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

Spatially Resolved Plasmonically Enhanced Photocurrent from Au Nanoparticles on a Si Nanowire

Semiconducting nanowires have been demonstrated as promising light-harvesting units with enhanced absorption compared to bulk films of equivalent volume. However, for small diameter nanowires, the ultrahigh aspect ratio constrains the absorption to be polarization selective by responding primarily to the transverse magnetic (TM) light. While this effect is useful for polarization-sensitive optoelectronic devices, practical light-harvesting applications demand efficient light absorption in both TM and transverse electric (TE) light. In this study, we engineer the polarization sensitivity and the charge carrier generation in a 50 nm Si nanowire by decorating the surface with plasmonic Au nanoparticles. Using scanning photocurrent microscopy (SPCM) with a tunable wavelength laser, we spatially and spectrally resolve the local enhancement in the TE photocurrent resulting from the plasmonic near-field response of individual nanoparticles and the broad-band enhancement due to surface-enhanced absorption. These results provide guidance to the development and the optimization of nanowire–nanoparticle light-harvesting systems

Direct Correlation of Structural Domain Formation with the Metal Insulator Transition in a VO₂ Nanobeam

The electrical resistance of single VO2 nanobeams was measured while simultaneously mapping the domain structure with Raman spectroscopy to investigate the relationship between structural domain formation and the metal−insulator transition. With increasing temperature, the nanobeams transformed from the insulating monoclinic M1 phase to a mixture of the Mott-insulating M2 and metallic rutile phases. Domain fractions were used to extract the temperature dependent resistivity of the M2 phase, which showed an activated behavior consistent with the expected Mott−Hubbard gap. Metallic monoclinic phases were also produced by direct injection of charge into devices, decoupling the Mott metal−insulator transition from the monoclinic to rutile structural phase transition

Ordered Stacking Fault Arrays in Silicon Nanowires

Correlated Raman microscopy and transmission electron microscopy were used to study the ordering of {111} planar defects in individual silicon nanowires. Detailed electron diffraction and polarization-dependent Raman analysis of individual nanowires enabled assessments of the stacking fault distribution, which varied from random to periodic, with the latter giving rise to local domains of 2H and 9R polytypes rather than the 3C diamond cubic structure. Some controversies and inconsistencies concerning earlier reports of polytypes in Si nanowires were resolved

Extraordinary Dynamic Mechanical Response of Vanadium Dioxide Nanowires around the Insulator to Metal Phase Transition

Nanomechanical resonators provide a compelling platform to investigate and exploit phase transitions coupled to mechanical degrees of freedom because resonator frequencies and quality factors are exquisitely sensitive to changes in state, particularly for discontinuous changes accompanying a first-order phase transition. Correlated scanning fiber-optic interferometry and dual-beam Raman spectroscopy were used to investigate mechanical fluctuations of vanadium dioxide (VO2) nanowires across the first order insulator to metal transition. Unusually large and controllable changes in resonator frequency were observed due to the influences of domain wall motion and anomalous phonon softening on the effective modulus. In addition, extraordinary static and dynamic displacements were generated by local strain gradients, suggesting new classes of sensors and nanoelectromechanical devices with programmable discrete outputs as a function of continuous inputs

Origin of Polytype Formation in VLS-Grown Ge Nanowires through Defect Generation and Nanowire Kinking

We propose layer-by-layer growth mechanisms to account for planar defect generation leading to kinked polytype nanowires. C_s-corrected scanning transmission electron microscopy enabled identification of stacking sequences of distinct polytype bands found in kinked nanowires, and Raman spectroscopy was used to distinguish polytype nanowires from twinned nanowires containing only the 3C diamond cubic phase. The faceting and atomic-scale defect structures of twinned 3C are compared with those of polytype nanowires to develop a common model linking nucleation pinning to nanowire morphology and phase

Extraordinary Dynamic Mechanical Response of Vanadium Dioxide Nanowires around the Insulator to Metal Phase Transition

Nanomechanical resonators provide a compelling platform to investigate and exploit phase transitions coupled to mechanical degrees of freedom because resonator frequencies and quality factors are exquisitely sensitive to changes in state, particularly for discontinuous changes accompanying a first-order phase transition. Correlated scanning fiber-optic interferometry and dual-beam Raman spectroscopy were used to investigate mechanical fluctuations of vanadium dioxide (VO₂) nanowires across the first order insulator to metal transition. Unusually large and controllable changes in resonator frequency were observed due to the influences of domain wall motion and anomalous phonon softening on the effective modulus. In addition, extraordinary static and dynamic displacements were generated by local strain gradients, suggesting new classes of sensors and nanoelectromechanical devices with programmable discrete outputs as a function of continuous inputs

Edge and Interface Resistances Create Distinct Trade-Offs When Optimizing the Microstructure of Printed van der Waals Thin-Film Transistors

Inks based on two-dimensional (2D) materials could be used to tune the properties of printed electronics while maintaining compatibility with scalable manufacturing processes. However, a very wide range of performances have been reported in printed thin-film transistors in which the 2D channel material exhibits considerable variation in microstructure. The lack of quantitative physics-based relationships between film microstructure and transistor performance limits the codesign of exfoliation, sorting, and printing processes to inefficient empirical approaches. To rationally guide the development of 2D inks and related processing, we report a gate-dependent resistor network model that establishes distinct microstructure-performance relationships created by near-edge and intersheet resistances in printed van der Waals thin-film transistors. The model is calibrated by analyzing electrical output characteristics of model transistors consisting of overlapping 2D nanosheets with varied thicknesses that are mechanically exfoliated and transferred. Kelvin probe force microscopy analysis on the model transistors leads to the discovery that the nanosheet edges, not the intersheet resistance, limit transport due to their impact on charge carrier depletion and scattering. Our model suggests that when transport in a 2D material network is limited by the near-edge resistance, the optimum nanosheet thickness is dictated by a trade-off between charged impurity screening and gate screening, and the film mobilities are more sensitive to variations in printed nanosheet density. Removal of edge states can enable the realization of higher mobilities with thinner nanosheets due to reduced junction resistances and reduced gate screening. Our analysis of the influence of nanosheet edges on the effective film mobility not only examines the prospects of extant exfoliation methods to achieve the optimum microstructure but also provides important perspectives on processes that are essential to maximizing printed film performance

Optical Control of Mechanical Mode-Coupling within a MoS₂ Resonator in the Strong-Coupling Regime

Two-dimensional (2-D) materials including graphene and transition metal dichalcogenides (TMDs) are an exciting platform for ultrasensitive force and displacement detection in which the strong light–matter coupling is exploited in the optical control of nanomechanical motion. Here we report the optical excitation and displacement detection of a ∼ 3 nm thick MoS₂ resonator in the strong-coupling regime, which has not previously been achieved in 2-D materials. Mechanical mode frequencies can be tuned by more than 12% by optical heating, and they exhibit avoided crossings indicative of strong intermode coupling. When the membrane is optically excited at the frequency difference between vibrational modes, normal mode splitting is observed, and the intermode energy exchange rate exceeds the mode decay rate by a factor of 15. Finite element and analytical modeling quantifies the extent of mode softening necessary to control intermode energy exchange in the strong coupling regime

Impact of Dopant Compensation on Graded <i>p</i>–<i>n</i> Junctions in Si Nanowires

The modulation between different doping species required to produce a diode in VLS-grown nanowires (NWs) yields a complex doping profile, both axially and radially, and a gradual junction at the interface. We present a detailed analysis of the dopant distribution around the junction. By combining surface potential measurements, performed by KPFM, with finite element simulations, we show that the highly doped (5 × 10¹⁹ cm^–3) shell surrounding the NW can screen the junction’s built in voltage at shell thickness as low as 3 nm. By comparing NWs with high and low doping contrast at the junction, we show that dopant compensation dramatically decreases the electrostatic width of the junction and results in relatively low leakage currents