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In the last two decades, much effort has been dedicated to studying curves and surfaces according to their angle with a given direction. However, most findings were obtained using a case-by-case approach, and it is often unclear what are consequences of the specificities of the ambient manifold and what could be generic. In this work, we propose a theoretical framework to unify parts of these findings. We study curves and surfaces by prescribing the angle they make with a parallel transported vector field. We show that the characterization of Euclidean helices in terms of their curvature and torsion is also valid in any Riemannian manifold. Among other properties, we prove that surfaces making a constant angle with a parallel transported direction are extrinsically flat ruled surfaces. We also investigate the relation between their geodesics and the so-called slant helices; we prove that surfaces of constant angle are the rectifying surface of a slant helix, i.e., the ruled surface with rulings given by the Darboux vector field of the directrix. We characterize rectifying surfaces of constant angle; in other words, when their geodesics are slant helices. As a corollary, we show tha

Surface doping of ZnO allows for tailoring the surface chemistry of the material while preserving the superb electronic structure of the bulk. Apart from obvious changes in adsorption energies and activation energies for catalysis, surface doping can alter the workfunction of the material and allow it to be tuned for specific photocatalytic and optoelectronic applications. We present first-principles electronic structure calculations for surface doping of Mn on the ZnO (0001) surface. Various dopant concentrations have been considered at the out-most (surface) layer of Zn atoms, while the interior of the material is kept at the ideal wurtzite structure. For each system, the surface energy and surface workfunction have been calculated. Both workfunction and surface energy drop with increasing Mn concentration for O-terminated surfaces, while more complex behaviour is observed in metal-terminated ones. We discuss trends in surface stability and surface electronic structure of this material and how they affect its properties.

As a departure from existing continuum approaches for describing the stability and evolution of surfaces of crystalline materials, this article provides a description of surface evolution based on the physics of the main feature imposed by the discrete nature of the material, namely, crystallographic surface steps. It is shown that the formation energy of surface steps depends on the sign of extensional strain of the crystal surface, and this behavior plays a crucial role in surface evolution. The nature of this dependence implies that there is no energetic barrier to nucleation of islands on the growth surface during deposition, and that island faces tend toward natural orientations which have no counterpart in unstrained materials. This behavior is expressed in terms of a small number of parameters that can be estimated through atomistic analysis of stepped surfaces. The continuum framework developed is then applied to study the time evolution of surface shape of an epitaxial film being deposited onto a substrate. The kinetic equation for mass transport is enforced in a weak form by means of a variational formulation [...].

The concept of catenary has been recently extended to the sphere and the hyperbolic plane by the second author [López, arXiv:2208.13694]. In this work, we define catenaries on any Riemannian surface. A catenary on a surface is a critical point of the potential functional, where we calculate the potential with the intrinsic distance to a fixed reference geodesic. Adopting semi-geodesic coordinates around the reference geodesic, we characterize catenaries using their curvature. Finally, after revisiting the space-form catenaries, we consider surfaces of revolution (where a Clairaut relation is established), ruled surfaces, and the Grušin plane.

The present contribution deals with surface terms appearing immediately in distributions and partition functions of statistical ensembles. It is shown that all ensembles under study, including ordinary canonical and grand canonical ensembles, involve surface terms. For a canonical ensemble both surface and volume terms correspond to a closed system. For a grand canonical ensemble the volume term corresponds to an open system, while the surface one - to a closed one. Finally, for the recently introduced open statistical ensemble the specific feature of which is the consideration of some surrounding region both volume and surface terms correspond to an open system. In conclusion, surface particles at solid/fluid boundary are interpreted as particles corresponding to number density oscillations near the surface; this completely agrees with the earlier introduced concept of surface tension at such a boundary.

Theory of adsorption on a surface with nanolocal defects is proposed. Two efficacy parameters of surface modification for nanotechnological purposes are introduced, where the modification is a creation of nanolocal artificial defects. The first parameter corresponds to applications where it is necessary to increase the concentration of certain particles on the modified surface. And the second one corresponds to the pattern transfer with the help of particle self-organization on the modified surface. The analytical expressions for both parameters are derived with the help of the thermodynamic and the kinetic approaches for two cases: jump diffusion and free motion of adsorbed particles over the surface. The possibility of selective adsorption of molecules is shown with the help of simulation of the adsorption of acetylene and benzene molecules in the pits on the graphite surface. The process of particle adsorption from the surface into the pit is theoretically studied by molecular dynamic technique. Some possible nanotechnological applications of adsorption on the surface with artificial defects are considered: fabrication of sensors for trace molecule detection, separation of isome

We investigate ruled surfaces in 3d Riemannian manifolds, i.e., surfaces foliated by geodesics. In 3d space forms, we find the striction curve, distribution parameter, and the first and second fundamental forms, from which we obtain the Gaussian and mean curvatures. We also provide model-independent proof for the known fact that extrinsically flat surfaces in space forms are ruled. This proof allows us to identify the necessary and sufficient condition the curvature tensor must satisfy for an extrinsically flat surface in a generic 3d manifold to be ruled. Further, we show that if a 3d manifold has an extrinsically flat surface tangent to any 2d plane and if they are all ruled surfaces, then the manifold is a space form. As an application, we prove that there must exist extrinsically flat surfaces in the Riemannian product of the hyperbolic plane, or sphere, with the reals, and that do not make a constant angle with the real direction.

We report on the electronic properties of GaN$(1\bar{1}00)$ and $(0001)$ surfaces after three different and subsequent device processing compatible cleaning steps: HCl etching, annealing at $400$ $^\circ$C in N$_2$ atmosphere, and O$_2$ plasma exposure. The surface electronic properties are quantified, in the dark and under ultraviolet illumination, using X-ray photoelectron spectroscopy and a Kelvin probe. We find that the cleaning steps largely affect the work function and the band bending of both GaN orientations. These modifications are attributed to the presence of different surface states as well as to the formation of adsorbates building up distinct surface dipoles. Besides these results, we detect that under ultraviolet illumination the work function of the surfaces exposed to HCl decreases by at least $0.2$ eV without screening of the band bending. We thus attribute the observed surface photovoltage to a photo-induced modification of the surface dipole. Overall, these results emphasize the strong dependence of the electronic properties of air-exposed GaN surfaces on adsorbates. As a result, we advocate the use of the common cleaning steps analyzed here to re-initialize at

We classify homomorphisms from the braid group on $n$ strands to the pure mapping class group of a nonoriantable surface of genus $g$. For $n\ge 14$ and $g\le 2\lfloor{n/2}\rfloor+1$ every such homomorphism is either cyclic, or it maps standard generators of the braid group to either distinct Dehn twists, or distinct crosscap transpositions, possibly multiplied by the same element of the centralizer of the image.

We apply the Green's function based full-potential screened Korringa-Kohn-Rostoker method in conjunction with the local density approximation to study the surface energies of the noble and the fcc transition and $sp$ metals. The orientation dependence of the transition metal surface energies can be well described taking into account only the broken bonds between first neighbors, quite analogous to the behavior we recently found for the noble metals [see cond-mat/0105207]. The (111) and (100) surfaces of the $sp$ metals show a jellium like behavior but for the more open surfaces we find again the noble metals behavior but with larger deviation from the broken-bond rule compared to the transition metals. Finally we show that the use of the full potential is crucial to obtain accurate surface energy anisotropy ratios for the vicinal surfaces.

We consider the extrinsic geometry of surfaces in simply isotropic space, a three-dimensional space equipped with a rank 2 metric of index zero. Since the metric is degenerate, a surface normal cannot be unequivocally defined based on metric properties only. To understand the contrast between distinct choices of an isotropic Gauss map, here we study surfaces with a Gauss map whose coordinates are eigenfunctions of the surface Laplace-Beltrami operator. We take into account two choices, the so-called minimal and parabolic normals, and show that when applied to simply isotropic invariant surfaces the condition that the coordinates of the corresponding Gauss map are eigenfunctions leads to planes, certain cylinders, or surfaces with constant isotropic mean curvature. Finally, we also investigate (non-necessarily invariant) surfaces with harmonic Gauss map and show this characterizes constant mean curvature surfaces.

Laser Texturing is one of the leading technologies applied to modify surface topography. To date, however, a standardized procedure to generate deterministic textures is virtually non-existent. In nature, especially in squamata, there are many examples of deterministic structured textures that allow species to control friction and condition their tribological response for efficient function. In this work, we draw a comparison between industrial surfaces and reptilian surfaces. We chose the python regius species as a bio-analogue with a deterministic surface. We first study the structural make up of the ventral scales of the snake (both construction and metrology). We further compare the metrological features of the ventral scales to experimentally recommended performance indicators of industrial surfaces extracted from open literature. The results indicate the feasibility of engineering a Laser Textured Surface based on the reptilian ornamentation constructs. It is shown that the metrological features, key to efficient function of a rubbing deterministic surface, are already optimized in the reptile. We further show that optimization in reptilian surfaces is based on synchronizing

We measure the electron conductivity of the surface states and the subsurface space charge layer originating from the Si(111)-4x1-In reconstruction as a function of temperature. The conductivity of the surface states drops sharply around 130 K with decreasing temperature, revealing a metal-insulator phase transition of the surface reconstruction. In contrast, the influence of the phase transition on the conductivity of the space charge layer is limited to temperatures above 60 K. This means that the surface Fermi level remains strongly pinned despite the phase transition, indicating the presence of free carriers in the surface states down to rather low temperatures.

Due to particle conservation, Canonical Molecular Dynamics (MD) simulations fail in the description of surface phase transitions involving coverage or lateral density changes. However, a step on the surface can act effectively as a source or a sink of atoms, in the simulation as well as in real life. A single surface step can be introduced by suitably modifying planar Periodic Boundary Conditions (PBC), to accommodate the generally inequivalent stacking of two adjacent layers. We discuss here how, through the introduction of two orthogonal surface steps, particle number conservation may no longer represent a fatal constraint for the study of these surface transitions. As an example, we apply the method for estimating temperature-induced lateral density increase of the reconstructed Au (001) surface; the resulting anisotropic cell change is consistent with experimental observations. Moreover, we implement this kind of scheme in conjunction with the variable curvature MD method, recently introduced by our group.

We investigate the spin and charge densities of surface states of the three-dimensional topological insulator $Bi_2Se_3$, starting from the continuum description of the material [Zhang {\em et al.}, Nat. Phys. 5, 438 (2009)]. The spin structure on surfaces other than the 111 surface has additional complexity because of a misalignment of the contributions coming from the two sublattices of the crystal. For these surfaces we expect new features to be seen in the spin-resolved ARPES experiments, caused by a non-helical spin-polarization of electrons at the individual sublattices as well as by the interference of the electron waves emitted coherently from two sublattices. We also show that the position of the Dirac crossing in spectrum of surface states depends on the orientation of the interface. This leads to contact potentials and surface charge redistribution at edges between different facets of the crystal.

The energies, widths, and shapes of features observed in the total energy distributions in field emission from W(100) and W(111) are compared with the results of a full-potential LAPW calculation of the surface density of states based on a supercell model of the crystal structure at the metal-vacuum interface. The Swanson hump on W(100) is attributed to two bands of surface states and surface resonances of dz^2 symmetry that are highly localised at the center of the surface Brillouin zone (Gamma_bar), and a second peak observed at lower energy is attributed to a band of surface resonances, also of dz^2 symmetry, centred at 0.11 A^(-1) along Gamma_bar to X_bar. The energy scale of the calculated total energy distribution is compressed by about 20% relative to the experimental data. The present calculation yields strong evidence that the broad asymmetric peak observed on W(111) is due to emission from a band of surface resonances. Further calculations for W(111) are proposed both to test the accuracy of the band model and to take into account the velocity factor that enters in a calculation of the emission current.

We report the results of an x-ray scattering study that reveals oxidation kinetics and formation of a previously unreported crystalline phase of SnO at the liquid-vapour interface of Sn. Our experiments reveal that the pure liquid Sn surface does not react with molecular oxygen below an activation pressure of \~5.0*10-6 Torr. Above that pressure a rough solid Sn oxide grows over the liquid metal surface. Once the activation pressure has been exceeded the oxidation proceeds at pressures below the oxidation pressure threshold. The observed diffraction pattern associated with the surface oxidation does not match any of the known Sn oxide phases. The data have an explicit signature of the face-centred cubic structure, however it requires lattice parameters that are about 9% smaller than those reported for cubic structures of high-pressure phases of Sn oxides. Keywords: X-ray scattering, diffraction, and reflection; Oxidation; Surface chemical reaction; Surface structure, morphology, roughness, and topography; Tin; Tin oxides; Liquid surfaces; Polycrystalline thin films

We present a clear and rigorous derivation of the Ewald-like method for calculation of the electrostatic energy of the systems infinitely periodic in two-dimensions and of finite size in the third dimension (slabs) which is significantly faster than existing methods. Molecular dynamics simulations using the transferable/polarizable model by Rustad et al. were applied to study the surface relaxation of the nonhydroxylated, hydroxylated, and solvated surfaces of alpha-Fe2O3 (hematite). We find that our nonhydroxylated structures and energies are in good agreement with previous LDA calculations on alpha-alumina by Manassidis et al. [Surf. Sci. Lett. 285, L517, 1993]. Using the results of molecular dynamics simulations of solvated interfaces, we define end-member hydroxylated-hydrated states for the surfaces which are used in energy minimization calculations. We find that hydration has a small effect on the surface structure, but that hydroxylation has a significant effect. Our calculations, both for gas-phase and solution-phase adsorption, predict a greater amount of hydroxylation for the (012) surface than for the (001) surface. Our simulations also indicate the presence of four-fold