The nature of the atomic surfaces of quasiperiodic self-similar structures

Luck, J. M.; Godrèche, Claude; Janner, A.; Janssen, Τ.

doi:10.1088/0305-4470/26/8/020

Cited by 117 publications

(73 citation statements)

References 26 publications

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“…43 Nevertheless, the renormalized chain is self-similar. In fact, the normalized eigenvector corresponding to the leading eigenvalue 1 = is ͑1/2, −1 /2, −2 /2͒, whose components give the relative frequencies of the renormalized sites ␣ , ␤ , and ␦, respectively.…”

Section: General Fibonacci Latticesmentioning

confidence: 99%

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Renormalization transformation of periodic and aperiodic lattices

Maciá

Rodríguez-Oliveros

2006

Phys. Rev. B

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In this work we introduce a similarity transformation acting on transfer matrices describing the propagation of elementary excitations through either periodic or Fibonacci lattices. The proposed transformation can act at two different scale lengths. At the atomic scale the transformation allows one to express the systems' global transfer matrix in terms of an equivalent on-site model one. Correlation effects among different hopping terms are described by a series of local phase factors in that case. When acting on larger scale lengths, corresponding to short segments of the original lattice, the similarity transformation can be properly regarded as describing an effective renormalization of the chain. The nature of the resulting renormalized lattice significantly depends on the kind of order ͑i.e., periodic or quasiperiodic͒ of the original lattice, expressing a delicate balance between chemical complexity and topological order as a consequence of the renormalization process.

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Section: General Fibonacci Latticesmentioning

confidence: 99%

“…51,52 Accordingly, one reasonably expects that its finite realizations would exhibit well-defined diffraction spectra, probably supported on singular continuous spectra, as has been previously discussed in the literature for other lattices exhibiting the singular feature ͉ i ͉ = 1 in the substitution matrix spectrum. 43,53 …”

Section: General Fibonacci Latticesmentioning

confidence: 99%

Renormalization transformation of periodic and aperiodic lattices

Maciá

Rodríguez-Oliveros

2006

Phys. Rev. B

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“…In turn, pure point lattice Fourier measures can be further split into four separate groups, namely, the so-called periodic ( 0 ), quasiperiodic ( I ), limit-quasiperiodic ( II ), and limit-periodic ( III ) pure point classes, respectively, by attending to …ner details in their related di¤raction patterns. [20,[22][23][24][25] The di¤raction spectra of 0 and I classes representatives both consist in Bragg peaks supported by a …nite Fourier module whose rank either equals the physical space dimension ( 0 class) or is larger than it ( I class). In the limit-quasiperiodic and limit-periodic classes II and III one also …nds a di¤raction spectrum consisting of a dense distribution of Bragg peaks.…”

Section: Introductionmentioning

confidence: 99%

“…However, this distribution is supported by a Fourier module with a countably in…nity of generators over the integers (i.e., the reciprocal space has in…nite dimensions). [20,26] The point-group symmetry of the di¤raction spectrum of the limit-periodic structures belonging to the III spectral class is compatible with periodicity (unlike QCs) and their overall atomic structure can be described in terms of a union of periodic substructures with ever increasing lattice constants, forming a sequence of successive sublattices. [27,28] Analogously, the di¤raction spectrum of the limit-quasiperiodic class representatives can be generated by a superposition of spectra of an in…nite number of quasiperiodic patterns.…”

Section: Introductionmentioning

confidence: 99%

Spectral Classification of One‐Dimensional Binary Aperiodic Crystals: An Algebraic Approach

Maciá

2017

Annalen der Physik

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A spectral classi…cation of general one-dimensional binary aperiodic crystals (BACs) based on both their di¤raction patterns and energy spectrum measures is introduced along with a systematic comparison of the zeroth-order energy spectrum main features for BACs belonging to di¤erent spectral classes, including Fibonacci-class, precious means, metallic means, mixed means and period doubling based representatives. These systems are described by means of mixed-type Hamiltonians which include both diagonal and o¤-diagonal terms aperiodically distributed. An algebraic approach highlighting chemical correlation e¤ects present in the underlying lattice is introduced.Close analytical expressions are obtained by exploiting some algebraic properties of suitable blocking schemes preserving the atomic order of the original lattice. The existence of a resonance energy which de…nes the basic anatomy of the zeroth-order energy spectra structure for the standard Fibonacci, the precious means and the Fibonacci-class quasicrystals is disclosed. This eigenstate is also found in the energy spectra of BACs belonging to other spectral classes, but for speci…c particular choices of the corresponding model parameters only. The transmission coe¢ cient of these resonant states is always bounded below, although their related Landauer conductance values may range from highly conductive to highly resistive ones, depending on the relative strength of the chemical bonds.

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“…In the one-dimensional (1D) case, they can be formed by stacking together dielectric layers of several different types according to the substitutional sequence under investigation (Cantor, Fibonacci, Rudin-Shapiro, Thue-Morse, etc.) [6]. The Fibonacci sequence is of particular importance, since it leads to the existence of two incommensurable periods in the spatial spectrum of the structure.…”

Section: Introductionmentioning

confidence: 99%

Photonic Analysis of Semiconductor Fibonacci Superlattices: Properties and Applications

Rahimi¹

2014

PIER M

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Abstract-In this paper, we theoretically study the phase treatment of reflected waves in onedimensional Fibonacci photonic quasicrystals composed of nano-scale fullerene and semiconductor layers. The dependence of the phase shift of reflected waves for TE mode and TM mode on the wavelength and incident angle is calculated by using the theoretical model based on the transfer matrix method in the infrared wavelength region. In the band gaps of supposed structures, it is found that the phase shift of reflected wave changes more slowly than within the transmission band gaps. Furthermore, the phase shift decreases with the incident angle increasing for TE mode, and increases with the incident angle increasing for TM mode. Also, for the supposed structures it is found that there is a band gap which is insensitive to the order of the Fibonacci sequence. These structures open a promising way to fabricate subwavelength tunable phase compensators, very compact wave plates and phase-sensitive interferometry for TE and TM waves.

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The nature of the atomic surfaces of quasiperiodic self-similar structures

Cited by 117 publications

References 26 publications

Renormalization transformation of periodic and aperiodic lattices

Renormalization transformation of periodic and aperiodic lattices

Spectral Classification of One‐Dimensional Binary Aperiodic Crystals: An Algebraic Approach

Photonic Analysis of Semiconductor Fibonacci Superlattices: Properties and Applications

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