruple-barrier structure / 8 / . Especially, it seems that a study of the resonant tunneling for the symmetrical quadruple-barrier structure has not been carried out yet, though an interesting and useful feature for the resonant tunneling may be veiled. Actually, some interesting feature has been found in this structure. In this note, the analytical expressions for the transmission coefficient and the resonance condition are derived for the first time. It is believed that they are useful for evaluating the energy variation of the transmission coefficient and estimating the resonance energies in the symmetrical rectangular quadruple-barrier structure, and will provide an important foundation for device fabrication, which will not be obtained in the equivalent quadruple-barrier structure or in the double-barrier structure.The model potential to be studied is a symmetrical one-dimensional rectangular quadruple-barrier structure as depicted in Fig . . . , and x8). The energy origin is taken at the conaucnon Dana eage of the region 1 material. It is assumed that an electron with energy E is incident from the left and transmits to the right along the x-direction. The electron mass is assumed to be constant m* over the entire region of the structure. By matching the wave functions and their first derivatives at the boundary we can derive the transmission coefficient T4S for 0 < E < Vo after a straightforward algebra:

Quasi‐resonance and confining phenomena in symmetrical rectangular quadruple‐barrier structures

“…It is confirmed that the variation of each resonance level is not so sensitive to changes of the outer barrier width L b l and is negligible, although the values of the transmission coefficient in the off-resonance energy range become smaller and the transmission spectrum becomes sharper with increasing outer barrier width. It leads to the result that the two independent resonance conditions (9) and (10) are almost retained even if the outer barrier width is not exactly equal to the inner one as in [9]. This result may be useful for new device fabrication.…”

“…For Lbl = Lbl, the probability density at the resonance energy obtained by the resonance condition (9) is invariant for various central well widths as shown Fig. 4b, because (9) gives the resonance condition for the double-barrier structure containing the outer well. It is seen for Lbl < Lba that the amplitude of the probability density is decreased as shown in Fig.…”

“…In the second-simplest structure containing at least one double-barrier structure, i.e., in the symmetrical quadruplebarrier structure with four identical barriers, there is a very interesting and useful feature of resonant tunneling: two independent resonance conditions exist [9].…”

Resonance Conditions and Independent Resonance Energy Selection in Asymmetrical Rectangular Triple-Barrier Structures

“…However, the well width decides definitely the resonance levels in the symmetrical triple-barrier as well as in the double-barrier structures: If one resonance level is selected, the other levels are decided according to the resonance condition with well width. On the other hand, the quadruple-barrier structures follow two independent resonance conditions and two independent resonance levels simultaneously [6].…”