Abstract
By using non-additive Tsallis entropy we demonstrate numerically that one-dimensional quasicrystals, whose energy spectra are multifractal Cantor sets, are characterized by an entropic parameter, and calculate the electronic specific heat, where we consider a non-additive entropy Sq. In our method we consider an energy spectra calculated using the one-dimensional tight binding Schrödinger equation, and their bands (or levels) are scaled onto the [0,1] interval. The Tsallis' formalism is applied to the energy spectra of Fibonacci and double-period one-dimensional quasiperiodic lattices. We analytically obtain an expression for the specific heat that we consider to be more appropriate to calculate this quantity in those quasiperiodic structures.
Highlights
Since the discovery of quasicrystals by Shechtman et al [1], awarded with the Nobel Prize, and the pioneering work of Merlin et al [2] on the nonperiodic Fibonacci and Thue–Morse GaAs–AlAs superlattices, quasicrystals have emerged as a new form of matter
In one dimension (1D), the Fibonacci sequence can directly be translated into a layered quasicrystal structure, which is feasible through atomic-precision growth via molecular-beam epitaxy (MBE) [2]
1D passive photonic quasicrystals have been realized by MBE and other techniques
Summary
Since the discovery of quasicrystals by Shechtman et al [1], awarded with the Nobel Prize, and the pioneering work of Merlin et al [2] on the nonperiodic Fibonacci and Thue–Morse GaAs–AlAs superlattices, quasicrystals have emerged as a new form of matter. The analysis of the thermodynamic properties based on the energy spectrum derived from a fractal structure was pioneered by Tsallis and collaborators [22,23] Their model was based on the most well-known and simple deterministic fractal geometry, the triadic Cantor set, and they showed that the specific heat of such a system exhibits a very particular behavior: it oscillates log-periodically around a mean value that equals the fractal dimension of the spectrum. Afterwards, Mauriz and collaborators [27,28] have presented a model based on the polariton’s and electron’s multifractal energy spectra for artificial structures following the Fibonacci, Thue–Morse and double-period sequences, in order to study their thermodynamic properties. In this work we investigate the theoretical behavior of some thermodynamical quantities, namely the specific heat, free energy and entropy, calculated by considering the nonadditivity effects arising on the system
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