Linear Periodic Potentials Research Articles

Towards the Analytical Generalization of the Transcendental Energy Equation, Group Velocity, and Effective Mass in One-Dimensional Periodic Potential Wells with a Computational Application to Common Coupled Potentials

The analytical generalization for N periodic potential wells coupled to a probe rectangular-like potential and a zero potential is extremely important in the study of one-dimensional periodic potentials in solid state physics, e.g., in the calculation of transport, optical, and magnetic properties. These findings raise the possibility of calculating equations for the generalization of N arbitrary potentials related to any potential V(x) using special functions as a solution. In this work, a novel analytical generalization of the transcendental energy equation, group velocity, and effective mass for N-coupled potentials to a probe one-dimensional potential V=V(x) was proposed. Initially, two well-known linear periodic potentials V=V(x) were employed to obtain analytical solutions for rectangular-like and Dirac-delta potentials. Python libraries were used to easily represent the equations for one or two rectangular-like potentials coupled with an arbitrary potential, highlighting the transcendental energy, group velocity, and effective mass. The results showed that the group velocity behavior changed its orientation due to the sign of the potential, whereas the width of the potential V(x) strongly influenced the group velocity behavior. The effective mass was also modified by the potential shapes, and their combinations, both effective mass and group velocity, exhibited similar physical behaviors to those found in ordinary rectangular-like potentials.

Moir\xe9 patterns arising from bilayer graphone/graphene superlattice

Moiré patterns from two-dimensional (2D) graphene heterostructures assembled via van der Waals interactions have sparked considerable interests in physics with the purpose to tailor the electronic properties of graphene. Here we report for the first time the observation of moiré patterns arising from a bilayer graphone/graphene superlattice produced through direct single-sided hydrogenation of a bilayer graphene on substrate. Compared to pristine graphene, the bilayer superlattice exhibits a rippled surface and two types of moiré patterns are observed: triangular and linear moiré patterns with the periodicities of 11 nm and 8–9 nm, respectively. These moiré patterns are revealed from atomic force microscopy and further confirmed by following fast Fourier transform (FFT) analysis. Density functional theory (DFT) calculations are also performed and the optimized lattice constants of bilayer superlattice heterostructure are in line with our experimental analysis. These findings show that well-defined triangular and linear periodic potentials can be introduced into the graphene system through the single-sided hydrogenation and also open a route towards the tailoring of electronic properties of graphene by various moiré potentials.

Preventing critical collapse of higher-order solitons by tailoring unconventional optical diffraction and nonlinearities

Self-trapped modes suffer critical collapse in two-dimensional cubic systems. To overcome such a collapse, linear periodic potentials or competing nonlinearities between self-focusing cubic and self-defocusing quintic nonlinear terms are often introduced. Here, we combine both schemes in the context of an unconventional and nonlinear fractional Schrödinger equation with attractive-repulsive cubic–quintic nonlinearity and an optical lattice. We report theoretical results for various two-dimensional trapped solitons, including fundamental gap and vortical solitons as well as the gap-type soliton clusters. The latter soliton family resembles the recently-found gap waves. We uncover that, unlike the conventional case, the fractional model exhibiting fractional diffraction order strongly influences the formation of higher band gaps. Hence, a new route for the study of self-trapped modes in these newly emergent higher band gaps is suggested. Regimes of stability and instability of all the soliton families are obtained with the help of linear-stability analysis and direct simulations.

Asymmetric localized states in periodic potentials with a domain-wall-like Kerr nonlinearity

We study the existence of one-dimensional localized states supported by linear periodic potentials and a domain-wall-like Kerr nonlinearity. The model gives rise to several new types of asymmetric localized states, including single- and double-hump soliton profiles, and multihump structures. Exploiting the linear stability analysis and direct simulations, we prove that these localized states are exceptional stable in the respective finite band gaps. The model applies to Bose–Einstein condensates loaded onto optical lattices, and in optics with period potentials, e.g., the photonic crystals and optical waveguide arrays, thereby the predicted solutions can be implemented in the state-of-the-art experiments.

Temperature Evolution of Quasi-one-dimensional C60 Nanostructures on Rippled Graphene.

We report the preparation of novel quasi-one-dimensional (quasi-1D) C60 nanostructures on rippled graphene. Through careful control of the subtle balance between the linear periodic potential of rippled graphene and the C60 surface mobility, we demonstrate that C60 molecules can be arranged into a quasi-1D C60 chain structure with widths of two to three molecules. At a higher annealing temperature, the quasi-1D chain structure transitions to a more compact hexagonal close packed quasi-1D stripe structure. This first experimental realization of quasi-1D C60 structures on graphene may pave a way for fabricating new C60/graphene hybrid structures for future applications in electronics, spintronics and quantum information.

Families of Surface Gap Solitons and Their Stability via the Numerical Evans Function Method

The nonlinear Schrödinger/Gross–Pitaevskii equation with a linear periodic potential and a nonlinearity coefficient $\Gamma$ with a discontinuity supports stationary localized solitary waves with frequencies inside spectral gaps, so-called surface gap solitons (SGSs). We compute families of one-dimensional SGSs using the arclength continuation method for a range of values of the jump in $\Gamma$. Using asymptotics, we show that when the frequency parameter converges to the bifurcation gap edge, the size of the allowed jump in $\Gamma$ converges to 0 for SGSs centered at any $x_c\in\mathbb{R}$. Linear stability of SGSs is next determined via the numerical Evans function method, in which the stable and unstable manifolds corresponding to the 0 solution of the linearized spectral ODE problem are evolved up to a common location where the determinant of their bases, i.e., the Evans function, is evaluated. Zeros of the Evans function coincide with eigenvalues of the linearized operator. Far from the SGS location the manifolds are spanned by exponentially decaying/increasing Bloch functions. As we show, evolution of the manifolds suffers from stiffness. A numerically stable formulation is possible in the exterior algebra formulation and with the use of Grassmanian preserving ODE integrators. Eigenvalues with a positive real part larger than a small constant are then detected via the use of the complex argument principle and a contour parallel to the imaginary axis. The location of real eigenvalues is found via a straightforward evaluation of the Evans function along the real axis, and several complex eigenvalues are located using Müller's method. The numerical Evans function method is described in detail in order to facilitate its use as a practical tool for locating eigenvalues. Our results show the existence of both unstable and stable SGSs (possibly with a weak instability), where stability is obtained even for some SGSs centered in the domain half with the less focusing nonlinearity. Direct simulations of the PDE for selected SGS examples confirm the results of Evans function computations.

Localized and periodic exact solutions to the nonlinear Schrödinger equation with spatially modulated parameters: Linear and nonlinear lattices

Using similarity transformations we construct explicit solutions of the nonlinear Schrödinger equation with linear and nonlinear periodic potentials. We present explicit forms of spatially localized and periodic solutions, and study their properties. We put our results in the framework of the exploited perturbation techniques and discuss their implications on the properties of associated linear periodic potentials and on the possibilities of stabilization of gap solitons using polychromatic lattices.

Delocalizing transition in one-dimensional condensates in optical lattices due to inhomogeneous interactions

It is shown that inhomogeneous nonlinear interactions in a Bose-Einstein condensate loaded in an optical lattice can result in a delocalizing transition in one dimension, which sharply contrasts to the known behavior of discrete and periodic systems with homogeneous nonlinearity. The transition can be originated either by decreasing the amplitude of the linear periodic potential or by the change of the mean value of the periodic nonlinearity. The dynamics of the delocalizing transition is studied.

Diffusion and current of Brownian particles in tilted piecewise linear potentials: amplification and coherence.

Overdamped motion of Brownian particles in tilted piecewise linear periodic potentials is considered. Explicit algebraic expressions for the diffusion coefficient, current, and coherence level of Brownian transport are derived. Their dependencies on temperature, tilting force, and the shape of the potential are analyzed. The necessary and sufficient conditions for the nonmonotonic behavior of the diffusion coefficient as a function of temperature are determined. The diffusion coefficient and coherence level are found to be extremely sensitive to the asymmetry of the potential. It is established that at the values of the external force, for which the enhancement of diffusion is most rapid, the level of coherence has a wide plateau at low temperatures with the value of the Péclet factor 2. An interpretation of the amplification of diffusion in comparison with free thermal diffusion in terms of probability distribution is proposed.