Abstract

1. Novel Approach to a Generalized Theory on Nanostructure DevicesA.C. Foster and W.D. BrandonThe University of North Carolina @ Pembroke Basic Theory Abstract Figure 2:Although the energy levels of a QD are discrete, they are much closer to continuous than a molecule. HOMO: highest occupied molecular orbital. LUMO: lowest unoccupied molecular orbital Nanoscience comprises a significant area in science's immediate frontier. Because of its prospects and its relevance to a wide variety of research fields, it is important to engage this area head-on, and from a variety of perspectives. In this investigation, some fundamental, salient properties of modern quantum devices (i.e. quantum dots, wells and wires) will be expounded upon. From there it will be shown that a generalized, yet very simple, model may be constructed without straying far beyond the rudimentary assertions made in classical rotational dynamics. This integrative approach is not only simple, but cogent enough to provoke deeper and purposefully unconventional thinking for both physics and chemistry majors. 1D-confined quantum wells, 2D-confined quantum wires, and 3D-confined quantum dots will ultimately be described mathematically, resulting in a basic framework with which to consider nanoscale devices. It can be observed from fig. 1 that in order for the aforementioned boundary condition to be satisfied an integer number of half-wavelengths must be within length l. That is: Where n = 1,2,3,etc. We can express the momentum of the particle as: Introduction Novel, Simple and Generalized Nanoscale Theory As a new type of material, semiconductor quantum dots (QDs) represent a borderline between chemistry and physics, exhibiting both molecular and bulk-like properties. Similar to atoms and molecules, the electronic spectra of QD show discrete bands- but they also exhibit conduction and valence bands as in bulk semiconductors. The electron–phonon coupling in QDs is weaker than in molecules, but stronger than in bulk semiconductors. Unlike either material, the QD properties can be tuned continuously by changing QD size and shape utilizing various synthesis techniques. The molecular and bulk points of view often lead to contradicting conclusions. For example, the molecular view suggests that the excitations in QDs should exhibit strong electron-correlation (excitonic) effects. In contrast, a finite-size limit of bulk properties indicates that the kinetic energy of quantum confinement should be significantly greater than excitonic effects. This and other qualitative differences have generated some controversy in the literature. The great potential of QDs for a variety of applications, including photovoltaics, spintronics, lasers, light-emitting diodes, and field-effect transistors makes it critical to settle these controversies. By integrating different viewpoints and presenting a simple unified atomistic picture of QDs, our semi-classical analysis shows some attractive features which we hope to address in future experimental investigations. Associating the QD confinement energy, Ec (i.e. the particle in a box) with the usual particle kinetic energy, (p^2)/2m it follows that A central feature of nanoscale physics is that electrons must occupy quantized energy levels. Resolution of these distinct energy states requires that the level spacing, DE must exceed any thermal noise present according to: wherekis Boltzman’a constant and Tis temperature. The energy difference (level spacing) at the Fermi energy for a box of size l, depends on the relevant space dimensions. Including spin degeneracy (electron interaction) we have the well-known result: These expressions derived using the de Broglie argument are quite applicable to the quantum dot. The quantum dot can be thought of as a 3-D particle in a sphere. It turns out the solution to the 3-D case is the same as the 1-D particle in a box, only the radius of the sphere plays the role of l. 1-D In the case of the quantum dot, electrons, which are typically in the valence band may gain enough energy to enter the conduction band. When this happens a hole is created. A hole is a “quasi-particle”, or that which represents the absence of an electron. In modeling quantum systems this is quite convenient and accurate. The electron-hole system is then modeled as a single particle, an exciton (an “emergent particle”). The excitonic energy is the sum of the confinement energies of both the electron and hole: 2-D 3-D Basic Theory: The de Broglie Argument We may circumvent mathematical complexities and the ad hoc theoretical nature in developing an appropriate Hamiltonian typical of various approaches taken in modeling nanoscale phenomena imposed from the requirement of the invariance of spatial translations, by incorporating generalized angular momentum and proceeding directly from a rotational basis according to: Louis de Broglie, in his 1924 PhD thesis, proposed the particle-wave duality of matter. This was an intuitive extension of Einstein‘s 1905 paper which postulated the particle-like nature, in addition to the accepted electromagnetic wave nature, of light in order to explain the photoelectric effect. This dualistic interpretation of matter and light, along with subsequent theoretical advances, has been overwhelmingly consistent with experiment. The de Broglie relation states: The exciton mass is defined in terms of electron mass, me and hole mass, mh as the system’s reduced mass: So that, Here we invoke a direct angular momentum quantization rule and kis a constant reflecting the effective mass and charge distribution of the exciton, which follows from the classical definition of moment of inertia, I=k ml2. This line of reasoning has a direct classical basis and it follows that the scaling parameter is: That is, the wavelength of a particle is the quotient of Planck's constant and the particles momentum. It is now quite important to ask “what is waving?” The generally accepted answer is that the probability of finding the particle is “waving” across space and time. This argument can now quite easily be used to find the possible momentums of a particle trapped in a one-dimensional “box” with length l. Since the probability of finding the particle outside of the box is zero, the wave function must be equal to zero at the bounds of the box. Assuming the wave function to be sinusoidal, possible wave functions include: The energy gained by a photon would be equal to the sum of the energy of the exciton before electron-hole recombination occurred and the band gap energy. The band gap energy is included because the exciton, by definition, exists in the conduction band. Additional energy is lost as the system returns to the valence band, the amount of which is the bandgap energy. In 1-D, 2-D and 3-D respectively. Note that, in addition, this approach closely follows the idealized expression for the exciton energy, Eexc ψ Difficulties of QD Theory The difficulty of accurately modeling the behavior of quantum dots is essentially because their scale lies in a range which is, at least for now, very grey for modern physics. They are the medium between, for instance, individual molecules and an infinite lattice. Quantum dots play by rules between quantum mechanics and bulk quantum mechanics. This issue is a recurring one which is the fundamental plague of modern physics; the behavior of the universe seems inconsistent across scale. Ultimately, a satisfactory theory of quantum dots will be a dialectic between theories of adjacent scales. Fig. 2 illustrates this conundrum. with k= 1 ; which corresponds classically to a point mass, or equivalently a thin ring of mass, rotating about a central position at distance l . Classically, this value reflects the equipartition of translational and rotational energies. x-axis x= l x = 0 References Semiconductor Physics and Applications; Minko Balkanski, Richard Fisher Wallis, Oxford University Press, 2000 Fig. 1: The first three possible wave functions for a 1-D particle in a box.