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.

The electronic properties for a one-dimensional random dimer model (RDM) with a- and b-type atoms are studied within a tight-binding on-site model. We carry out a perturbative calculation on the energy spectrum for two different situations (a)εa-εb=t and (b) εa-εb=t, where εa and εb are the site energies of a- and b-type atoms, respectivley, t is a nearest-neighbor matrix element. Let △E(i)=∣Ei-εs∣, where s=a or b, Ei is the ith eigenenergy; we find that △E(i) for Ei around εa or εb equals 1.3m/N and 8.5m2/N2 for cases (a) and (b), respectively, where m is the period of wave functions and N is the number of total states. Interestingly, by using the results of △E(i), we find /N and 0.34/N extended electronic states in RDM for cases (a) and (b), respectively.

We study the properties of Fibonacci numbers and the transparency of clusters for electrons at some values of the energy. For the mth Fibonacci number Fm, a set of divisors are obtained by Fm /k=[Fm /k], 1＜k≤Fm . Interestingly, the numerical and analytical results show that any new divisors of the mth Fibonacci sequence will appear periodically in the following Fibonacci sequence. Furthermore, in the mixing Fibonacci system, we perform computer simulations and analytical calculations to study the transparent properties and spatial distributions of electronic states with the energies determined by the divisors of Fibonacci systems. The results show that the transmission coefficients are unity and the corresponding wave functions have periodiclike features. We report that an infinite number of one-dimensional disordered lattices, which are composed of some specific Fibonacci clusters, exhibit an absence of localization.

We study numerically the optical transmission of one-dimensional binary quasiperiodic dielectric multilayers, which are arranged in Fibonacci sequences along two opposite directions and possess a mirror symmetry. We find that the transmission coefficient is unity for all sequences studied at the central wavelength λ=λ0, whereλ0 4nA(B)dA(B), with nA(B) and dA(B) being the index of refraction and thickness of two kinds of layer, respectively. As the number of layers in the sequence increases, more and more perfect transmission peaks appear. We observe a scaling of the transmission spectra with increasing sequence length. These phenomena will find applications in fabrication of multiwavelength narrow-band optical filters.

electronic transport properties in one-dimensional systems with two kinds of mesoscopic ring defects: squarelike mesoscopic ring (SMR) defects and siamese-twins-like mescoscopic ring (STMR) defects. By using the transfer-matrix method, the resonant energies (where the transmission coefficient T=1) are derived successfully for both system. For the one SMR defect system, two resonant energies are found as a function of the magnetic fluxφthreading the ring defect, while for the latter case, two magnetic-flux-dependent and one magnetic-flux-independent resonant energies are predicted in the system, furthermore, if φ takes some specific values, one of theφ-dependent resonant energies may be the same as theφ-independent resonant energy. The word ‘‘resonant’’ is used to describe this situation. When a finite concentration of SMR or STMR defects are randomly embedded in a perfect chain, the numerical results confirm all the analytical predictions. Finally, for the ‘‘resonant’’ case, we show numerically a rather wide perfect transmission region which is almost ten times as wide as that of the ‘‘unresonant’’ case.