Kemnitz’ conjecture revisited

AbstractA conjecture of Kemnitz remained open for some 20 years: each sequence of 4n-3 lattice points in the plane has a subsequence of length n whose centroid is a lattice point. It was solved independently by Reiher and di Fiore in the autumn of 2003. A refined and more general version of Kemnitz’ conjecture is proved in this note. The main result is about sequences of lengths between 3p-2 and 4p-3 in the additive group of integer pairs modulo p, for the essential case of an odd prime p. We derive structural information related to their zero sums, implying a variant of the original conjecture for each of the lengths mentioned. The approach is combinatorial

Light-Emitting GaAs Nanowires on a Flexible Substrate

Semiconductor nanowire-based devices are among the most promising structures used to meet the current challenges of electronics, optics and photonics. Due to their high surface-to-volume ratio and excellent optical and electrical properties, devices with low power, high efficiency and high density can be created. This is of major importance for environmental issues and economic impact. Semiconductor nanowires have been used to fabricate high performance devices, including detectors, solar cells and transistors. Here, we demonstrate a technique for transferring large-area nanowire arrays to flexible substrates while retaining their excellent quantum efficiency in emission. Starting with a defect-free self-catalyzed molecular beam epitaxy (MBE) sample grown on a Si substrate, GaAs core–shell nanowires are embedded in a dielectric, removed by reactive ion etching and transferred to a plastic substrate. The original structural and optical properties, including the vertical orientation, of the nanowires are retained in the final plastic substrate structure. Nanowire emission is observed for all stages of the fabrication process, with a higher emission intensity observed for the final transferred structure, consistent with a reduction in nonradiative recombination via the modification of surface states. This transfer process could form the first critical step in the development of flexible nanowire-based light-emitting devices

Mobility Enhancement by Sb-mediated Minimisation of Stacking Fault Density in InAs Nanowires Grown on Silicon

We report the growth of InAs_1–<i>x</i>Sb_<i>x</i> nanowires (0 ≤ <i>x</i> ≤ 0.15) grown by catalyst-free molecular beam epitaxy on silicon (111) substrates. We observed a sharp decrease of stacking fault density in the InAs_1–<i>x</i>Sb_<i>x</i> nanowire crystal structure with increasing antimony content. This decrease leads to a significant increase in the field-effect mobility, this being more than three times greater at room temperature for InAs_0.85Sb_0.15 nanowires than InAs nanowires