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Absence of localization in a model with correlation measure as a random lattice
Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics . Linköping University, The Institute of Technology.
Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics . Linköping University, The Institute of Technology.
2004 (English)In: Physical Review B, Condensed matter and materials physics, Vol. 69, no 9, 094204- p.Article in journal (Refereed) Published
Abstract [en]

A coherent picture of localization in one-dimensional aperiodically ordered systems is still missing. We show the presence of purely singular continuous spectrum for a discrete system whose modulation sequence has a correlation measure which is absolutely continuous, such as for a random sequence. The system showing these properties is modeled by the Rudin-Shapiro sequence, whose correlation measure even has a uniform density. The absence of localization is also supported by a numerical investigation of the dynamics of electronic wave packets showing weakly anomalous diffusion and an extremely slow algebraic decay of the temporal autocorrelation function.

Place, publisher, year, edition, pages
2004. Vol. 69, no 9, 094204- p.
National Category
Natural Sciences
Identifiers
URN: urn:nbn:se:liu:diva-14652DOI: 10.1103/PhysRevB.69.094204OAI: oai:DiVA.org:liu-14652DiVA: diva2:24116
Available from: 2007-09-20 Created: 2007-09-20
In thesis
1. Spectra and Dynamics of Excitattions in Long-Range Correlated Strucutures
Open this publication in new window or tab >>Spectra and Dynamics of Excitattions in Long-Range Correlated Strucutures
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [sv]

Vad karaktäriserar en kristall? Svaret på denna till synes enkla fråga blir kanske att det är en anordning av atomer uppradade i periodiska mönster. Så ordnade strukturer kan studeras genom att det uppträder så kallade Braggtoppar i röntgendiffraktionsmönstret. Om frågan gäller elektrontäthetsfördelningen, kanske svaret blir att denna är periodisk och grundar sig på elektronvågor som genomtränger hela kristallen. I och med att nya typer av ordnade system, så kallade kvasikristaller, upptäcks och framställs på artificiell väg blir svaren på dessa frågor mer intrikata.

En kristall behöver inte bestå av enheter upprepade periodiskt i rummet, och den klassiska metoden att karaktärisera strukturer via röntgendiffraktionsmönstret kanske inte alls är den allena saliggörande. I denna avhandling visas att ett ordnat gitter vars röntgendiffraktionsmönster saknar inre struktur, dvs är av samma diffusa typ som vad ett oordnat material uppvisar, fortfarande kan ha elektronerna utsträckta över hela strukturen. Detta implicerar att det inte finns något enkelt samband mellan diffraktionsmönstret från gittret och dess fysikaliska egenskaper såsom t ex lokalisering av vågfunktionerna. Man talar om lokalisering när en vågfunktion är begränsad inom en del av materialet och inte utsträckt över hela dess längd, vilket är av betydelse när man vill avgöra huruvida ett material är en isolator, halvledare eller ledare. Det vittnar samtidigt om behovet av att söka efter andra karakteristika när man försöker beskriva skillnaden mellan ett ordnat och ett oordnat material, där den senare kategorin kan uppvisa lokalisering. Resultaten utgör en klassificering av det svåröverskådliga området aperiodiska gitter i en dimension. Det leder till hypotesen att ideala kvasikristaller, genererade med bestämda regler, har kontinuerligt energispektrum av fraktal natur.

I reella material spelar korrelation en viktig roll. Vid icke-linjär återkoppling till gittret kan man erhålla intrinsiskt lokaliserade vågor, som i många avseenden beter sig som partiklar, solitoner, vilka har visat sig ha viktiga tillämpningar inom bl a optisk telekommunikation. Sådana vågors roll for lagring och transport av energi har undersökts i teoretiska modeller for optiska vågledare och kristaller där ljuset har en förmåga att manipulera sig självt.

Abstract [en]

Spectral and dynamical properties of electrons, phonons, electromagnetic waves, and nonlinear coherent excitations in one-dimensional modulated structures with long-range correlations are investigated from a theoretical point of view.

First a proof of singular continuous electron spectrum for the tight-binding Schrödinger equation with an on-site potential, which, in analogy with a random potential, has an absolutely continuous correlation measure, is given. The critical behavior of such a localization phenomenon manifests in anomalous diffusion for the time-evolution of electronic wave packets. Spectral characterization of elastic vibrations in aperiodically ordered diatomic chains in the harmonic approximation is achieved through a dynamical system induced by the trace maps of renormalized transfer matrices. These results suggest that the zero Lebesgue measure Cantor-set spectrum (without eigenvalues) of the Fibonacci model for a quasicrystal is generic for deterministic aperiodic superlattices, for which the modulations take values via substitution rules on finite sets, independent of the correlation measure.

Secondly, a method to synthesize and analyze discrete systems with prescribed long-range correlated disorder based on the conditional probability function of an additive Markov chain is effectively implemented. Complex gratings (artificial solids) that simultaneously display given characteristics of quasiperiodic crystals and amorphous solids on the Fraunhofer diffraction are designated. A mobility edge within second order perturbation theory of the tight-binding Schrödinger equation with a correlated disorder in the dichotomic potential realizes the success of the method in designing window filters with specific spectral components.

The phenomenon of self-localization in lattice dynamical systems is a subject of interest in various physical disciplines. Lattice solitons are studied using the discrete nonlinear Schrödinger equation with on-site potential, modeling coherent structures in, for example, photonic crystals. The instability-induced dynamics of the localized gap soliton is found to thermalize according to the Gibbsian equilibrium distribution, while the spontaneous formation of persisting intrinsic localized modes from the extended out-gap soliton reveals a phase transition of the solution.

Place, publisher, year, edition, pages
Institutionen för fysik, kemi och biologi, 2007. 52 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1109
Keyword
quasicrystals, deterministic aperiodic superlattices, Markov chains, electronic structure and diffusion, elastic vibrations, Fraunhofer diffraction, discrete solitons, phase transitions
National Category
Condensed Matter Physics
Identifiers
urn:nbn:se:liu:diva-9727 (URN)978-91-85831-57-9 (ISBN)
Public defence
2007-09-14, Planck, Fysikhuset, Campus Valla, Linköpings universitet, Linköping, 14:15 (English)
Opponent
Supervisors
Available from: 2007-09-20 Created: 2007-09-20 Last updated: 2014-01-13

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Kroon, LarsRiklund, Rolf

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