The H\"older-Poincar\'e Duality for Lq,pL_{q,p}-cohomology

We prove the following version of Poincare duality for reduced $L_{q,p}$-cohomology: For any $1<q,p<\infty$, the $L_{q,p}$-cohomology of a Riemannian manifold is in duality with the interior $L_{p',q'}-cohomology for$1/p+1/p'=1$,$1/q+1/q'=1$.Comment: 21 page

Phonon Raman spectra of colloidal CdTe nanocrystals: effect of size, non-stoichiometry and ligand exchange

Resonant Raman study reveals the noticeable effect of the ligand exchange on the nanocrystal (NC) surface onto the phonon spectra of colloidal CdTe NC of different size and composition. The oleic acid ligand exchange for pyridine ones was found to change noticeably the position and width of the longitudinal optical (LO) phonon mode, as well as its intensity ratio to overtones. The broad shoulder above the LO peak frequency was enhanced and sharpened after pyridine treatment, as well as with decreasing NC size. The low-frequency mode around 100 cm-1 which is commonly related with the disorder-activated acoustical phonons appears in smaller NCs but is not enhanced after pyridine treatment. Surprisingly, the feature at low-frequency shoulder of the LO peak, commonly assigned to the surface optical phonon mode, was not sensitive to ligand exchange and concomitant close packing of the NCs. An increased structural disorder on the NC surface, strain and modified electron-phonon coupling is discussed as the possible reason of the observed changes in the phonon spectrum of ligand-exchanged CdTe NCs

Micro-Raman and micro-transmission imaging of epitaxial graphene grown on the Si and C faces of 6H-SiC

Micro-Raman and micro-transmission imaging experiments have been done on epitaxial graphene grown on the C- and Si-faces of on-axis 6H-SiC substrates. On the C-face it is shown that the SiC sublimation process results in the growth of long and isolated graphene ribbons (up to 600 μm) that are strain-relaxed and lightly p-type doped. In this case, combining the results of micro-Raman spectroscopy with micro-transmission measurements, we were able to ascertain that uniform monolayer ribbons were grown and found also Bernal stacked and misoriented bilayer ribbons. On the Si-face, the situation is completely different. A full graphene coverage of the SiC surface is achieved but anisotropic growth still occurs, because of the step-bunched SiC surface reconstruction. While in the middle of reconstructed terraces thin graphene stacks (up to 5 layers) are grown, thicker graphene stripes appear at step edges. In both the cases, the strong interaction between the graphene layers and the underlying SiC substrate induces a high compressive thermal strain and n-type doping

Optical study of oriented double-Se₈-ring clusters and luminescent Se₂⁻ anions in LTA at extremely high selenium loading density

Abstract Recently, LTA-Se(1–8) samples with 1–8 Se atoms per cavity (simplified unit cell, large cavity + sodalite cage) obtained via adsorption at the temperature of ∼450 °C were reported. It was shown that single Se8 or single Se₁₂ ring are formed in the large LTA cavities, Se₈/Se₁₂ ring concentration ratio decreasing with an increase in the Se loading density. Contrary, in the present work, using Se vapour adsorption at ∼550 °C, we succeeded in encapsulation of ∼17 Se atoms per cavity (LTA-Se(17)) with a significant increase in the Se₈/Se₁₂ concentration ratio manifesting double Se₈-ring cluster formation in the most of the LTA large cavities, which is a step towards cluster crystal fabrication. According to our polarization/orientation Raman spectroscopic study of LTA-Se(17) single crystals, the orientations of the Se₈ and Se₁₂ appeared to be similar to those in previously investigated LTA-Se(1–8). Importantly, luminescent Se₂⁻ anions, oriented along the LTA 4-fold axes and located in the sodalite cages, are detected via Raman polarization/orientation dependencies of LTA-Se(17). Bright Se₂⁻ light emission with a maximum at ∼1.56 eV and vibronic structure is observed in the 1.3–1.8 eV spectral range. We show that the anions experience a compression in LTA which is slightly relaxing with a decrease in temperature producing an anomalous Raman band downshift. The compression of Se₂⁻ in LTA is weaker/stronger than that in sodalite/cancrinite, luminescence band photon energy depending on its strength. High concentration of regularly arranged Se₂⁻ in LTA suggests considering LTA-Se(17) as an important novel light-emitting material