Letters to the Editor

The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to either of the editors-in-chief, Chandler Davis or Marjorie Senechal.     

Quantum Dot Research: Current State and Future Prospects

This article reviews the current state of research involving semiconductor quantum dots, provides a brief review of the theory behind their unique properties, and an introduction explaining the importance of quantum dot research. The characteristic shifting of the band gap energy with quantum dot size, as predicted from the density of states for low-dimensional structures, allows experimental measurements to determine the extent to which quantum confinement effects play a role in the resulting properties. A few of the current techniques used to measure the presence and physical characteristics of quantum dots and their energy levels is reviewed, including transmission electron microscopy, optical transmission, and Raman and photoluminescence spectroscopy. Finally, some of the more exciting applications for quantum dots currently being researched for use in the field of optoelectronics are reviewed, including quantum dot infrared photodetectors, quantum dot lasers, and quantum dot solar cells. Comments are made on the current progress and the future prospects of quantum dot research and device applications.     

Novel perturbation expansion for the Langevin equation

We discuss the randomly driven systemdx/dt= -W(x) +f(t), wheref(t) is a Gaussian random function or stirring force withf(t)f(t)=2(t–t), andW(x) is of the formgx ¹⁺². The parameter is a measure of the nonlinearity of the equation. We show how to obtain the correlation functionsx(t)f(t)···x(t( n)) _f as a power series in. We obtain three terms in the expansion and show how to use Padé approximants to analytically continue the answer in the variable. By using scaling relations, we show how to get a uniform approximation to the equal-time correlation functions valid for allg and.     

Scaling of the energy gap withT c in Bi2Sr2CaCu2O8+δ

The results of a high-resolution photoemission study using synchrotron radiation of two single crystals of Bi₂Sr₂CaCu₂O₈₊, with different critical temperaturesT _c due to a variation in oxygen stoichiometry are reported. Within experimental accuracy, the energy gap 2 is found to scale withT _c , amounting to a reduced gap parameter of 2/k _BT_c7.4. Employing resonant photoemission at the O–2s and Cu–3p thresholds, two spectral peaks at binding energies of 180 meV and 320 meV were identified as predominantly O–2p-and Cu–3d 4s-derived states, respectively.     

The hall effect in some liquid alloys

The Hall coefficient has been measured for (a) liquid CuTe, AgTe and AuTe₂ and (b) liquid Cu-Sn alloy. The results are discussed in terms of the possible semiconducting nature of liquids comprising group (a) and in terms of bound state formation around the Sn atoms for those in group (b).     

Clutter interference and the integration time of echoes in the echolocating bat, Eptesicus fuscus

The ability of the echolocating bat, Eptesicus fuscus, to detect a sonar target is affected by the presence of other targets along the same axis at slightly different ranges. If echoes from one target arrive at about the same delay as echoes from another target, clutter interference occurs and one set of echoes masks the other. Although the bat's sonar emissions and the echoes themselves are 2 to 5 ms long, echoes (of approximately equal sensation levels--around 15 dB SL) only interfere with each other if they arrive within 200 to 400 microseconds of the same arrival time. This figure is an estimate of the integration time of the bat's sonar receiver for echoes. The fine structure of the clutter-interference data reflects the reinforcement and cancellation of echoes according to their time separation. When clutter interference first occurs, the waveforms of test and cluttering echoes already overlap for much of their duration. The masking effect underlying clutter interference appears specifically due to overlap, not between raw echo waveforms, but between the patterns of mechanical excitation created when echoes pass through bandpass filters equivalent to auditory-nerve tuning curves. While the time scale of clutter interference is substantially shorter than the duration of echo waveforms, it still is much longer than the eventual width of a target's range-axis image expressed in terms of echo delay.     

New class of magnetoresistance oscillations: interaction of a two-dimensional electron gas with leaky interface phonons

We report on a new class of magnetoresistance oscillations observed in a high-mobility two-dimensional electron gas (2DEG) in GaAs-Al(x)Ga(1--x)As heterostructures. Appearing in a weak magnetic field ( B < 0.3 T) and only in a narrow temperature range ( 2 K < T < 9 K), these oscillations are periodic in 1/B with a frequency proportional to the electron Fermi wave vector, k(F). We interpret the effect as a magnetophonon resonance of the 2DEG with leaky interface-acoustic phonon modes carrying a wave vector q = 2k(F). Calculations show a few branches of such modes existing on the GaAs-Al(x)Ga(1--)xAs interface, and their velocities are in quantitative agreement with the observation.     

Influence of connectivity and porosity on ligand-based luminescence in zinc metal-organic frameworks

Applications of metal-organic frameworks (MOFs) require close correlation between their structure and function. We describe the preparation and characterization of two zinc MOFs based on a flexible and emissive linker molecule, stilbene, which retains its luminescence within these solid materials. Reaction of trans-4,4'-stilbene dicarboxylic acid and zinc nitrate in N,N-dimethylformamide (DMF) yielded a dense 2-D network, 1, featuring zinc in both octahedral and tetrahedral coordination environments connected by trans-stilbene links. Similar reaction in N,N-diethylformamide (DEF) at higher temperatures resulted in a porous, 3-D framework structure, 2. This framework consists of two interpenetrating cubic lattices, each featuring basic zinc carboxylate vertices joined by trans-stilbene, analogous to the isoreticular MOF (IRMOF) series. We demonstrate that the optical properties of both 1 and 2 correlate with the local ligand environments observed in the crystal structures. Steady-state and time-resolved spectroscopic measurements reveal that the stilbene linkers in the dense structure 1 exhibit a small degree of interchromophore coupling. In contrast, the stilbenoid units in 2 display very little interaction in this low-density 3-D framework, with excitation and emission spectra characteristic of monomeric stilbenes, similar to the dicarboxylic acid in dilute solution. In both cases, the rigidity of the stilbene linker increases upon coordination to the inorganic units through inhibition of torsion about the central ethylene bond, resulting in luminescent crystals with increased emission lifetimes compared to solutions of trans-stilbene. The emission spectrum of 2 is found to depend on the nature of the incorporated solvent molecules, suggesting use of this or related materials in sensor applications.