Influence of the shape and size of a quantum struture on its energy levels
Abstract
In this study the importance of the luminescent properties of lowdimensional quantum structures are investigated focusing on the change in the exciton binding energy with a change in the size of the low dimensional QuantumWell or Wire. With a reduction in dimensionality, moving from bulk semiconductor materials through Quantum Wells, Wires and ultimately Quantum Dots, the band structure as well as the density of states for these lowdimensional structures change appreciably going from quasicontinuous in bulk semiconductors to discrete in Quantum Dots. This leads to an increase in the energy gap (compared to the bulk material), with a decrease in size for a lowdimensional structure. An interacting electronhole pair in a Quantum WellWire is studied within the framework of the EffectiveMass Approximation. A mathematical technique is presented which investigates the quasitwodimensional, quasionedimensional behavior of a confined exciton inside a semiconductor as the bulk material is reduced in dimensions to form a Quantum Well and Wire. The technique is applied to an infinite WellWire confining potential. The Envelope Function Approximation is employed in the approach, involving a three parameter variational calculation in which the symmetry of the component of the wave function representing the relative motion is allowed to vary from the one to the two and threedimensional limits. A quasi–twodimensional behavior occurs on reducing the well width as the average electronhole distance decrease leading to an increase in the binding energy. However, when the well width is smaller than a critical value, the leakage of the wave function into the barriers becomes more important and the binding energy is reduced until it reaches the value appropriate to the bulk barrier material for which L = 0. As the electronic industry progress from microtechnologies to nanotechnologies whereby devices are designed in the nanometer range, it becomes increasingly necessary to address the concern of the exciton losing its enhanced effects in the ultra small quantum structures, due to the increased penetration of the exciton wave function into the barrier regions in the direction of diminishing spatial confinement. A trial wave function is employed; written as a product of three wave functions. The first two are corresponding to the single particle wave function of an electron and a hole in the Quantum WellWire and the third represents a free exciton whose radius is adjusted as a variational parameter. This method can be suitably adapted for any particular choice of variational wave function. The choice of this wave function is only limited by the users’ qualitative knowledge of the system under consideration and how this knowledge is imbedded into this trial wave function. Results to this numerical calculation are presented. Quantitative comparisons with previous calculations for quantum wells was made (in the wire limit where Lz → ∞) and it was found that there exists a good agreement between this infinite and other finite as well as infinite  potential models up to a point of 100 Å. A plot of the binding energy vs. the variational parameter λ revealed that the electron in the exciton has a very similar behavior than the electron in the Hydrogen atom (or for that matter any particle trapped inside a radial decreasing (i.e. V~1/r) potential field). However on reducing the size and dimensions of the quantum structure, it seems that the screening of the other electrons surrounding the hole start to play a very important role and the shape of a plot of binding energy versus λ is very similar to that of an alpha particle trapped in an atomic nucleus. It is concluded from this that for accurately predicting the behavior of systems like these it is important to include in such a model not only the different dielectric constants for the barrier and the wellwire materials, but also to include the change in dielectric constant due to a change in size, i.e. ε = ε (L), i.e. to take into account the decrease in the amount of electrons in the valence band due to a decrease in size of the Quantum WellWire.
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