Optical properties of structures composed of periodic,quasi-periodic,and aperiodic sequences of particulate monolayers

The spectra of the coherent transmission and reflection coefficients of multilayers consisting of the periodic, Fibonacci (quasi-periodic), and Thue–Morse (aperiodic) sequences of plane-parallel monolayers of monodisperse spherical alumina and silica particles are investigated using the quasi-crystalline approximation (QCA) and the transfer matrix method (TMM). The additional opportunities for the transmission and reflection spectra manipulation in comparison with the periodic sequence of monolayers are demonstrated. Photonic band gaps in the spectra of the particulate structures are shifted to the short-wavelength range in comparison with those for systems of homogeneous layers. The shift is larger for the Thue–Morse sequence. The widths of the photonic band gaps for particulate systems are narrower than the ones for multilayers consisting of homogeneous layers of an equivalent volume of matter. The results can be used to create optical, optoelectronics, and photonics devices—for example, multispectral filters, light emitting diodes, solar cells, displays.     

Quasiperiodic magnonic superlattices with mirror symmetry

We address the spin wave modes propagating in Fibonacci, Thue–Morse, and double period quasiperiodic magnonic superlattices. These structures are made of layers of a metamagnetic material alternating with layers of a nonmagnetic material, presenting mirror symmetry. Our calculations are carried out in the magnetostatic regime for the antiferromagnetic phase. Our model takes into account the presence of an external applied magnetic field, which is perpendicular to the interfaces of the superlattice, as well as the crystalline anisotropic contribution to the inner magnetic field. The magnetostatic bulk and surface modes are obtained by using the transfer matrix technique. The metamagnetic material considered here is FeBr₂, however, our results can be extended to other materials. Our numerical results show the behavior of these modes, for small frequencies of the energy spectra. The results reported here can be experimentally observed by light scattering techniques.     

Unbounded Trace Orbits of Thue–Morse Hamiltonian

It is well known that, an energy is in the spectrum of Fibonacci Hamiltonian if and only if the corresponding trace orbit is bounded. However, it is not known whether the same result holds for the Thue–Morse Hamiltonian. In this paper, we give a negative answer to this question. More precisely, we construct two subsets \(\Sigma _{II}\) and \(\Sigma _{III}\) of the spectrum of the Thue–Morse Hamiltonian, both of which are dense and uncountable, such that each energy in \(\Sigma _{II}\cup \Sigma _{III}\) corresponds to an unbounded trace orbit. Exact estimates on the norm of the transfer matrices are also obtained for these energies: for \(E\in \Sigma _{II}\cup \Sigma _{III}, \) the norms of the transfer matrices behave like

$$\begin{aligned} e^{c_1\gamma \sqrt{n}}\le \Vert T_{ n}(E)\Vert \le e^{c_2\gamma \sqrt{n}}. \end{aligned}$$

However, two types of energies are quite different in the sense that each energy in \(\Sigma _{II}\) is associated with a two-sided pseudo-localized state, while each energy in \(\Sigma _{III}\) is associated with a one-sided pseudo-localized state. The difference is also reflected by the local dimensions of the spectral measure: the local dimension is 0 for energies in \(\Sigma _{II}\) and is larger than 1 for energies in \(\Sigma _{III}.\) As a comparison, we mention another known countable dense subset \(\Sigma _I\). Each energy in \(\Sigma _I\) corresponds to an eventually constant trace map and the associated eigenvector is an extended state. In summary, the Thue–Morse Hamiltonian exhibits “mixed spectral nature”.

     

Perfect transmission of spin waves in a one-dimensional magnonic quasicrystal

Existence of the perfect transmission resonances of spin waves in a one-dimensional Thue–Morse magnonic quasicrystal is proposed. We find that occurrence of the perfect transmission just corresponds to the repeated bandedges in the bandedge map. Frequencies at the perfect transmissions for system with arbitrary generation order can be predicted by the bandedge map of the second and third order systems using two iterative schemes. These results show that density of the perfect transmission resonance increases exponentially for increasing the order of the system. However, the perfect transmissions are kept for higher order systems even if the peaks become denser.     

On the Hausdorff Dimension of the Spectrum of the Thue–Morse Hamiltonian

We show that the Hausdorff dimension of the spectrum of the Thue–Morse Hamiltonian has a common positive lower bound for all coupling.     

Efficient band-pass and stop-band filtering by GaAs–Al c Ga1− c As generalized Thue–Morse multibarrier systems

The influence of Al concentration and barrier width on the transmission spectra in GaAs–Al _c Ga_{1? c} As heterostructure is studied for two different kinds of the generalized Thue–Morse multibarrier systems (GTMS). These structures provide interesting filter properties with regard to effective band-pass or stop-band (of extremely low width) circuitry. The first kind of GTMS is suitable for application as a band-pass electronic filter, while the second kind as stop-band filter. High Al concentration and large barrier width result in sharp resonances in the transmission spectrum.     

Fibonacci序列的统计属性和电子输运系数

对按膨胀规律A→AB和B→A生成的Fibonacci序列,采用一维随机行走模型数值计算了序列的自相关函数以及自行定义的准标准偏差.利用Hurst分析法研究了序列的再标度范围函数及其Hurst指数,并将结果与一维随机二元序列进行了对比.发现这些统计量有奇特的准周期振荡行为以及小于05的Hurst指数,直接论证了Fibonacci序列具有关联、标度不变及自相似等性质.从Anderson紧束缚模型出发,采用传输矩阵方法研究了Fibonacci序列的电子输运特性,讨论了输运系数对能量及其序列长度的依赖关系.研究 关键词： Fibonacci序列 统计属性 电子输运系数     

Band gaps in the terahertz frequency range in quasiperiodic one-dimensional magnonic crystals

In this work we investigate magnonic band gaps, in the terahertz (THz) frequency range, in periodic and quasiperiodic (Fibonacci sequence) magnonic crystals formed by layers of Cobalt (Co) and Permalloy (Py). Our theoretical model is based on a magnetic Heisenberg Hamiltonian in the exchange regime, together with a transfer-matrix treatment within the random-phase approximation (RPA). For periodic arrangements the bulk band structure is analogous to those found in photonic crystals, while for quasiperiodic multilayers it presents additional pass bands similar to those found in doped electronic materials.     

Transport and Conductance in Fibonacci Graphene Superlattices with Electric and Magnetic Potentials

We investigate the electron transport and conductance properties in Fibonacci quasi-periodic graphene superlattices with electrostatic barriers and magnetic vector potentials.It is found that a new Dirac point appears in the band structure of graphene superlattice and the position of the Dirac point is exactly located at the energy corresponding to the zero-averaged wave number.The magnetic and electric potentials modify the energy band structure and transmission spectrum in entirely diverse ways.In addition,the angular-dependent transmission is blocked by the potential barriers at certain incident angles due to the appearance of the evanescent states.The effects of lattice constants and different potentials on angular-averaged conductance are also discussed.     

Ballistic phonon transport through a Fibonacci array of acoustic nanocavities in a narrow constriction

We investigate the ballistic phonon transport through a Fibonacci array of acoustic nanocavities in a narrow constriction of a semiconductor nanowire at low temperatures. It is found that the transmission spectrum of such a system consists of quasiband gaps and narrow resonances caused by the coupling of phonon waves. Both phonon transmission and thermal conductance exhibit the similarity due to the Fibonacci sequence structure. The similarity is sensitive to the number n and parameters of nanocavities. The results are compared with those in a periodic acoustic nanocavities.     

Quantum dots in aperiodic order

We study numerically with a Green-function technique one-dimensional arrays of quantum dots with two different models. The arrays are ordered according to the Fibonacci, the Thue–Morse, and the Rudin–Shapiro sequences. As a comparison, results from a periodically ordered chain and also from a random chain are included. The focus is on how the conductance (calculated within the Landauer–Büttiker formalism) depends on the Fermi level. In the first model, we find that in some cases rather small systems (≈60 dots) behave in the same manner as very large systems (>16,000 dots) and this makes it possible in these cases to interpret our results for the small systems in terms of the spectral properties of the infinite systems. In particular, we find that it is possible to see some consequences of the singular continuous spectra that some of the systems possess, at least for temperatures up to 100 mK. In the second model, we study the phenomenon ohmic addition, i.e. when the resistances of the constrictions add up to the total resistance. It results that of the systems studied, it is only the Rudin–Shapiro system that has this behaviour for large structures, while the resistances of the Fibonacci and the Thue–Morse systems might reach a limiting value (as a periodic system does).     

Coherent transport in linear arrays of quantum dots: The effects of period doubling and of quasi-periodicity

We evaluate the phase-coherent transport of electrons along linear structures of varying length, which are made from two types of potential wells set in either a periodic or a Fibonacci quasi-periodic sequence. The array is described by a tight-binding Hamiltonian and is reduced to an effective dimer by means of a decimation-renormalization method, extended to allow for connection to external metallic leads, and the transmission coefficient is evaluated in a T-matrix-scattering approach. Parallel behaviors are found for the energy dependence of the density of electron states and of the transmittivity of the array. In particular, we explicitly show that on increasing its length the periodic array undergoes a metal–insulator transition near single occupancy per dot, whereas prominent pseudo-gaps emerge away from the band center in the Fibonacci-ordered array.     

Scaling behavior in a Fibonacci-sequenced system

The rudiments of dynamical systems theory are employed to analyze the transmission of light through a two-element medium in which the elements are arranged in a Fibonacci sequence. A dynamical map, introduced by previous authors, is extended so that the enlarged map generates direct predictions about the behavior of the transmission coefficient for phases in the neighborhood of certain critical values. These values correspond to unique periodic orbits of the map. Lowest-order calculations are performed analytically to study the properties of scaling near the critical phases. A scale factor is defined to describe this behavior. The study examines three cases in which the map has a fixed point, a 3-cycle, and a 6-cycle. The first two cases have the property that their scale factors are given by exactly the same Fibonacci number. In contrast, the third case has the property that its scale factor depends explicitly on a parameter of the physical system. Speculative remarks are added in conclusion to argue for the occurrence of a type of scaling whose features originate in the abstract structure of the Fibonacci sequence and are independent of the particular choice of physical system.     

Quantum diffusion in semi-infinite periodic and quasiperiodic systems

This paper studies quantum diffusion in semi-infinite one-dimensional periodic lattice and quasiperiodic Fibonacci lattice. It finds that the quantum diffusion in the semi-infinite periodic lattice shows the same properties as that for the infinite periodic lattice. Different behaviour is found for the semi-infinite Fibonacci lattice. In this case, there are still C（t） - t^-δ and d（t） - t^β. However, it finds that 0 〈δ 〈 1 for smaller time, and δ = 0 for larger time due to the influence of surface localized states. Moreover, β for the semi-infinite Fibonacci lattice is much smaller than that for the infinite Fibonacci lattice. Effects of disorder on the quantum diffusion are also discussed.     

Computationally efficient recurrence relations for one-dimensional Franck–Condon overlap integrals

Calculations based on analytical expressions for the harmonic oscillator Franck–Condon factors often yield numerically unstable and erroneous results for large values of the oscillator quantum numbers. This instability arises from inherent machine precision limits and large number round-off associated with the products and ratios of factorial and gamma functions in these expressions; the analytical expressions themselves are exact. This paper presents, first, efficient, exact recurrence relations to evaluate Franck–Condon factors for the harmonic oscillator model. The recurrence relations, which are similar to those originally found by Manneback, Wagner and Ansbacher avoid the direct use of the factorial and gamma functions. Second, a variational strategy for the evaluation of Franck–Condon factors for the Morse oscillator is proposed. The Schrödinger equation for the Morse model is solved variationally with a large enough basis set of one-dimensional harmonic oscillator functions to get good agreement with the analytic eigenvalues of the Morse potential itself. The eigenvectors of this analysis are then used together with the associated harmonic oscillator Franck–Condon overlap matrix elements to evaluate the overlap for the Morse potential. This approach allows one, in principle, to estimate Franck–Condon overlap up to states near to the dissociation limit of the Morse oscillator.     

Energy spectrum of the graphene-based Fibonacci superlattice

Energy spectra of the graphene-based Fibonacci superlattice (SL) in the presence of the band gap in graphene have been investigated. The lattice consists of rectangular barriers, which are arranged along axis Ox. The quasi-periodic modulation is performed due to the difference in the values of the mass term of the Hamiltonian in various SL elements. It is shown that effective splitting of allowed bands (and thereby the formation of a series of gaps) under the effect of the quasi-periodic factor is implemented with both oblique and normal incidence of the electron wave on the SL surface. The energy spectra have a clearly pronounced periodic character over the entire energy scale. The bands split in separate fragments of the spectrum (conventionally periods) according to the Fibonacci inflation rule in each new generation. The band gap associated with a new Dirac point is formed in all Fibonacci generations similarly to the periodic graphene-based SLs. The location of the Dirac point is independent of the SL period; it is very sensitive to the potential barrier height and to the width ratio between the quantum well and the barrier and depends weakly on the mass term in the Hamiltonian. The dependence of the spectra on the incidence angle of the electron wave is insignificant.     

Fibonacci链的电子结构特性

对由递推关系S_m+1={S_m|S_m-1}生成的Fibonacci链,从Anderson紧束缚模型出发,用负本征值理论及三对角高阶厄米矩阵本征值理论,对电子的态密度和能级结构进行数值研究,直观简洁地论证其三分叉的能带结构.用重整化群方法,结合散射理论,研究链中电子的局域长度和输运系数,发现具有不同局域属性的能态.一些特定的能量区间值存在扩展态,其相应的输运系数接近1.绝大部分能量对应的电子具有很小或几乎为零的局域长度,说明链中存在相当数量的局域态.定性得出电子输运系数随Fibonacci链参数变化的规律.     

The electron transport characters in a nanostructure with the periodic magnetic-electric barriers

This paper detailedly studies the transmission probability, the spin polarization and the conductance of the ballistic electron in a nanostructure with the periodic magnetic-electric barriers. These observable quantities are found to be strongly dependent not only on the magnetic configuration, the incident electron energy and the incident wave vector, but also on the number of the periodic magnetic-electric barriers. The transmission coefficient and the spin polarization show a periodic pattern with the increase of the separation between two adjacent magnetic fields, and the resonance splitting increases as the number of periods increases. Surprisingly, it is found that a polarization can be achieved by spin-dependent resonant tunnelling in this structure, although the average magnetic field of the structure is zero.     

The electron transport characters in a nanostructure with the periodic magnetic-electric barriers

This paper detailedly studies the transmission probability, the spin polarization and the conductance of the ballistic electron in a nanostrueture with the periodic magnetic-electric barriers These observable quantities are found to be strongly dependent not only on the magnetic configuration, the incident electron energy and the incident wave vector, but also on the number of the periodic magnetic-electric barriers The transmission coefficient and the spin polarization show a periodic pattern with the increase of the separation between two adjacent magnetic fields, and the resonance splitting increases as the number of periods increases. Surprisingly, it is found that a polarization can be achieved by spin-dependent resonant tunnelling in this structure, although the average magnetic field of the structure is zero.