Hysteresis Effects on Quantum Wires: Do They Exist or Not?

Semiconductor devices, especially those named heterostructures, are very attractive systems to study the quantum behavior of nature. Quantum-mechanical effects are particularly important for designing new classes of devices for specific applications. As a general rule, the final goal is to obtain smaller devices (for best integration) able to operate at higher frequencies for transmission of more information (and lower prices). Three steps are fundamental in the engineering of complex semiconductor devices before testing: to model, to grow, and to process. Heterostructures are complex devices of different semiconductor films for which the typical widths are of the order of some atomic layers. The reality of growing these structures opened new and enthusiastic possibilities in physics (basic and applied), as well as in microand nano-electronics. Results are very clear: direct applications to different fields of science and new hightechnology products. The benefits of these artificial manmade crystals are easy to find anywhere: see for example Refs. [1–5] We want to study complex structures in which carriers (electrons) are restricted to move in one-dimension (1D), different from the usual three-dimensional (3D) bulk materials, due to confinement effects. In particular, we intend to analyze what are called quantum wires (QW), in which only one dimension remains unconfined for the electrons movement. Quantum wires have a great potential for many applications in microelectronics, such as novel opto-electronics devices and transistors. They can be obtained in two manners: either from a growth profile [6] or from a technologic process on a sample [7]. Firstly, we describe the electrostatic potential for a rectangular quantum wire of charge. Then, we introduce the potential term due to accumulation of attracted charges that leads to the self-consistent SchrödingerPoisson system of equations in which we take into account the Fermi energy as constant. Finally, we introduce the exchange correlation energy in our formalism and discuss the physical implications of that term. When the exchange correlation is neglected, no hysteresis effect is observed, even for a very asymmetric wire shape. However, we observe bi-stability when exchange correlation is taken into account, even for absolutely symmetric cases such as square quantum wires. The electrostatic potential for (fixed) positive charges can be obtained, as usual, from the Poisson equation. Analytical solutions are found when a linear uniform distribution of charge along the wire direction is considered. We shall call this potential φdop(x, y), in allusion to the doping concentration fixed by the growing conditions. That term is given by the following expression: