One-dimensional Periodic Potential Research Articles

Exploring the effects of a one-dimensional periodic potential on a three-dimensional topological insulator

High mobility two-dimensional systems with superposed 1D lateral periodic potentials exhibit characteristic commensurability (Weiss) oscillations that reflect the interplay of the cyclotron radius at the Fermi level and the superlattice period. Here, we impose a one-dimensional periodic potential on strained HgTe, which is a strong 3D topological insulator. By tuning the Fermi level with top gates, the effects of the artificial potential can be studied in the bulk gap, where only Dirac surface states exist, in the conduction band, and in the valence band, where Dirac electrons and holes coexist. On the electron side, we observe clear commensurability oscillations whose period is governed by the carrier density of the top-surface Dirac electrons. Unexpectedly, weak commensurability oscillations are also observed in the valence band with a period that depends on both electron and hole density. Published by the American Physical Society 2024

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.

Electronic band gaps and transport properties in superlattice based on semi-Dirac material

We investigate the energy band structure and transport properties of electrons in superlattice structure based on semi-Dirac material with one-dimensional periodic potentials of square barriers. The influence of different parameters on transmission probability in superlattice structure is studied. It is found that two new Dirac points are formed in the superlattice structure by adjusting the gap parameter. The position of the new Dirac points can be regulated by changing the size of the potential barrier, the ratio of well width to barrier width, and the gap parameter. Some forbidden bands appear in the transmission probability, the location and width of the forbidden bands can be regulated by changing the system parameters.

Generalized One-Dimensional Periodic Potential Wells Tending to the Dirac Delta Potential

The solution of a quantum periodic potential in solid state physics is relevant because one can understand how electrons behave in a corresponding crystal. In this paper, the analytical solution of the energy formulation for a one-dimensional periodic potential that meets certain boundary conditions is developed in a didactic and detailed way. In turn, the group speed and effective mass are also calculated from the transcendental energy equation of a potential V=V(x). From this, a comparison is made with periodic potentials with known analytical solutions, such as the Dirac delta, as well as rectangular and triangular potentials. Finally, some limits are proposed in which these periodic potentials of the form V=V(x) approach the periodic Dirac delta potential of positive intensity. Therefore, the calculations described in this paper can be used as the basis for more-complex one-dimensional potentials and related simulations.

Vibrational resonance in a multistable system with position-dependent mass

The occurrence of vibrational resonance (VR) in a dual-frequency-driven multistable system with a spatially varying mass modelling particle with position-dependent mass (PDM) and evolving in a one-dimensional symmetric periodic potential has been investigated and reported in this paper. We numerically compute and analyze the response amplitude, the effects of the PDM parameters (m 0, a) on the potential structure, the occurrence of VR and the bifurcation of the equilibrium points. It is shown that the PDM parameters, besides controlling VR, can induce unconventional resonance patterns through the variation of the potential well depth. The resonant states can be influenced through the cooperation of the PDM parameters and the external forcing leading the system from multiresonance state into single and double resonance states. The contributions of the fixed rest mass m 0 on the VR and the PDM-induced resonance are determined by threshold conditions imposed by the magnitude of the mass nonlinear strength a.

Tuning the spin of Dirac fermions on the surface of topological insulators in proximity to a helical spin density wave

We investigate the spin tunability of Dirac fermions on the surface of a 3D topological insulator in proximity to a helical spin density wave, acting as an applied one-dimensional periodic potential for spins produced by spiral multiferroic oxide. It is observed that the spin mean values of Dirac fermion undergo oscillations under the influence of such a periodic potential created by the exchange field of magnetization. The tunability of spin is strongly affected by the strength, orientation and period of the exchange field. In particular, the mean values of spin are anisotropic around the Dirac point, depending strongly on the amplitude and spatial period of the periodic potential. We also find that the spin expectation values change significantly by changing the plane of magnetization. Interestingly, the in-plane components of spin mean values perform pronounced oscillations, whereas the out of plane component does not oscillate at all. The oscillations of planar components of spin are originated from the spin-momentum locking on the surface of topological insulator.

Autonomous ratcheting by stochastic resetting.

We propose a generalization of the stochastic resetting mechanism for a Brownian particle diffusing in a one-dimensional periodic potential: randomly in time, the particle gets reset at the bottom of the potential well it was in. Numerical simulations show that in mirror asymmetric potentials, stochastic resetting rectifies the particle's dynamics, with a maximum drift speed for an optimal average resetting time. Accordingly, an unbiased Brownian tracer diffusing on an asymmetric substrate can rectify its motion by adopting an adaptive stop-and-go strategy. Our proposed ratchet mechanism can model the directed autonomous motion of molecular motors and micro-organisms.

Paradoxical effects for a one-dimensional periodic potential embedded in a two-dimensional system

Paradoxical effects for a one-dimensional periodic potential embedded in a two-dimensional system

Activity-assisted barrier crossing of self-propelled colloids over parallel microgrooves.

We report a systematic study of the dynamics of self-propelled particles (SPPs) over a one-dimensional periodic potential landscape U_{0}(x), which is fabricated on a microgroove-patterned polydimethylsiloxane (PDMS) substrate. From the measured nonequilibrium probability density function P(x;F_{0}) of the SPPs, we find that the escape dynamics of the slow rotating SPPs across the potential landscape can be described by an effective potential U_{eff}(x;F_{0}), once the self-propulsion force F_{0} is included into the potential under the fixed angle approximation. This work demonstrates that the parallel microgrooves provide a versatile platform for a quantitative understanding of the interplay among the self-propulsion force F_{0}, spatial confinement by U_{0}(x), and thermal noise, as well as its effects on activity-assisted escape dynamics and transport of the SPPs.

Driven particle in a one-dimensional periodic potential with feedback control: Efficiency and power optimization.

A Brownian particle moving in a staircaselike potential with feedback control offers a way to implement Maxwell's demon. An experimental demonstration of such a system using sinusoidal periodic potential carried out by Toyabe etal. [Nat. Phys. 6, 988 (2010)1745-247310.1038/nphys1821] has shown that information about the particle's position can be converted to useful work. In this paper, we carry out a numerical study of a similar system using Brownian dynamics simulation. A Brownian particle moving in a periodic potential under the action of a constant driving force is made to move against the drive by measuring the position of the particle and effecting feedback control by altering potential. The work is extracted during the potential change and from the movement of the particle against the external drive. These work extractions come at the cost of information gathered during the measurement. Efficiency and work extracted per cycle of this information engine are optimized by varying control parameters as well as feedback protocols. Both these quantities are found to crucially depend on the amplitude of the periodic potential as well as the width of the region over which the particle is searched for during the measurement phase. For the case when potential flip (i.e., changing the phase of the potential by 180^{∘}) is used as the feedback mechanism, we argue that the square potential offers a more efficient information-to-work conversion. The control over the numerical parameters and averaging over large number of trial runs allow one to study the nonequilibrium work relations with feedback for this process with precision. It is seen that the generalized integral fluctuation theorem for error-free measurements holds to within the accuracy of the simulation.

Anomalous relaxation from a non-equilibrium steady state: An isothermal analog of the Mpemba effect

The Mpemba effect denotes an anomalous relaxation phenomenon where a system initially at a hot temperature cools faster than a system that starts at a less elevated temperature. We introduce an isothermal analog of this effect for a system prepared in a non-equilibrium steady state that then relaxes towards equilibrium. Here, the driving strength, which determines the initial non-equilibrium steady state, takes the role of the temperature in the original version. As a paradigm, we consider a particle initially driven by a non-conservative force along a one-dimensional periodic potential. We show that for an asymmetric potential relaxation from a strongly driven initial state is faster than from a more weakly driven one at least for one of the two possible directions of driving. These results are first obtained through perturbation theory in the strength of the potential and then extended to potentials of arbitrary strength through topological arguments.

Scattering of two particles in a one-dimensional lattice

This study concerns the two-body scattering of particles in a one-dimensional periodic potential. A convenient ansatz allows for the separation of center-of-mass and relative motion, leading to a discrete Schr\"odinger equation in the relative motion that resembles a tight-binding model. A lattice Green's function is used to develop the Lippmann-Schwinger equation, and ultimately derive a multiband scattering $K$ matrix which is described in detail in the two-band approximation. Two distinct scattering lengths are defined according to the limits of zero relative quasimomentum at the top and bottom edges of the two-body collision band. Scattering resonances occur in the collision band when the energy is coincident with a bound state attached to another higher or lower band. Notably, repulsive on-site interactions in an energetically closed lower band lead to collision resonances in an excited band.

Asymptotic behavior of correlation functions of one-dimensional polar-molecules on optical lattices

We combine a slave-spin approach with a mean-field theory to develop an approximate theoretical scheme to study the density, spin, and, pairing correlation functions of fermionic polar molecules. We model the polar molecules subjected to a one-dimensional periodic optical lattice potential using a generalized t–J model, where the long-range part of the interaction is included through the exchange interaction parameter. For this model, we derive a set of self-consistent equations for the correlation functions, and evaluate them numerically for the long-distance behavior. We find that the pairing correlations are related to spin correlations through the density and the slave-spin correlations. Further, our calculations indicates that the long-range character of the interaction can be probed through these correlation functions. In the absence of exact solutions for the one-dimensional t–J model, our approximate theoretical treatment can be treated as a useful tool to study one dimensional long-range correlated fermions.

ENERGY BAND STRUCTURE OF AN ELECTRON IN A ONE-DIMENSIONAL PERIODIC POTENTIAL

It has been observed that electron in a perfect crystal moves in a spatially periodic field of force due to the ions and the averaged effect of all the electrons. This work shows the investigative work done to determine the energy band structure of an electron in a one-dimensional periodic potential. The application of the Kronig-Penney model was applied to an electron state in a delta-like potential. To fully understand the Kronig-Penney model, the concept of Bloch’s theorem was first introduced to describe the conduction of electrons in solids. It has been found that the periodic potential introduces gaps in the reduced representation with an increasing number of potential well/barrier strengths. It has been observed that the regions of non-propagating states, which give rise to energy band gaps, become larger with decreasing values.

Mobility of a spatially modulated electron liquid on the helium surface

We present the results of large-scale numerical simulations of the mobility of a two-dimensional electron liquid on the helium surface in the presence of a one-dimensional periodic potential. Even where the potential is much weaker than the electron-electron interaction, it can strongly change the mobility. The effect depends on the interrelation between the potential period and the mean interelectron distance. It is most pronounced where the period is close to the period of the Wigner crystal that would form if the liquid were cooled to a lower temperature. The results suggest, in particular, that the correlation length in the electron liquid can be found by measuring the mobility in a weak periodic potential. The simulations are based on the microscopic model of the electron scattering by the excitations in helium.

Nonequilibrium Transport and Phase Transitions in Driven Diffusion of Interacting Particles

Abstract Driven diffusive systems constitute paradigmatic models of nonequilibrium physics. Among them, a driven lattice gas known as the asymmetric simple exclusion process (ASEP) is the most prominent example for which many intriguing exact results have been obtained. After summarising key findings, including the mapping of the ASEP to quantum spin chains, we discuss the recently introduced Brownian ASEP (BASEP) as a related class of driven diffusive system with continuous space dynamics. In the BASEP, driven Brownian motion of hardcore-interacting particles through one-dimensional periodic potentials is considered. We study whether current–density relations of the BASEP can be considered as generic for arbitrary periodic potentials and whether repulsive particle interactions other than hardcore lead to similar results. Our findings suggest that shapes of current–density relations are generic for single-well periodic potentials and can always be attributed to the interplay of a barrier reduction, blocking, and exchange symmetry effect. This implies that in general up to five different phases of nonequilibrium steady states are possible for such potentials. The phases can occur in systems coupled to particle reservoirs, where the bulk density is the order parameter. For multiple-well periodic potentials, more complex current–density relations are possible, and more phases can appear. Taking a repulsive Yukawa potential as an example, we show that the effects of barrier reduction and blocking on the current are also present. The exchange symmetry effect requires hardcore interactions, and we demonstrate that it can still be identified when hardcore interactions are combined with weak Yukawa interactions. The robustness of the collective dynamics in the BASEP with respect to variations of model details can be a key feature for a successful observation of the predicted current–density relations in actual physical systems.

Extension of generalized Kramers theory in the spatial diffusion regime: Diffusion of Brownian particles in periodic potential

Extension of generalized Kramers theory in the spatial diffusion regime to extensive noise parameters is illustrated through the internal Ornstein–Uhlenbeck noise-induced diffusion of a particle in one-dimensional periodic potential. In the current over population method, the Fokker–Planck equation does not have a non-equilibrium stationary-state solution for a large scope of Ornstein–Uhlenbeck noise parameters. Instead, an effective non-equilibrium stationary-state transition probability density can be obtained and can serve as the long-term mean transition probability density for the Langevin simulation of the escape rate. The analytical results of the diffusion coefficient in the spatial diffusion regime are in good agreement with the Langevin simulation results at a low temperature. This theoretical approach can be applied to Gaussian noise processes in potential barrier crossing problems, including the processes in the presence of external noises.

Dirac point formation revealed by Andreev tunneling in superlattice-graphene/superconductor junctions

A graphene superlattice is formed by a one-dimensional periodic potential and is characterized by the emergence of new Dirac points in the electronic structure. The group velocity of graphene's massless Dirac fermions at the new points is drastically reduced, resulting in a measurable effect in the conductance spectroscopy. We show here that tunnel spectroscopy using a superconducting hybrid junction is more sensitive to the formation of Dirac points in the spectrum of graphene superlattices due to the additional contribution of Andreev processes. We examine the transport properties of a graphene-based superlattice--superconductor hybrid junction and demonstrate that a superlattice potential can coexist with proximity-induced superconducting correlations. Both effects contribute to change graphene's spectrum for subgap energies and, as a result, the normalized tunneling conductance features sharp changes for voltages proportional to the energy separation between the original and the newly generated Dirac points.

Klein collimation by rippled graphene superlattice

The hybridization of and orbitals of carbon atoms in graphene depends on the surface curvature. Considering a single junction between flat and rippled graphene subsystems, it is found an accumulation of charge in the rippled subsystem due to Klein penetration phenomenon that gives rise to n–p junction. Using this fact, we show that the momentum distribution of electrons in ballisitically propagating beam can be selective without a waveguide, or external electric, and/or magnetic fields in graphene strip under experimentally feasible one-dimensional periodic potential. Such a potential is created with the aid of superlattice that consists of periodically repeated graphene pieces with different hybridizations of carbon orbits, produced by variation of the graphene surface curvature. The charge redistribution and selected transmission of electrons, caused by the superlattice, allows to control the electron focusing in the considered system by simply changing the element properties in the superlattice.

Modulation of Magneto-Intersubband Oscillations in a One-Dimensional Lateral Superlattice

Low-temperature magnetotransport in a quasi-two-dimensional electron system based on a selectively doped GaAs quantum well with two occupied quantum-confinement subbands with one-dimensional periodic modulation of a potential is investigated. It is shown that commensurability oscillations of the resistance in this electron system coexist with magneto-intersubband oscillations. It is found that, in addition to commen-surability oscillations, the one-dimensional periodic potential in the two-subband electron system leads to the modulation of the magneto-intersubband oscillation amplitude. The experimental results are explained by the formation of Landau bands in the quasi-two-dimensional electron system with one-dimensional periodic modulation of the potential.