Periodic orbits in polygonal billiards

We review some properties of periodic orbit families in polygonal billiards and discuss in particular a sum rule that they obey. In addition, we provide algorithms to determine periodic orbit families and present numerical results that shed new light on the proliferation law and its variation with the genus of the invariant surface. Finally, we deal with correlations in the length spectrum and find that long orbits display Poisson fluctuations.     

矩形弹子球中的量子谱和经典周期轨道

利用SU(2)相干态的表示,我们构造了二维矩形弹子球中与经典周期轨道对应的波函数.经典周期轨道和量子波函数之间的关系可以通过物理图像清晰的表示出来.另外,利用周期轨道理论,我们计算了二维矩形弹子球体系的量子谱的傅立叶变换ρ(L).变换谱|ρN(L)|2对L图像中的峰可以和粒子在二维矩形腔中运动的经典轨迹的长度相比较.量子谱中的每一条峰正好对应一条经典周期轨道的长度,表明量子力学和经典力学的对应关系.     

Ground state of an arbitrary triangle with a Calogero-Sutherland-Moser potential

We construct the expression for the ground state eigenfunction of the Schrödinger equation for a particle inside an arbitrary planar triangle under the influence of a potential.     

Negative Magneto-Resistance Beyond Weak Localization in Three-Dimensional Billiards:Effect of Arnold Diffusion

We investigate a semiclassical conductance for ballistic open three-dimensional (3-d) billiards. For partially or completely broken-ergodic 3-d billiards such as SO(2) symmetric billiards, the dependence of the conductance on the Fermi wavenumber is dramatically changed by the lead orientation. Application of a symmetry-breaking weak magnetic field brings about mixed phase-space structures of 3-d billiards which ensures a novel Arnold diffusion that cannot be seen in 2-d billiards. In contrast to the 2-d case, the anomalous increment of the conductance should inevitably include a contribution arising from Arnold diffusion as well as a weak localization correction. Discussions are devoted to the physical condition for observing this phenomenon.     

Scars of periodic orbits in the stadium action billiard

Compact billiards in phase space, or action billiards, are constructed by truncating the classical Hamiltonian in the action variables. The corresponding quantum mechanical system has a finite Hamiltonian matrix. In previous papers we defined the compact analog of common billiards, i.e., straight motion in phase space followed by specular reflections at the boundaries. Computation of their quantum energy spectra establishes that their properties are exactly those of common billiards: the short-range statistics follow the known universality classes depending on the regular or chaotic nature of the motion, while the long-range fluctuations are determined by the periodic orbits. In this work we show that the eigenfunctions also follow qualitatively the general characteristics of common billiards. In particular, we show that the low-lying levels can be classified according to their nodal lines as usual and that the high excited states present scars of several short periodic orbits. Moreover, since all the eigenstates of action billiards can be computed with great accuracy, Bogomolny's semiclassical formula for the scars can also be tested successfully.     

Stochastic modeling of a billiard in a gravitational field: Power law behavior of Lyapunov exponents

We consider the motion of a point particle (billiard) in a uniform gravitational field constrained to move in a symmetric wedge-shaped region. The billiard is reflected at the wedge boundary. The phase space of the system naturally divides itself into two regions in which the tangent maps are respectively parabolic and hyperbolic. It is known that the system is integrable for two values of the wedge half-angle ₁ and ₂ and chaotic for ₁<< ₂. We study the system at three levels of approximation: first, where the deterministic dynamics is replaced by a random evolution; second, where, in addition, the tangent map in each region is, replaced by its average; and third, where the tangent map is replaced by a single global average. We show that at all three levels the Lyapunov exponent exhibits power law behavior near ₁ and ₂ with exponents 1/2 and 1, respectively. We indicate the origin of the exponent 1, which has not been observed in unaccelerated billiards.     

Quantum chaos and magnetotransport in lateral superlattices: The transition from weak to strong modulation potential

Magnetotransport in lateral superlattices covers a variety of apparently different physics dependent on the amplitude of the modulation potential relative to the Fermi energy ranging from the weakly modulated 2DEG to the antidot lattice. Here we investigate the transition between weak modulation and antidots within the quantum-mechanical theory by studying (i) the single-particle spectrum with respect to quantum signatures of classical chaos and (ii) the change of characteristic features in the magnetoresistance starting from the Kubo formula.     

Semi-classical smoothing in a non-monotonic field

It is pointed out that the semi-classical method of Jennings, Bhaduri and Brack for smoothing the total energy of a system of non-interacting fermions in a potential wellV(r) breaks down in the fourth-order term if this potential is non-monotonic. This means that the method as it stands can only be used by neglecting the Coulomb force on protons. We propose a modified treatment for the fourth-order term which appears to be entirely satisfactory.     

Quantum transport and classical dynamics in open billiards

We report numerical results of an investigation of quantum transport for a weakly opened integrable circle and chaotic stadium billiards with a pair of conducting leads. While the statistics of spacings of resonance energies commonly follow the Wigner (GOE)-like distribution, the electric conductance as a function of the Fermi wavenumber shows characteristic noisy fluctuations associated with a typical set of classical orbits unique for both billiards. The wavenumber autocorrelation for the conductance is stronger in the stadium than the circle billiard, which we show is related to the length spectrum of classical short orbits. We propose an explanation of these contrasts in terms of the effect of phase decoherence due to the underlying chaotic dynamics.     

Periodic orbits in magnetic billiards

We propose a simple method to calculate periodic orbits in two-dimensional systems with no symbolic dynamics. The method is based on a line by line scan of the Poincaré surface of section and is particularly useful for billiards. We have applied it to the Square and Sinai's billiards subjected to a uniform orthogonal magnetic field and we obtained about 2000 orbits for both systems using absolutely no information about their symbolic dynamics. Received 21 September 1999 and Received in final form 13 April 2000     

Homoclinic orbits in three-dimensional Shilnikov-type chaotic systems

In this paper, the Padé approximant and analytic solution in the neighborhood of the initial value are introduced into the process of constructing the Shilnikov type homoclinic trajectories in three-dimensional nonlinear dynamical systems. The PID controller system with quadratic and cubic nonlinearities, the simplified solar-wind-driven-magnetosphere-ionosphere system, and the human DNA sequence system are considered. With the aid of presenting a new condition, the solutions of solving the boundary-value problems which are formulated for the trajectory and evaluating the initial amplitude values become available. At the same time, the value of the bifurcation parameter is obtained directly, which is almost consistent with the numerical result.     

Magnetotransport in a graphene monolayer with two tunable magnetic barriers

The magnetotransport property for a monolayer graphene with two turnable magnetic barriers has been investigated by the transfer-matrix method. We show that the parameters of barrier height, width, and interval between two barriers affect the electron wave decaying length, which determine the conductance with parallel or antiparallel magnetization configuration, and consequently the tunneling magnetoresistance (TMR) for the system. Interestingly, a graphene attached by two barriers with different heights can produce a resonant TMR peak at low energy region one order of magnitude larger than that for the system with two same height barriers because that the asymmetry of magnetic barriers block the electron transmission in the case of antiparallel magnetization configuration. The results obtained here may be useful in understanding of electron tunneling in graphene and in designing of graphene-based nanodevices.     

A peculiar Maxwell’s Demon observed in a time-dependent stadium-like billiard

The dynamics of a driven stadium-like billiard is considered using the formalism of discrete mappings. The model presents a resonant velocity that depends on the rotation number around fixed points and external boundary perturbation which plays an important separation rule in the model. We show that particles exhibiting Fermi acceleration (initial velocity is above the resonant one) are scaling invariant with respect to the initial velocity and external perturbation. However, initial velocities below the resonant one lead the particles to decelerate therefore unlimited energy growth is not observed. This phenomenon may be interpreted as a specific Maxwell’s Demon which may separate fast and slow billiard particles.     

Ballistic magnetoresistance in different nanocontact configurations: a basis for future magnetoresistance sensors

We report ballistic magnetoresistance (BMR) values in magnetic nanocontacts for Ni, Co, and Fe. The samples range from atomic nanocontacts (smaller than 1 nm cross-section) to stable electrodeposited nanocontacts (up to 30 nm cross-section). The experiments are done at room temperature and up to 4 kOe applied field. We obtain values of stable BMR up to 700%. By manipulating the resistance and the contact shape electrochemically in situ we can have any desired value of BMR. We also discuss BMR in Ni microclusters contacted through pinholes on thin oxides with nanometer thick Ni and Co films with BMR up to 15%. All the experiments show that the BMR is a very local effect of the size and shape of the nanocontact. In this respect 2D and 3D domain wall calculations are presented. The experiments reported here show that magnetic nanocontacts have potential for development of highly compacted sensor.     

A variational approach to connecting orbits in nonlinear dynamical systems

We propose a variational method for determining homoclinic and heteroclinic orbits including spiral-shaped ones in nonlinear dynamical systems. Starting from a suitable initial curve, a homotopy evolution equation is used to approach a true connecting orbit. The procedure is an extension of a variational method that has been used previously for locating cycles, and avoids the need for linearization in search of simple connecting orbits. Examples of homoclinic and heteroclinic orbits for typical dynamical systems are presented. In particular, several heteroclinic orbits of the steady-state Kuramoto–Sivashinsky equation are found, which display interesting topological structures, closely related to those of the corresponding periodic orbits.     

Homoclinic orbits and mixed-mode oscillations in far-from-equilibrium systems

Nonlinear autonomous dynamical systems with ahomoclinic tangency to a periodic orbit are investigated. We study the bifurcation sequences of the mixed-mode oscillations generated by the homoclinicity, which are shown to belong to two different types, depending on the nature of the Liapunov numbers of the basic periodic orbit. A detailed numerical analysis is carried out to show how the existence of a tangent homoclinic orbit allows us to understand in a quantitative way a particular and regular sequence of cool flame-ignition oscillations observed in a thermokinetic model of hydrocarbon oxidation. Chaotic cool flame oscillations are also observed in the same model. When the control parameter crosses a critical value, this chaotic set of trajectories becomes globally unstable and forms a Cantor-like hyperbolic repellor, and the ignition mechanism generates ahomoclinic tangency to the Cantor set of trajectories. The complex bifurcation diagram may be globally reconstructed from a one-dimensional dynamical system, thanks to the strong contractivity of thermokinetics. It is found that a symbolic dynamics with three symbols is necessary to classify the periodic windows of the complex bifurcation sequence observed numerically in this system.     

300% magnetocurrent in a room temperature operating spin-valve transistor

Here we present the realization of a room temperature operating spin-valve transistor with huge magnetocurrent (MC=300%) at low fields. This spin-valve transistor employs hot-electron transport across a Ni₈₁Fe₁₉/Au/Co spin valve. Hot electrons are injected into the spin valve across a Si–Pt Schottky barrier. After traversing the spin valve, these hot electrons are collected using a second Schottky barrier (Si–Au), which provides energy and momentum selection. The collector current is found to be extremely sensitive to the spin-dependent scattering of hot electrons in the spin valve, and therefore on the applied magnetic field. We also illustrate the role of the collector diode characteristics in determining the magnetocurrent under collector bias.     

Magnetotransport properties and magnetocaloric effect in La0.67Ca0.33Mn1−xTMxO3 (TM=Cu,Zn) perovskite manganites

The magnetic and transport properties of the perovskites La_0.67Ca_0.33Mn_1-xTM_xO₃ were found to be sufficiently changed with the substitution of Mn-sites by other 3d transition-metal cations (TM=Cu,Zn; x=0.15). The values of T_C, T_M−I, and T_CMR were surveyed when Mn was replaced by Cu and Zn. The magnetic field induced resistivity and magnetic entropy change of these samples showed abrupt changes near T_C (194.2 and 201.5 K for Cu and Zn-doped case respectively) and attained the highest values among the doped cases (up to 20% Cu). The maximum values (obtained at H=4 kOe) of magnetoresistance ratio (CMR) were 27.8%, and 24.5% and of magnetic entropy change (−ΔS_M) were 3.9 and 3.2 J/kg K for Cu and Zn-doped, respectively.     

Growth protocols and characterization of epitaxial graphene on SiC elaborated in a graphite enclosure

The epitaxial growth of graphene by the sublimation of Si-terminated silicon carbide (SiC) is studied inside a graphite enclosure in a radio-frequency furnace by comparing different in situ processes involving hydrogen etching or not and different growth conditions. For the growth under vacuum, even with the surface preparation of hydrogen etching, the morphology of the synthesized graphene is found full of voids and defects in the form of a multilayer graphene film. For the growth under Ar, the hydrogen etching plays a vital role to improve the graphene quality in terms of surface roughness, the number of graphene layers and the domain size. For the graphene samples grown with the proposed protocol, the original combination of micro-probe Raman spectroscopy and simultaneous optical transmission and reflection measurements reveals a detailed spatially resolved image of the graphene domains with monolayer domain size of ~5×5 µm² on about 2/3 of the total sample surface. The magnetotransport data yield charge-carrier mobilities up to 2900 cm²/Vs as found for high quality graphene on the Si-face of SiC. The observed magnetoquantum oscillations in the magnetoresistance confirm the expected behavior of single-layer graphene.