Two-dimensional Penrose Lattice Research Articles

Magnetic Response and Valence Fluctuations in the Extended Anderson Lattice Model with Quasiperiodicity

We study valence fluctuations in the extended Anderson model on two-dimensional Penrose lattice, using the real-space dynamical mean-field theory combined with the continuous-time Monte Carlo method. Calculating $f$-electron number, $c$-$f$ spin correlations, and magnetic susceptibility at each site, we find site-dependent formations of the singlet state and valence distribution at low temperatures, which are reflected by the quasiperiodic lattice structure. The bulk magnetic susceptibility is also addressed.

Valence fluctuations and electric reconstruction in the extended Anderson model on the two-dimensional Penrose lattice

We study the extended Anderson model on the two-dimensional Penrose lattice, combining the real-space dynamical mean-field theory with the non-crossing approximation. It is found that the Coulomb repulsion between localized and conduction electrons does not induce a valence transition, but the crossover between the Kondo and mixed valence states in contrast to the conventional periodic system. In the mixed-valence region close to the crossover, nontrivial valence distributions appear characteristic of the Penrose lattice, demonstrating that the mixed-valence state coexists with local Kondo states in certain sites. The electric reconstruction in the mixed valence region is also addressed.

Local Electron Correlations in a Two-Dimensional Hubbard Model on the Penrose Lattice

We study electron correlations in the half-filled Hubbard model on two-dimensional Penrose lattice. Applying the real-space dynamical mean-field theory to large clusters, we discuss how low-temperature properties are affected by the quasiperiodic structure. By calculating the double occupancy and renormalization factor at each site, we clarify the existence of the Mott transition. The spatially-dependent renormalization characteristic of geometrical structure is also addressed.

Calculation of Conduction Spectra in Quantum Dot Composed of Penrose Lattice

Conductance through finite two-dimensional Penrose lattices (PLs) are calculated as a function of gate voltage. To investigate effects of lattice defects and lattice vibrations, three types of PLs are taken into account; (A) PL without lattice defects, (B) PL with lattice defects, and (C) PL with lattice vibrations, which is applied the electric field across the conductor. When energy levels of PL coincides with the chemical potential of the electrodes with increasing gate voltage, electron transfer through PL takes place. However, electron transfer is forbidden at certain states; These states have confined states, which have multiple-degeneracy in their electronic states. In addition, the states are localized due to interference of their wave-functions though the wave-function of each state is extended. We would like to point out that rigidity of confined states with respect to lattice defects and lattice vibrations.

Critical currents in superconductors with quasiperiodic pinning arrays: One-dimensional chains and two-dimensional Penrose lattices

We study the critical depinning current J_c, as a function of the applied magnetic flux Phi, for quasiperiodic (QP) pinning arrays, including one-dimensional (1D) chains and two-dimensional (2D) arrays of pinning centers placed on the nodes of a five-fold Penrose lattice. In 1D QP chains of pinning sites, the peaks in J_c(Phi) are shown to be determined by a sequence of harmonics of long and short periods of the chain. This sequence includes as a subset the sequence of successive Fibonacci numbers. We also analyze the evolution of J_c(Phi) while a continuous transition occurs from a periodic lattice of pinning centers to a QP one; the continuous transition is achieved by varying the ratio gamma = a_S/a_L of lengths of the short a_S and the long a_L segments, starting from gamma = 1 for a periodic sequence. We find that the peaks related to the Fibonacci sequence are most pronounced when gamma is equal to the golden mean. The critical current J_c(Phi) in QP lattice has a remarkable self-similarity. This effect is demonstrated both in real space and in reciprocal k-space. In 2D QP pinning arrays (e.g., Penrose lattices), the pinning of vortices is related to matching conditions between the vortex lattice and the QP lattice of pinning centers. Although more subtle to analyze than in 1D pinning chains, the structure in J_c(Phi) is determined by the presence of two different kinds of elements forming the 2D QP lattice. Indeed, we predict analytically and numerically the main features of J_c(Phi) for Penrose lattices. Comparing the J_c's for QP (Penrose), periodic (triangular) and random arrays of pinning sites, we have found that the QP lattice provides an unusually broad critical current J_c(Phi), that could be useful for practical applications demanding high J_c's over a wide range of fields.

Long range antiferromagnetic order in Ising model simulations in a two-dimensional Penrose lattice

Long-range antiferromagnetic order was found in simulated annealing calculations for an Ising spin system on a two-dimensional Penrose lattice under Ruderman–Kittel–Kasuya–Yosida-like alternating exchange interactions at low temperatures. Two models of magnetic interactions resulted in the long-range antiferromagnetic order; a model of an antiferromagnetic interaction for the shortest distance and ferromagnetic interaction for the second and the third shortest distances, and a reversed sign model. Both models showed sharp magnetic diffraction spots with the same peak widths as the ordinary lattice diffraction, which indicates the long-range magnetic order extending over the whole system. The contrasts of the magnetic diffraction patterns were different from the patterns arising from the spin-independent scattering amplitudes from the Penrose lattice sites. The Patterson map analysis also confirmed the long-range nature of the antiferromagnetic order. The reversed-sign interaction model resulted in a stronger antiferromagnetic character than the non-reversed one.

Electronic Transmission Properties of Two-Dimensional Quasi-Lattice

In the framework of the tight binding model, the electronictransmission properties of two-dimensional Penrose lattices with freeboundary conditions are studied using the generalized eigenfunctionmethod (Phys. Rev. B 60(1999)13 444). The electronic transmissioncoefficients for Penrose lattices with different sizes and widths arecalculated, and the result shows strong energy dependence because ofthe quasiperiodic structure and quantum coherent effect. Around theFermi level E = 0, there is an energy region with zero transmissionamplitudes, which suggests that the studied systems are insulating.The spatial distributions of several typical electronic states withdifferent transmission coefficients are plotted to display thepropagation process.

Photonic band-gap effect, localization, and waveguiding in the two-dimensional Penrose lattice

We report experimental observation of a full photonic band gap in a two-dimensional Penrose lattice made of dielectric rods. Tightly confined defect modes having high quality factors were observed. Absence of the translational symmetry in Penrose lattice was used to change the defect frequency within the stop band. We also achieved the guiding and bending of electromagnetic waves through a row of missing rods. Propagation of photons along highly localized coupled-cavity modes was experimentally demonstrated and analyzed within the tight-binding approximation.

An investigation of the momentum dependence of the electronic spectrum for a disordered system

The spectral properties of disordered systems are investigated using twodimensional examples in the tight-binding approximation. The spectra are examined as functions of the momentum wavevector for two-dimensional systems. A particular example is the directional dependence of the spectrum for a quasicrystal, represented by a two-dimensional Penrose lattice. The spectrum changes dramatically, as expected, when the momentum vector moves off a symmetry axis of the system. The value of this approach for analysing the electronic properties of structurally disordered systems is discussed.

Monte Carlo study of the XY-model on Penrose lattices

We have carried out a Monte Carlo study of the nearest-neighbour ferromagnetic XY-model on the two-dimensional (2D) Penrose lattice and on periodically stacked (three-dimensional) 2D Penrose lattices. For the 2D case we have examined the magnetization, specific heat, linear susceptibility, helicity modulus and the derivative of the helicity modulus with respect to inverse temperature. The behaviour of all of these quantities points to a Kosterlitz-Thouless transition occurring in the temperature range , with critical exponents that are consistent with those obtained for crystalline (e.g., square) lattices. For the 3D stacking of the 2D Penrose lattices, examination of the magnetization, specific heat and linear susceptibility reveals a conventional second-order phase transition. Through cumulant analysis and finite-size scaling we obtain a critical temperature and critical exponents and , in agreement with previous studies of the XY-model on the 3D cubic lattice.

A new technique to calculate the electronic structure of two-dimensional quasicrystals

Abstract The electronic structure of a two-dimensional Penrose lattice has been investigated in the tight-binding limit using a new theoretical approach based on multiple-scattering theory. We have studied in particular the vertex model of the lattice and calculated the density and integrated density of electronic states. The method used is effectively for an infinite structure and leads to no distortions from edge effects or periodic continuation. The vertex model, which is based on fat and thin rhombi, has three nearest interatomic distances: the short diagonal of a thin rhombus, the edge of a rhombus, and the short diagonal of a fat rhombus. We have investigated the effect of interactions involving these various distances in the density of states, giving results which are similar to those using other methods. It is stressed that the method is capable of considerable generalization.

Spin-glass behavior of frustrated two-dimensional Penrose lattice in the classical planar model.

Via extensive Monte Carlo studies we show that the frustrated XY Hamiltonian on a 2-D Penrose lattice admits of a spin glass phase at low temperature. Studies of the Edwards-Anderson order parameter, spin glass susceptibility, and local (linear) susceptibility point unequivocally to a paramagnetic to spin glass transition as the temperature is lowered. Specific heat shows a rounded peak at a temperature above the spin glass transition temperature, as is commonly observed in spin glasses. Our results strongly suggest that the critical point exponents are the same as obtained by Bhatt and Young in the ${\pm}J$ Ising model on a square lattice. However, unlike in the latter case, the critical temperature is clearly finite (nonzero). The results imply that a quasiperiodic 2-D array of superconducting grains in a suitably chosen transverse magnetic field should behave as a superconducting glass at low temperature.

Electronic transport of quasicrystals with random phason strain: The two-dimensional Penrose lattice.

The electronic density of states, eigenstates, length dependence, and Fermi-energy dependence of the dc conductance in the two-dimensional Penrose lattice are calculated by using an exact diagonalization, Landauer formula and the forced oscillator method in finite size, zero-temperature systems with and without random phasons. Two typical behaviors of the conductance are found according to a choice of the Fermi energy. One is that the randomness decreases the localization length and decreases conductance. The other is that the randomness smears out the density of states and the channels for the electronic transport open, i.e., the conductance increases. Based on this observation, a plausible explanation for the abnormal transport properties in quasicrystals is proposed.

Computer Simulation of Growth Process of Binary Quasi Crystal

We performed Monte Carlo simulation of the aggregation process of a two-component Lennard-Jones system on the two-dimensional (2D) Penrose lattice for the purpose of studying the details of the creation and growth process of quasi crystals. We adopted the potential parameter values proposed by Widom et al. [Phys. Rev. Lett. 58 (1987) 706]. The present simulation clarified that the stability of the formed quasi crystals depends on temperature and the strength of the potential and that the transition from solid state to liquid state occurs when the value of the inverse of the dimensionless temperature is approximately 4.0.

Electronic transport of a two-dimensional Penrose lattice with random phason strain

The electronic density of states and conductance of a two-dimensional Penrose lattice with random phason strain were calculated to explain the anomalous behaviour of conductance in quasicrystals. Two different kinds of typical behavior were observed. The random phason strain enhances localization of the electron at some energy, and the conductance decreases. In addition, random phason strain increases the conductance with increasing density of states in the energy gaps at some other energy.

Band structure effects of transport properties in icosahedral quasicrystals.

Transport properties and optical conductivity are calculated for the crystalline approximant AlMn alloy. The number of electrons at the Fermi energy is very small and, based on the band structure, the anomalously small dc conductivity and temperature dependent thermoelectric power are explained. A model calculation for the two-dimensional Penrose lattice with random phason strain shows that it becomes more conductive than the perfect Penrose lattice. We propose the mechanism of the interband transition by phason randomness and inelastic scattering for the origin of the anomalous conductivity in real quasicrystals.

Reformulation of the percolation problem on a quasilattice: Estimates of the percolation threshold, chemical dimension, and amplitude ratio.

Percolation on a two-dimensional Penrose lattice is formulated using the distance between sites as the criteria for direct connections between occupied sites. The percolation threshold, chemical dimension, and amplitude ratio of the mean cluster size are estimated for this problem.

Phonon modes and melting properties of two-dimensional Penrose lattices.

We study the phonon modes and melting properties of two-dimensional Penrose lattices with fivefold rotational symmetry. We use a similarity transformation to reduce the numerical calculations, which also helps us determine analytically the degeneracies of the phonon modes. It is found that two-thirds of the modes are doubly degenerate and the remaining one-third nondegenerate. Using the mean-square atomic-displacement criterion of Lindemann, we have studied the melting properties of two-dimensional Penrose lattices and found that the boundary atoms always have a lower melting temperature than those in the bulk.

Monte Carlo Simulation of Diffusion Process on Quasi-Lattice

The characteristics of the diffusion process of quasicrystals are studied for one-dimensional and two-dimensional models by the conventional Monte Carlo simulation. For the one-dimensional model, the tracer diffusion coefficients of an isolated particle are calculated for the three types of lattice models, i.e., single crystal, binary solid solution and quasicrystal. The characteristics of diffusivity of the one-dimensional quasicrystal model are derived from comparisons among the three types of models. For the two-dimensional model, the characteristics of diffusivity of a quasicrystal are discussed in comparison with those of the honeycomb lattice. In particular, the concentration dependence of diffusivity on the two-dimensional Penrose lattice is calculated and the effects of interatomic potential are discussed.

Electronic properties of the Penrose lattice. I. Energy spectrum and wave functions.

The electronic structure of the two-dimensional Penrose lattice is studied by numerical diagonalization of a tight-binding Hamiltonian for finite systems with up to 3571 sites. We have analyzed the smoothness of the energy spectrum and localization behavior of the wave functions by level statistics and a generalized participation ratio. The results show that the energy spectrum contains a singular part, and most of the wave functions are critical, i.e., neither extended nor localized. These behaviors of the electronic properties are discussed based on their quasiperiodic lattice structure.