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American Journal of Physics

Man-made quantum wells: A new perspective on the finite square-well problem

Robert M. Kolbas, N. Holonyak

Published May 1, 1984• 86 citations
Publisher

Abstract

The observation of man-made quantum size effects is now possible due to advances in semiconductor technology. Interest in quantum-well heterostructures and superlattices has added a new dimension to both the application and perspective of the traditional discussion of a particle in a box. This paper addresses these new perspectives and presents in an elementary manner the techniques employed to model single and multiple square wells in semiconductors. A brief overview of semiconductor microlayer devices is followed by a discussion of the physics of quantum-well heterostructures. Since the behavior of these structures is described by the finite square well, a simple iterative technique to solve for the bound states of single or multiple (coupled) square wells is presented. The iterative technique is quite general and powerful, yet the physical ideas and the mathematics are straightforward and simple. Finally, computer simulation is employed to illustrate the interaction between multiple coupled wells; the results are compared with those obtained via the usual Kronig–Penney model.

DOI
10.1119/1.13649

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The observation of man-made quantum size effects is now possible due to advances in semiconductor technology. Interest in quantum-well heterostructures and superlattices has added a new dimension to both the application and perspective of the traditional discussion of a particle in a box. This paper addresses these new perspectives and presents in an elementary manner the techniques employed to model single and multiple square wells in semiconductors. A brief overview of semiconductor microlayer devices is followed by a discussion of the physics of quantum-well heterostructures. Since the behavior of these structures is described by the finite square well, a simple iterative technique to solve for the bound states of single or multiple (coupled) square wells is presented. The iterative technique is quite general and powerful, yet the physical ideas and the mathematics are straightforward and simple. Finally, computer simulation is employed to illustrate the interaction between multiple coupled wells; the results are compared with those obtained via the usual Kronig–Penney model.
Nanotechnologically fabricated structures out of compound semiconductor materials make up the functional parts in LEDs, in semiconductor lasers and in electronics components. The small size of the structures evidently leads to requiring the modeling to look at phenomena at the quantum level. Especially for optoelectronical devices, one needs to treat the interaction between light and semiconductor material. This research adds to the understanding of inter-pretations of light-matter interaction phenomena and makes it possible to find totally new and undiscovered functionalities in semiconductor nanostructures. \nThis thesis presents work on modeling and creates new theoretical foundations for phenom-ena appearing in semiconductor quantum structures. Of special interest has been to generate a theoretical description to such phenomena that have been caused by an optical field and in which one sees changes, e.g., in angular momentum or spatial distribution of excited states. Basically, two kinds of structures are examined, quantum rings and wells. A quantum ring is a toroidal object of which volume is very small, but its diameter can be significant when com-pared to the wavelength of light. Another class of systems under study is a double quantum well structure, where there are two quantum wells on both sides of a tunnel barrier. In this system, the type-I and type-II semiconductor band-structure properties become combined in spatial coordinates, since the electrons and holes can occupy either the same or the separate quantum wells. \nThe two most interesting results are the angle-of-emission dependent photoluminescence from the quantum rings and the coherent control of vertical transport of desired quasiparticles through material interfaces in the quantum-well system in a selective manner. Since the ob-tained results on quantum rings are closely connected to the orbital-angular-momentum coupling between light and matter, they can prove to be important in different quantum infor-mation schemes where this angular-momentum aspect of light has been found to be highly beneficial. In quantum optics, there already are visible trends towards this direction. The predicted coherent control in a double-quantum-well system may mean a large technological progress, since by utilizing this effect one can study transport properties between materials over an interface, one of the most interesting examples of such phenomena being the transfer of quantum correlations through the interface without changing the local densities of electrons.
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Examining and controlling the interaction between semiconductor quantum qubits and their environment can boost semiconductor quantum technologies, which have many applications in table-top quantum computing hardware. Electron beams in electron microscopes have opened up a new avenue for the quantum-coherent probing of semiconductor excitations and strong-coupling effects. Here, I provide a brief overview of recent advancements in electron-beam probes for investigating quantum coherence in semiconductors and two-dimensional materials, complemented by my perspective on using electron beams to manipulate the entanglement and correlations between quantum systems.
The ground-state energies and wave functions of a particle moving in one-dimensional potentials of the form V(x)=x2p, p=1,2,..., are obtained nearly exactly by iterative solution of the Schrödinger equation. These solutions are used as the ‘‘experimental benchmark’’ against which approximate solutions for the general case are tested. Both finite depth wells and infinitely deep wells are studied, as is the three-dimensional version (spherical symmetry, angular momentum quantum number l=0). The approach is aimed at student exploration of the physics of the family of potential wells, with emphasis upon connecting the familiar extreme cases, V(x)=x2 (simple harmonic oscillator) and the limiting case xk, k→∞ (square well).
The celebrated Kronig-Penney model traditionally has been formulated with square well potentials representing atomic centres. Here, we use a slightly more realistic potential, the truncated harmonic oscillator, in lieu of square well potentials, and solve the model analytically. We derive the energy dispersion and wave functions for this model. This configuration has some important similarities and differences compared to the usual model. In particular, we write the governing equation in a form suggestive of the tight-binding approximation, as can be done for the usual model. In this way, it is straightforward to derive an expression for the tunneling amplitude used in tight-binding in terms of the harmonic oscillator potential parameters.