Search

Let $q$ be an integer. A $D(q)$-$m$-tuple is a set of $m$ distinct positive integers ${a_1, a_2, . . . , a_m}$ such that $a_ia_j + q$ is a perfect square for all $1 \leq i < j \leq m$. By counting integer solutions $x \in [1, b]$ of congruences $x^2 \equiv q (\mod b)$ with $b \leq N$, we count $D(q)$-pairs with both elements up to $N$, and give estimates on asymptotic behaviour. We show that for prime $q$, the number of such $D(q)$-pairs and $D(q)$-triples grows linearly with $N$. Up to a factor of $2$, the slope of this linear function is the quotient of the value of the $L$-function of an appropriate Dirichlet character (usually a Kronecker symbol) and of $\zeta(2)$., Comment: 20 pages

For a nonzero rational number $q$, a rational $D(q)$-$n$-tuple is a set of $n$ distinct nonzero rationals $\{a_1, a_2, \dots, a_n\}$ such that $a_ia_j+q$ is a square for all $1 \leqslant i < j \leqslant n$. We investigate for which $q$ there exist infinitely many rational $D(q)$-quintuples. We show that assuming the Parity Conjecture for the twists of several explicitly given elliptic curves, the density of such $q$ is at least $295026/296010\approx 99.5\%$., Comment: 16 pages

The nature of the "forbidden" local- and long-range polar order in nominally nonpolar paraelectric phases of ferroelectric materials has been an open question since the discovery of ferroelectricity in oxide perovskites (ABO3). A currently considered model suggests locally correlated displacements of B-site atoms along a subset of <111> cubic directions. Such offsite displacements have been confirmed experimentally, however, being essentially dynamic in nature they cannot account for the static nature of the symmetry-forbidden polarization implied by the macroscopic experiments. Here, in an atomically resolved study by aberration corrected scanning transmission electron microscopy (STEM) complemented by Raman spectroscopy, we reveal, directly visualize and quantitatively describe static, 2-4 nm large polar nanoclusters in the nominally nonpolar cubic phases of (Ba,Sr)TiO3 and BaTiO3. These results have implications on understanding of the atomic-scale structure of disordered materials, the origin of precursor states in ferroelectrics, and may help answering ambiguities on the dynamic-versus-static nature of nano-sized clusters., Comment: the main text and supplementary information

For a rational number $q$, a rational $D(q)$-$n$-tuple is a set of $n$ distinct nonzero rationals $\{a_1, a_2, \dots, a_n\}$ such that $a_ia_j+q$ is a rational square for all $1 \leqslant i < j \leqslant n$. For every $q$ we find all rational $m$ such that there exists a $D(q)$-quadruple with product $abcd=m$. We describe all such quadruples using points on a specific elliptic curve depending on $(q,m).$, Comment: 15 pages

Two-dimensional (2D) metallic states induced by oxygen vacancies at oxide surfaces and interfaces provide new opportunities for the development of advanced applications, but the ability to control the behavior of these states is still limited. We used Angle Resolved Photoelectron Spectroscopy combined with density functional theory to study the reactivity of states induced by the oxygen vacancies at the (001)-(1x4) surface of anatase TiO2, where both 2D metallic and deeper lying in-gap states (IGs) are observed. Remarkably, the two states exhibit very different evolution when the surface is exposed to molecular O2: while IGs are almost completely quenched, the metallic states are only weakly affected. The energy scale analysis for the vacancy migration and recombination resulting from the DFT calculations confirms indeed that only the IGs originate from and remain localized at the surface, whereas the metallic states originate from subsurface vacancies, whose migration and recombination at the surface is energetically less favorable rendering them therefore insensitive to oxygen dosing.

We have grown untwinned YBa$_2$Cu$_3$O$_{7-\delta}$ (YBCO) films on (110) MgO substrates that were pre-annealed at high temperature in oxygen atmosphere. The annealing results in surface reconstruction with shallow facets, which induce the suppression of the YBCO twinning domains, and the preferential alignment of the CuO chains along one of the in-plane directions of the substrate. Because of the large mismatch between the in-plane lattice parameters of film and substrate, the strain induced by the MgO into the YBCO layer is strong and very peculiar. The YBCO film is compressed, with respect to the bulk, and presents a unidirectional buckling of the atomic planes, along the chains' direction, due to a deformation of the copper-oxygen octahedra. The YBCO films, which can be grown with thicknesses down to few unit cells and oxygen doping levels spanning most of the superconducting dome, are patterned into nanowires with dimensions down to 50 nm. The anisotropies due to the untwinning state are preserved in these structures; moreover, additional anisotropies appear, in ultrathin structures where strain effects become more pronounced. Such untwinned and compressively strained films can therefore be used as a platform to study the interplay between strain and the various local orders in the normal state of YBCO., Comment: 11 pages, 8 figures

We report on the growth and characterization of ultrathin YBa$_2$Cu$_3$O$_{7-\delta}$ (YBCO) films on MgO (110) substrates, which exhibit superconducting properties at thicknesses down to 3 nm. YBCO nanowires, with thicknesses down to 10 nm and widths down to 65 nm, have been also successfully fabricated. The nanowires protected by a Au capping layer show superconducting properties close to the as-grown films, and critical current densities, which are only limited by vortex dynamics. The 10 nm thick YBCO nanowires without the Au capping present hysteretic current voltage characteristics, characterized by a voltage switch which drives the nanowires directly from the superconducting to the normal state. Such bistability is associated in NbN nanowires to the presence of localized normal domains within the superconductor. The presence of the voltage switch, in ultrathin nanostructures characterized by high sheet resistance values, though preserving high quality superconducting properties, make our nanowires very attractive devices to engineer single photon detectors., Comment: 12 pages, 8 figures