A. E. Botha

Devil's staircases and continued fractions in Josephson junctions

The detailed numerical simulations of the IV-characteristics of Josephson junction under external electromagnetic radiation show devils staircases within different bias current intervals. We have found that the observed steps form very precisely continued fractions. Increasing of the amplitude of radiation shifts the devils staircases to higher Shapiro steps. The algorithm of appearing and detection of the subharmonics with increasing radiation amplitude is proposed. We demonstrate that subharmonic steps registered in the famous experiments by A. H. Dayem and J. J. Wiegand [Phys. Rev 155, 419 (1967)] and J. Clarke [Phys. Rev. B 4, 2963 (1971)] also form continued fractions.

Optimized shooting method for finding periodic orbits of nonlinear dynamical systems

An alternative numerical method is developed to find stable and unstable periodic orbits of nonlinear dynamical systems. The method exploits the high efficiency of the Levenberg–Marquardt algorithm for medium-sized problems and has the additional advantage of being relatively simple to implement. It is also applicable to both autonomous and non-autonomous systems. As an example of its use, it is employed to find periodic orbits in the Rössler system, a coupled Rössler system, as well as an eight-dimensional model of a flexible rotor-bearing; problems which have been treated previously via two related methods. The results agree with the previous methods and are seen to be more accurate in some cases. A simple implementation of the method, written in the Python programming language, is provided as an Appendix.

Structured chaos in a devil's staircase of the Josephson junction

The phase dynamics of Josephson junctions (JJs) under external electromagnetic radiation is studied through numerical simulations. Current-voltage characteristics, Lyapunov exponents, and Poincaré sections are analyzed in detail. It is found that the subharmonic Shapiro steps at certain parameters are separated by structured chaotic windows. By performing a linear regression on the linear part of the data, a fractal dimension of D = 0.868 is obtained, with an uncertainty of ±0.012. The chaotic regions exhibit scaling similarity, and it is shown that the devils staircase of the system can form a backbone that unifies and explains the highly correlated and structured chaotic behavior. These features suggest a system possessing multiple complete devils staircases. The onset of chaos for subharmonic steps occurs through the Feigenbaum period doubling scenario. Universality in the sequence of periodic windows is also demonstrated. Finally, the influence of the radiation and JJ parameters on the structured chaos is investigated, and it is concluded that the structured chaos is a stable formation over a wide range of parameter values.

Manifestation of resonance-related chaos in coupled Josephson junctions

Abstract Manifestation of chaos in the temporal dependence of the electric charge is demonstrated through the calculation of the maximal Lyapunov exponent, phase–charge and charge–charge Lissajous diagrams and correlation functions. It is found that the number of junctions in the stack strongly influences the fine structure in the current–voltage characteristics and a strong proximity effect results from the nonperiodic boundary conditions. The observed resonance-related chaos exhibits intermittency. The criteria for a breakpoint region with no chaos are obtained. Such criteria could clarify recent experimental observations of variations in the power output from intrinsic Josephson junctions in high temperature superconductors.

Modeling of LC-shunted intrinsic Josephson junctions in high-Tc superconductors

Resonance phenomena in a model of intrinsic Josephson junctions shunted by LC-elements (L-inductance, C-capacitance) are studied. The phase dynamics and IV-characteristics are investigated in detail when the Josephson frequency approaches the frequency of the resonance circuit. A realization of parametric resonance through the excitation of a longitudinal plasma wave, within the bias current interval corresponding to the resonance circuit branch, is demonstrated. It is found that the temporal dependence of the total voltage of the stack, and the voltage measured across the shunt capacitor, reflect the charging of superconducting layers, a phenomenon which might be useful as a means of detecting such charging experimentally. Thus, based on the voltage dynamics, a novel method for the determination of charging in the superconducting layers of coupled Josephson junctions is proposed. A demonstration and discussion of the influence of external electromagnetic radiation on the IV-characteristics and charge-time dependence is given. Over certain parameter ranges the radiation causes an interesting new type of temporal splitting in the charge-time oscillations within the superconducting layers.

Characteristic distribution of finite-time Lyapunov exponents for chimera states.

Our fascination with chimera states stems partially from the somewhat paradoxical, yet fundamental trait of identical, and identically coupled, oscillators to split into spatially separated, coherently and incoherently oscillating groups. While the list of systems for which various types of chimeras have already been detected continues to grow, there is a corresponding increase in the number of mathematical analyses aimed at elucidating the fundamental reasons for this surprising behaviour. Based on the model systems, there are strong indications that chimera states may generally be ubiquitous in naturally occurring systems containing large numbers of coupled oscillators – certain biological systems and high-Tc superconducting materials, for example. In this work we suggest a new way of detecting and characterising chimera states. Specifically, it is shown that the probability densities of finite-time Lyapunov exponents, corresponding to chimera states, have a definite characteristic shape. Such distributions could be used as signatures of chimera states, particularly in systems for which the phases of all the oscillators cannot be measured directly. For such cases, we suggest that chimera states could perhaps be detected by reconstructing the characteristic distribution via standard embedding techniques, thus making it possible to detect chimera states in systems where they could otherwise exist unnoticed.

Multiband k-p Riccati equation for electronic structure and transport in type-II heterostructures

An alternative method is proposed and implemented to calculate electronic structure and quantum transport properties of type-II heterojunctions. By deriving a multiband k.p Riccati equation for the envelope function matrix, it is shown how to obtain the reflection matrix through a simple numerical integration of the Riccati equation. Numerical instability, which is usually associated with type-II systems due to the simultaneous presence propagating and evanescent states, is avoided by working with the logarithmic derivative of the envelope function matrix. The theory is implemented numerically for a 6-band k.p matrix Hamiltonian in which the electron spin components have been decoupled. Preliminary results are presented for InAs/GaSb/InAs quantum wells. First, the calculated transmission versus energy curves are compared to those obtained from the microscopic empirical pseudo-potential calculation of Edwards and Inkson [Semicond. Sci. Technol. 9 (1994) 178]. For GaSb layer widths of 18, 55 and 152A, the transmission spectra agree almost exactly. Minor discrepancies are discussed. Second, the net current density is calculated at room temperature as a function of applied voltage. For a GaSb layer width of 74A, the self-consistently calculated net current density and peak-to-valley ratios are found to be in semi-quantitative agreement with the experiment of Yu et al. [Appl. Phys. Lett. 57 (1990) 2675]. Other than its applications to electron tunneling, which can include spin-dependent tunneling (spintronics), the theory may also be applied to emerging field of multi-channel inverse scattering; in which desired transmission or reflection properties are used as input for designing heterostructure potential profiles.

General R-matrix approach for integrating the multiband k.p equation in layered semiconductor structures

Abstract A Fortran 90 code is provided for calculating the electron reflection and transmission coefficients in semiconductor heterostructures within the 14-band k ⋅ p approximation. The code may easily be adapted for use with any k ⋅ p model, including magnetic field and/or strain effects, for example. Numerical instability, which is problematic in type-II systems due to the simultaneous presence of propagating and evanescent states, is reduced by developing a novel log-derivative R -matrix approach based on the Jost solution to the k ⋅ p equation. Program summary Program title: multiband-kp Catalogue identifier: AEKG_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEKG_v1_0.html Program obtainable from: CPC Program Library, Queenʼs University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 7088 No. of bytes in distributed program, including test data, etc.: 90 237 Distribution format: tar.gz Programming language: Fortran 90 Computer: HP 128-node cluster (8 Intel 3.0 GHz Xeon processors per node) Operating system: RedHat Enterprise Linux 5.1 RAM: 11 MB Classification: 7.3, 7.9 External routines: LAPACK [1], ODE [2] Nature of problem: Calculating the electron transmission (or reflection) coefficient for single, double or multiple semiconductor quantum wells. Solution method: Makes use of a log-derivative reflection matrix approach which is based on obtaining the Jost solution to the multiband envelope function k ⋅ p equation. Restrictions: Accuracy depends on the limitations of the k ⋅ p model. In this implementation a “bare” 14-band model is used. Unusual features: By default Intelʼs math-kernel-library (MKL) [3] runs in serial mode. MKL also has built in parallel matrix algorithms which can be invoked without explicit parallelization in the source code. In this case all of the 8 CPUs in one node are used by the LAPACK subroutines. Running time: The given sample output, for the transmission coefficient at 750 different energies, required 376.51 CPU seconds (less than 7 minutes on a single CPU). References: [1] E. Anderson, et al., LAPACK Usersʼ Guide, 3rd edition, Society for Industrial and Applied Mathematics, Philadelphia, 1999. [2] L.F. Shampine, M.K. Gordan, Computer Solution to Ordinary Differential Equations: The Initial Value Problem, W.H. Freeman and Company, San Francisco, 1975. [3] http://software.intel.com/en-us/articles/intel-math-kernel-library-documentation/ .

Devil's staircase and the absence of chaos in the dc- and ac-driven overdamped Frenkel-Kontorova model

The devils staircase structure arising from the complete mode locking of an entirely nonchaotic system, the overdamped dc+ac driven Frenkel-Kontorova model with deformable substrate potential, was observed. Even though no chaos was found, a hierarchical ordering of the Shapiro steps was made possible through the use of a previously introduced continued fraction formula. The absence of chaos, deduced here from Lyapunov exponent analyses, can be attributed to the overdamped character and the Middleton no-passing rule. A comparative analysis of a one-dimensional stack of Josephson junctions confirmed the disappearance of chaos with increasing dissipation. Other common dynamic features were also identified through this comparison. A detailed analysis of the amplitude dependence of the Shapiro steps revealed that only for the case of a purely sinusoidal substrate potential did the relative sizes of the steps follow a Farey sequence. For nonsinusoidal (deformed) potentials, the symmetry of the Stern-Brocot tree, depicting all members of particular Farey sequence, was seen to be increasingly broken, with certain steps being more prominent and their relative sizes not following the Farey rule.

Evaluation of Kirkwood-Buff integrals via finite size scaling: a large scale molecular dynamics study

Solvation of bio-molecules in water is severely affected by the presence of co-solvent within the hydration shell of the solute structure. Furthermore, since solute molecules can range from small molecules, such as methane, to very large protein structures, it is imperative to understand the detailed structure-function relationship on the microscopic level. For example, it is useful know the conformational transitions that occur in protein structures. Although such an understanding can be obtained through large-scale molecular dynamic simulations, it is often the case that such simulations would require excessively large simulation times. In this context, Kirkwood-Buff theory, which connects the microscopic pair-wise molecular distributions to global thermodynamic properties, together with the recently developed technique, called finite size scaling, may provide a better method to reduce system sizes, and hence also the computational times. In this paper, we present molecular dynamics trial simulations of biologically relevant low-concentration solvents, solvated by aqueous co-solvent solutions. In particular we compare two different methods of calculating the relevant Kirkwood-Buff integrals. The first (traditional) method computes running integrals over the radial distribution functions, which must be obtained from large system-size NVT or NpT simulations. The second, newer method, employs finite size scaling to obtain the Kirkwood-Buff integrals directly by counting the particle number fluctuations in small, open sub-volumes embedded within a larger reservoir that can be well approximated by a much smaller simulation cell. In agreement with previous studies, which made a similar comparison for aqueous co-solvent solutions, without the additional solvent, we conclude that the finite size scaling method is also applicable to the present case, since it can produce computationally more efficient results which are equivalent to the more costly radial distribution function method.