On the Quantum Confinement of the Exciton

April 26, 2015 - PRLog -- .
Introduction
Today, the quantum confinement of the exciton to the Bohr diameter forms the basis for much of the theory governing the optical phenomenon observed in nanostructures. Common nanostructures [1] are the QD and QW corresponding to the quantum dot and wire.  Recently, QCs were observed [2,3] upon irradiating a nickel coated surface of silicon. QCs stand for quantum cones depicted in the AFM image shown in the thumbnail.

The QCs are thought [2] formed by a two-stage mechanism. First, melting of the thin nickel film upon irradiation by a laser beam is followed by formation of nickel islands by surface tension. The second stage consists of melting the structure and mass transfer along the interface between silicon and nickel due to surface tension gradient, the so-called Marangoni effect.

The optical phenomenon observed in the PL spectrum upon forming the QCs suggests [2] the exciton (electron and hole pair) undergoes quantum confinement provided the Bohr diameter fits inside the nanostructure. PL stands for photoluminescence. Since the QC diameter decreases from the substrate to the tip, the QC may be considered a graded band gap device. The thumbnail shows a QC with a gradually increasing band gap Eg in terms of the Bohr diameter d depending on the height z from the substrate. The band gap Eg is given, Eg = h²ζ²m*/(2π²d²), where h = Planck’s constant, ζ = 2.4048, and m* is given by 1 / (m*) =1 / (me*) + 1 / (mh*). Here, me* and mh* are the effective electron and hole masses.

Problem
Excitons do form in the QCs, but are not directly caused by laser irradiation. In this regard, there may be an alternative explanation for QC optical emission.

Alternative
Laser irradiation provides a source of EM energy that is first absorbed by the nickel coated silicon surface. EM stands for electromagnetic. The excitons do form in the silicon, but only after QED induces the absorbed EM energy to create openings in the nickel layer. QED stands for quantum electrodynamics. Only then does melted silicon flowing through the openings form the QCs. Further laser irradiation creates the excitons in the QCs.

QED is based on the QM interpretation of the atom as a harmonic oscillator that precludes the atoms at the nanoscale from having the heat capacity to conserve EM energy by an increase in temperature. QM stands for quantum mechanics. See numerous QED applications at http://www.nanoqed.org, 2010 – 2015.

Theory
The QED induced EM radiation is confined almost totally in the surface of the QCs corresponding to their TIR mode. TIR stands for total internal reflection. What this means is TIR confines almost all of the absorbed EM energy to the QC surface because like QDs and QWs, QCs have high surface to volume ratios. The QED photons created in the QC surfaces have TIR wavelength λ = πd and frequency υ = (c/n)/λ, where c is the velocity of light and n the refractive index of QC. The QCs therefore emit EM radiation having Planck energy E = hυ = hc/πnd. The Planck energy E is sufficient to create excitons in silicon that upon recombination produce the observed QC optical phenomenon.

Mechanism
Only in the latter stages does QC formation proceed by melting of silicon as nickel islands first need to form on the silicon. Initially, the absorbed laser radiation charges the nickel atoms by removing electrons to place the nickel layer under Coulomb repulsion, the compression of which lifts the nickel off the silicon to form the islands. Over this time, QM precludes temperature increases in the nickel layer because the interface with the silicon opens to form a vacuum and allow the layer to be surrounded by a lower refractive index - a necessary condition for the TIR confinement of the nickel layer. Once Coulomb repulsion produces openings in the nickel islands, the underlying silicon melts and flows through the openings to form the QCs..

Analysis
Exciton confinement in QCs based on the Bohr diameter theory [1] developed for the QD and QW gives vanishingly small diameters at optical frequencies. Indeed, Fig. 1 (b) of [2] shows the peak of the PL response ( E = 2.9 eV and λ = 430 nm) requires a Bohr diameter d < 1 nm. The AFM image above generally show larger QC diameters, say 150 nm at the base with a height of 50 nm. Hence, PL emission occurs at the very tip of the QCs by the Bohr theory.

In contrast, QED theory requires TIR confinement at the QC surface. Assuming a refractive index n = 4 for silicon, the peak PL emission at λ = 430 nm therefore occurs at a QC diameter d = 34 nm, or at about 11 nm from the QC tip and 39 nm above the base.

Conclusions
The formation of excitons in QCs requires EM radiation that is produced upon the QED conversion of the laser energy absorbed in the surface of the QC, the recombination of the excitons producing the optical phenomenon in the observed PL spectrum.

Quantum confinement by Bohr theory suggests peak PL spectrum emission occurs at the very tip of the QC. In contrast, QED theory places the peak PL further away from the tip that appears more consistent with observed QC emissions.

Smaller QC diameters are expected for Bohr confinement of the exciton  compared to QED because the electron has far lower energy than the photon under TIR confinement. Moreover, QED confinement constrains absorbed EM energy to the well defined surface of the nanostructure, a condition absent in the nebulous confinement of the electron in the Bohr theory of the exciton.

QED confinement by TIR depends on the high surface to volume ratios of nanostructures; whereas, exciton confinement by Bohr theory does not.

References
[1] J. Li  & L.-W. Wang, "Comparison between quantum confinement effect of
quantum wires and dots," Chem Mater 6,4012–4015, 2004.

[2] A. Medvid, et al., "Formation mechanisms of nano and microcones
by laser radiation on surfaces of Si, Ge, and SiGe crystals, Nanoscale Research Letters, 8, 264, 2013.

[3] A. Medvid, et al., "Exciton quantum confinement effect in nanostructures
formed by laser radiation on the surface of CdZnTe ternary compound,"
Phys Status Solidi C, 6, 209–212, 2009.