Electron States in Biased Heterostructures

Abstract

In this contribution we are mainly interested in the effects of an externally applied electric field on the electronic states of semiconductors quantum wells and superlattices. Such structures are simply viewed as a sequence of different semiconductor layers grown sucessively along a well defined crystalline growth direction. The carriers’ motion along this growth direction (hereafter called the z direction) is then strongly modified by the presence of the various thin (≈100Å) layers, whereas the translation symmetry (at the level of the bulk unitary cells) remains for the in-plane (x,y) directions. Single quantum wells present states which are localized in space around the well region for the z motion. Double quantum wells (DQW) consist of two well layers separated by a finite barrier layer. The localized states of the isolated quantum wells interact through the barrier region (tunnel interaction) to form the DQW eigenstates. The resulting new states are delocalized over the DQW region. For example, two identical wells (same thickness and material composition) give rise to symmetrical and antisymmetrical z-dependent DQW wavefunctions (with respect to the center of the central barrier). For wells of different thicknesses (asymmetrical DQW) the tunnel coupling is less efficient and each resulting DQW state presents a preferential localization around one of the two wells, reminiscent of the localized states of the isolated wells. In other words, the tunnel coupling is less effective to mix states of distinct quantum wells wich are misaligned in energy. The application of an external electric field F//z acts on the DQW states as follows. The electrostatic potential eFz (we take its origin at the center of the intermediate barrier) raises (lowers) in energy the eigenstates of the right (left) well. Thus, by increasing the bias one is capable to externally align or bring off resonance the different states pertaining to the separated wells, and thus to force a localization or delocalization of the DQW eigenstates. Semiconductor superlattices (SL) consist of a periodic repetition of a fundamental cell composed by a sequence of wells and barriers. The simplest one consists of a well and a barrier (with eventually different thicknesses). The states of the isolated thin wells (with only one electron bound state) are all at the same energy at zero bias. The tunnel coupling lifts this (infinite) degeneracy and gives rise to an electron miniband with a finite energy width. The Bloch states are completely delocalized. In the presence of an electric field the energy levels of neighbouring wells are not at the same energy. Then, like for DQW, this energy misalignement inhibits the inter well coupling and the eigenstates of the biased SL are no longer completely delocalized. We shall show that they become localized within a Δz region which decreases continuously with increasing electric field until the wavefunction concentrates around one SL period (0394z ≈ well width if the electrostatic potential drop within a SL period becomes much greater than the tunnel coupling). This increasing spatial localization of the states with increasing applied electric field is counterintuitive and corresponds to one of the most striking results for superlattices.