A cosmological model for corrugated graphene sheets

Defects play a key role in the electronic structure of graphene layers flat or curved. Topological defects in which an hexagon is replaced by an n-sided polygon generate long range interactions that make them different from vacancies or other potential defects. In this work we review previous models for topological defects in graphene. A formalism is proposed to study the electronic and transport properties of graphene sheets with corrugations as the one recently synthesized. The formalism is based on coupling the Dirac equation that models the low energy electronic excitations of clean flat graphene samples to a curved space. A cosmic string analogy allows to treat an arbitrary number of topological defects located at arbitrary positions on the graphene plane. The usual defects that will always be present in any graphene sample as pentagon–heptagon pairs and Stone-Wales defects are studied as an example. The local density of states around the defects acquires characteristic modulations that could be observed in scanning tunnel and transmission electron microscopy.     

Effects of topological defects and local curvature on the electronic properties of planar graphene

A formalism is proposed to study the electronic and transport properties of graphene sheets with corrugations as the one recently synthesized. The formalism is based on coupling the Dirac equation that models the low energy electronic excitations of clean flat graphene samples to a curved space. A cosmic string analogy allows to treat an arbitrary number of topological defects located at arbitrary positions on the graphene plane. The usual defects that will always be present in any graphene sample as pentagon–heptagon pairs and Stone–Wales defects are studied as an example. The local density of states around the defects acquires characteristic modulations that could be observed in scanning tunnel and transmission electron microscopy.     

Graphene wormholes: A condensed matter illustration of Dirac fermions in curved space

We study the properties of graphene wormholes in which a short nanotube acts as a bridge between two graphene sheets, where the honeycomb carbon lattice is curved from the presence of 12 heptagonal defects. By taking the nanotube bridge with very small length compared to the radius, we develop an effective theory of Dirac fermions to account for the low-energy electronic properties of the wormholes in the continuum limit, where the frustration induced by the heptagonal defects is mimicked by a line of fictitious gauge flux attached to each of them. We find in particular that, when the effective gauge flux from the topological defects becomes maximal, the zero-energy modes of the Dirac equation can be arranged into two triplets, that can be thought as the counterpart of the two triplets of zero modes that arise in the dual instance of the continuum limit of large spherical fullerenes. We further investigate the graphene wormhole spectra by performing a numerical diagonalization of tight-binding Hamiltonians for very large lattices realizing the wormhole geometry. The correspondence between the number of localized electronic states observed in the numerical approach and the effective gauge flux predicted in the continuum limit shows that graphene wormholes can be consistently described by an effective theory of two Dirac fermion fields in the curved geometry of the wormhole, opening the possibility of using real samples of the carbon material as a playground to experiment with the interaction between the background curvature and the Dirac fields.     

Landau levels in graphene layers with topological defects

In this work, we study the low-energy electronic spectrum of a graphene layer structure with a disclination in the presence of a magnetic field. We make this study using the continuum approach, where we use the geometric theory of topological defects to introduce a disclination in a graphene layer, and the electrons are described by the massless Dirac equation in this curved background. The bound states energy spectrum and eigenfunctions are also obtained and an explicit dependence was found on the parameter that characterizes the topological defect and on the magnetic field.     

Dirac Equation in Space–Time with Torsion

The Dirac equation in a curved space–time endowed with compatible affine connection is reconsidered. After a detailed decomposition of the total action, the equation is obtained by varying with respect to the Dirac spinor and the torsion field. The result is a known Dirac-like equation with constraints that can be interpreted as the equation of a self-interacting spin 1/2 particle in curved space–time. The scheme is then translated into the language of the 2-spinor formalism of curved space–time based on the choice of a null tetrad frame. The spinorial equation so obtained coincides with the standard one in case of no torsion, while in general it remains a nonlinear equation describing a self-interacting spin 1/2 particle. The nonlinearity is produced by the interaction of the particle with its own current that remains conserved as in the free torsion case.     

Generalization of the Kelvin equation for arbitrarily curved surfaces

Capillary condensation, which takes place in confined geometries, is the first-order vapor-to-liquid phase transition and is explained by the Kelvin equation, but the equation's applicability for arbitrarily curved surface has been long debated and is severe problem. Recently, we have proposed generic dynamic equations for moving surfaces. Application of the equations to the vapor/fluid interfaces in chemical equilibrium conditions nearly trivially solves the generalization problem for the Kelvin equation. The equations are universally true for any surfaces: atomic, molecular, micro or macro scale, real or virtual, Riemannian or pseudo-Riemannian, active or passive.     

Quantum Projector Method on Curved Manifolds

A generalized stochastic method for projecting out the ground state of the quantum many-body Schrödinger equation on curved manifolds is introduced. This random-walk method is of wide applicability to any second order differential equation (first order in time), in any spatial dimension. The technique reduces to determining the proper quantum corrections for the Euclidean short-time propagator that is used to build up their path-integral Monte Carlo solutions. For particles with Fermi statistics the Fixed-Phase constraint (which amounts to fixing the phase of the many-body state) allows one to obtain stable, albeit approximate, solutions with a variational property. We illustrate the method by applying it to the problem of an electron moving on the surface of a sphere in the presence of a Dirac magnetic monopole.     

Curved graphene nanoribbons and tunneling current

The density of states and the electronic spectrum of long-wave electrons in a curved graphene nanoribbon is calculated on the basis of the Dirac equation in curved space-time. Using this density of states, we obtain the current-voltage characteristics of tunnel junctions of nanoribbons with metal and quantum dots. The dependence of the curved nanoribbon on the geometric parameters is found.     

Investigation of Fermions in Non-commutative Space by Considering Kratzer Potential

The two-dimensional Dirac equation for a fermion moving under Kratzer potential in the presence of an external magnetic field is analytically being solved for the energy eigenvalues and eigenfunctions. Subsequently, we have obtained the Wigner function corresponding to the eigenfunctions.     

On the electronic spectrum in curved graphene nanoribbons

Using the formalism of the Dirac equation for curved space-time in the Friedmann model of a non-stationary universe, we calculate the electronic spectrum and density of states in curved graphene nanoribbons. Based on the obtained density of states we further study the current-voltage characteristics of the nanoribbonmetal tunnel junction. The dependence on the geometric characteristics of the nanoribbon has been revealed, showing a great influence of such parameters as the number of carbon atoms and the characteristic frequency of distortion.     

Electromagnetic solitons in a system of graphene planes with Anderson impurities

We consider the Maxwell equations for the electromagnetic-field propagation in a system of graphene planes with Anderson impurities. A phenomenological equation is obtained in the form of an analog of the classical 1 + 1-dimensional sine-Gordon equation. Electrons are considered within the quantum formalism taking into account the dispersion-law variations in the presence of an impurity subsystem. The phenomenological equation is analyzed numerically. It was found that the formation of a forbidden band in the graphene spectrum influenced the propagation of ultrashort optical pulses.     

洛仑兹破缺标量场的霍金隧穿辐射

把洛仑兹破缺的标量场方程推广到弯曲时空中,并通过Aether-like项对标量场方程进行修正,该项所产生的效应也会影响到黑洞时空视界附近处的物理效应.接着,进一步在半经典近似下得到了修正的Hamilton-Jacobi方程,然后用这一修正的Hamilton-Jacobi方程研究了史瓦西黑洞的隧穿辐射特征,并讨论了洛仑兹破缺对黑洞霍金辐射和黑洞熵的影响.结果表明,u~α=δ_t~αu~t,δ_r~αu~r形式的Aether-like项的效应可能使黑洞温度增加,而黑洞熵降低.该工作可以帮助我们更深刻地理解弯曲时空中的洛仑兹破缺效应的物理性质.     

Soliton dynamics in the complex modified KdV equation with a periodic potential

We study the existence and stability of stationary and moving solitary waves in a periodically modulated system governed by an extended cmKdV (complex modified Korteweg-de Vries) equation. The proposed equation describes, in particular, the co-propagation of two electromagnetic waves with different amplitudes and orthogonal linear polarizations in a liquid crystal waveguide, the stronger (nonlinear) wave actually carrying the soliton, while the other (a nearly linear one) creates an effective periodic potential. A variational analysis predicts solitons pinned at minima and maxima of the periodic potential, and the Vakhitov-Kolokolov criterion predicts that some of them may be stable. Numerical simulations confirm the existence of stable stationary solitary waves trapped at the minima of the potential, and show that persistently moving solitons exist too. The dynamics of pairs of interacting solitons is also studied. In the absence of the potential, the interaction is drastically different from the behavior known in the NLS (nonlinear Schrödinger) equation, as the force of the interaction between the cmKdV solitons is proportional to the sine, rather than cosine, of the phase difference between the solitons. In the presence of the potential, two solitons placed in one potential well form a persistently oscillating bound state.     

Radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems

In graded index medium, the ray goes along a curved path determined by Fermat principle, and the curved ray-tracing is very difficult and complex. To avoid the complicated and time-consuming computation of curved ray trajectory, the methods not based on ray-tracing technique need to be developed for the solution of radiative transfer in graded index medium. For this purpose, in this paper the streaming operator along a curved ray trajectory in original radiative transfer equation for graded index medium is transformed and expressed in spatial and angular ordinates and the radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems are derived. The conservative and the non-conservative forms of radiative transfer equation for three-dimensional graded index medium are given, which can be used as base equations to develop the numerical simulation methods, such as finite volume method, discrete ordinates method, and finite element method, for radiative transfer in graded index medium in cylindrical and spherical coordinate systems.     

单层石墨烯片的非线性板模型

基于实验得到的非线性本构关系和板理论,本文建立了包含三次及五次非线性项的单层石墨烯片的板动力学模型.针对四边简支矩形板,使用Ritz法研究了在板中点作用集中力时的静力弯曲,以及边界均匀受力时的静力屈曲问题.结果显示,基于非线性本构关系的板模型能很好的描述单层石墨烯片的力学行为,而且模型中的五次非线性项对结构的弯曲变形有显著影响.     

Numerical solution for curved crack problem in elastic half-plane using hypersingular integral equation

A hypersingular integral equation for the curved crack problems of an elastic half-plane is introduced. Formulation of the equation is based on the usage of a modified complex potential. The potential is generally expressed in the form of a Cauchy-type integral. The modified complex potential is composed of the principal part and the complementary part. The principal part of the complex potential is actually equivalent to the original complex potential for the curved crack in an infinite plate. The role of the complementary part is to eliminate the boundary traction along the boundary of the half-plane caused by the principal part. From the assumed boundary traction condition, a hypersingular integral equation is obtained for the curved crack problems of an elastic half-plane. The curve length coordinate method is used to obtain a final solution. Several numerical examples are presented that prove the efficiency of the suggested method.     

格子玻尔兹曼方法模拟弯流道中粒子的惯性迁移行为

本文发展了一个能够模拟微流场环境下粒子惯性迁移行为的三维耦合模型.该模型采用基于动理论的格子玻尔兹曼方法(LBM)描述流体流动,采用牛顿动力学模型描述粒子的平动和转动,采用基于LBM反弹格式的运动边界法实现流体与粒子模型的耦合.模拟了重力作用下粒子的沉降过程和Couette流条件下粒子的转动过程,通过将模拟结果与文献中的基准解进行对比定量验证了模型的可靠性.模拟了不同大小的球形粒子在环形流道中的迁移,成功复现了经典的流道截面二次流形成过程,分析了粒径大小对粒子在流道中平衡位置的影响机理.结果表明,粒子在弯流道中的平衡位置与粒径大小密切相关,小半径粒子的平衡位置靠近流道外侧而大半径粒子则靠近流道内侧.通过实验对模拟结果进行了定性验证.本模型为深入研究微流场环境下粒子的运动特性以及开发微流控粒子分选器件提供了参考依据.     

Comparing the relativistic and non-relativistic transmission coefficients of the electron transport through a quantum heterostructure

The Letter deals with the transport of relativistic electron through a quantum heterostructure in the presence of a constant bias voltage. The transmission coefficient associated to the relativistic moving through the considered system is computed using the transfer matrix formalism for piecewise constant potentials. The results are compared to those which correspond to the non-relativistic case, in which the Schrödinger equation is used. The comparison is performed for different values of the system length and the effect of the bias voltage is also studied.     

Transmission through biased graphene strip

We solve the 2D Dirac equation describing graphene in the presence of a linear vector potential. The discretization of the transverse momentum due to the infinite mass boundary condition reduced our 2D Dirac equation to an effective massive 1D Dirac equation with an effective mass equal to the quantized transverse momentum. We use both a numerical Poincaré map approach, based on space discretization of the original Dirac equation, and a direct analytical method. These two approaches have been used to study tunneling phenomena through a biased graphene strip. The numerical results generated by the Poincaré map are in complete agreement with the analytical results.