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

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

Nanowire Kinking Modulates Doping Profiles by Reshaping the Liquid–Solid Growth Interface

Dopants modify the electronic properties of semiconductors, including their susceptibility to etching. In semiconductor nanowires doped during growth by the vapor–liquid–solid (VLS) process, it has been shown that nanofaceting of the liquid–solid growth interface influences strongly the radial distribution of dopants. Hence, the combination of facet-dependent doping and dopant selective etching provides a means to tune simultaneously the electronic properties and morphologies of nanowires. Using atom-probe tomography, we investigated the boron dopant distribution in Au catalyzed VLS grown silicon nanowires, which regularly kink between equivalent ⟨112⟩ directions. Segments alternate between radially uniform and nonuniform doping profiles, which we attribute to switching between a concave and convex faceted liquid–solid interface. Dopant selective etching was used to reveal and correlate the shape of the growth interface with the observed anisotropic doping

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

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

Nonvolatile Modulation of Light Emission from Delocalized and Defect-Bound Excitons in Monolayer MoS₂ Using the Remnant Polarization in the Ferroelectric Polymer P(VDF-TrFE): Implications for Optical Memory

Excitons in semiconductors are a potential alternative to charge for manipulation and storage of information. The reduced electrostatic screening in monolayers of two-dimensional (2D) semiconductors increases the binding energies of delocalized (free) excitons, while also encouraging the formation of defect-bound excitons that can be leveraged in devices for neuromorphic and quantum computing. We report the nonvolatile modulation of light emission from delocalized and defect-bound excitons in monolayer MoS2 using the remnant polarization of P(VDF-TrFE) ferroelectric polymer islands. Photoluminescence emission intensities of delocalized excitons at room-temperature and bound excitons at cryogenic temperatures are enhanced by polarization that depletes free electrons and reduces Coulomb screening. The opposite polarization suppresses emission, providing a simple means of electrically encoding and optically reading information. Two distinct defect-bound exciton bands are identified by their spectral positions and their distinct variations with temperature and polarization, suggesting that they have distinct origins or interactions with the surrounding environment. The selective reconfiguration of free and bound excitonic emission suggests a novel device scheme toward electrically reconfigurable, optically accessible optical memory devices

Identification of an Intrinsic Source of Doping Inhomogeneity in Vapor–Liquid–Solid-Grown Nanowires

The vapor–liquid–solid (VLS) process of semiconductor nanowire growth is an attractive approach to low-dimensional materials and heterostructures because it provides a mechanism to modulate, in situ, nanowire composition and doping, but the ultimate limits on doping control are ultimately dictated by the growth process itself. Under widely used conditions for the chemical vapor deposition growth of Si and Ge nanowires from a Au catalyst droplet, we find that dopants incorporated from the liquid are not uniformly distributed. Specifically, atom probe tomographic analysis revealed up to 100-fold enhancements in dopant concentration near the VLS trijunction in both B-doped Si and P-doped Ge nanowires. We hypothesize that radial and azimuthal inhomogeneities arise from a faceted liquid–solid interface present during nanowire growth, and we present a simple model to account for the distribution. As the same segregation behavior was observed in two distinct semiconductors with different dopants, the observed inhomogeneity is likely to be present in other VLS grown nanowires

Atom Probe Tomography of <i>a</i>-Axis GaN Nanowires: Analysis of Nonstoichiometric Evaporation Behavior

GaN nanowires oriented along the nonpolar <i>a</i>-axis were analyzed using pulsed laser atom probe tomography (APT). Stoichiometric mass spectra were achieved by optimizing the temperature, applied dc voltage, and laser pulse energy. Local variations in the measured stoichiometry were observed and correlated with facet polarity using scanning electron microscopy. Fewer N atoms were detected from nonpolar and Ga-polar surfaces due to uncorrelated evaporation of N₂ ions following N adatom diffusion. The observed differences in Ga and N ion evaporation behaviors are considered in detail to understand the influence of intrinsic materials characteristics on the reliability of atom probe tomography analysis. We find that while reliable analysis of III–N alloys is possible, the standard APT procedure of empirically adjusting analysis conditions to obtain stoichiometric detection of Ga and N is not necessarily the best approach for this materials system

Metal-Free Carbon-Based Nanomaterial Coatings Protect Silicon Photoanodes in Solar Water-Splitting

The decreasing cost of silicon-based photovoltaics has enabled significant increases in solar electricity generation worldwide. Silicon photoanodes could also play an important role in the cost-effective generation of solar fuels, but the most successful methods of photoelectrode passivation and performance enhancement rely on a combination of precious metals and sophisticated processing methods that offset the economic arguments for silicon. Here we show that metal-free carbon-based nanomaterial coatings deposited from solution can protect silicon photoanodes carrying out the oxygen evolution reaction in a range of working environments. Purified semiconducting carbon nanotubes (CNTs) act as a hole extraction layer, and a graphene (Gr) capping layer both protects the CNT film and acts as a hole exchange layer with the electrolyte. The performance of semiconducting CNTs is found to be superior to that of metallic or unsorted CNTs in this context. Furthermore, the insertion of graphene oxide (GO) between the n-Si and CNTs reduces the overpotential relative to photoanodes with CNTs deposited on hydrogen-passivated silicon. The composite photoanode structure of n-Si/GO/CNT/Gr shows promising performance for oxygen evolution and excellent potential for improvement by optimizing the catalytic properties and stability of the graphene protective layer