liu.seSearch for publications in DiVA
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Spectra and Dynamics of Excitattions in Long-Range Correlated Strucutures
Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics . Linköping University, The Institute of Technology.
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 [en]
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: urn:nbn:se:liu:diva-9727ISBN: 978-91-85831-57-9 (print)OAI: oai:DiVA.org:liu-9727DiVA: diva2:24120
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
List of papers
1. Localization-delocalization in aperiodic systems
Open this publication in new window or tab >>Localization-delocalization in aperiodic systems
2002 (English)In: Physical Review B, Condensed matter and materials physics, ISSN 1098-0121, Vol. 66, no 9, 094204- p.Article in journal (Refereed) Published
Abstract [en]

The question of localization in a one-dimensional tight-binding model with aperiodicity given by substitutions is discussed. Since the localization properties of the well-known Rudin-Shapiro chain is still far from well understood, partly due to the absence of rigorous analytical results, we introduce a sequence that has several features in common with the Rudin-Shapiro sequence. We derive a trace map for this system and prove analytically that the electron spectrum is singular continuous. Despite the extended (non-normalizable) nature of the corresponding wave functions, the states show strong localization for finite approximations of the chain. Similar localization properties are found for the Rudin-Shapiro chain, where earlier results have indicated a pure point spectrum. We compare the properties for two other physical systems, ordered according to the two discussed sequences; stationary electron transmission is studied through finite chains using a dynamical map, optical properties of dielectric multilayer structures are investigated.

National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-14650 (URN)10.1103/PhysRevB.66.094204 (DOI)
Available from: 2007-09-20 Created: 2007-09-20
2. Renormalization of aperiodic model lattices: spectral properties
Open this publication in new window or tab >>Renormalization of aperiodic model lattices: spectral properties
2003 (English)In: Journal of Physics A: Mathematical and General, ISSN 1751-8113, Vol. 36, 4519-4532 p.Article in journal (Refereed) Published
Abstract [en]

Many of the published results for one-dimensional deterministic aperiodic systems treat rather simplified electron models with either a constant site energy or a constant hopping integral. Here we present some rigorous results for more realistic mixed tight-binding systems with both the site energies and the hopping integrals having an aperiodic spatial variation. It is shown that the mixed Thue–Morse, period-doubling and Rudin–Shapiro lattices can be transformed to on-site models on renormalized lattices maintaining the individual order between the site energies. The character of the energy spectra for these mixed models is therefore the same as for the corresponding on-site models. Furthermore, since the study of electrons on a lattice governed by the Schrödinger tight-binding equation maps onto the study of elastic vibrations on a harmonic chain, we have proved that the vibrational spectra of aperiodic harmonic chains with distributions of masses determined by the Thue–Morse sequence and the period-doubling sequence are purely singular continuous.

National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-14651 (URN)10.1088/0305-4470/36/16/303 (DOI)
Available from: 2007-09-20 Created: 2007-09-20
3. Absence of localization in a model with correlation measure as a random lattice
Open this publication in new window or tab >>Absence of localization in a model with correlation measure as a random lattice
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.

National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-14652 (URN)10.1103/PhysRevB.69.094204 (DOI)
Available from: 2007-09-20 Created: 2007-09-20
4. Spectral analysis and synthesis of 1D dichotomous long-range correlated systems: From diffraction gratings to quantum wire
Open this publication in new window or tab >>Spectral analysis and synthesis of 1D dichotomous long-range correlated systems: From diffraction gratings to quantum wire
Show others...
2008 (English)In: Physica A: Statistical Mechanics and its Applications, ISSN 0378-4371, E-ISSN 1873-2119, Vol. 387, no 19-20, 4733-4739 p.Article in journal (Refereed) Published
Abstract [en]

Spectral properties of 1D systems with long-range correlated disorder and their response to an applied field are examined. An algorithm based on the additive multi-step Markov chains is used to analyze and synthesize layered systems consisting of two randomly alternated elements. Using an equation connecting the correlation and memory functions enables one to reveal the microscopic structure, which can be expressed in terms of the Markov chain conditional probability function. Specifically, a method of designing complex gratings with prescribed characteristics that simultaneously display periodic, quasi-periodic and random properties is emphasized. The tight-binding Schrödinger equation with a weak correlated disorder in the dichotomic potential exhibiting sharp transition in conductivity is studied.

Keyword
Markov chain, Diffraction grating, Correlation function, Lyapunov exponent, Mobility edge
National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-14653 (URN)10.1016/j.physa.2008.03.038 (DOI)
Available from: 2007-09-20 Created: 2007-09-20 Last updated: 2017-12-13
5. Bifurcation picture and stability of the gap and out-gap discrete solitons
Open this publication in new window or tab >>Bifurcation picture and stability of the gap and out-gap discrete solitons
2007 (English)In: Low temperature physics (Woodbury, N.Y., Print), ISSN 1063-777X, E-ISSN 1090-6517, Vol. 33, no 5, 481-483 p.Article in journal (Refereed) Published
Abstract [en]

The dynamics of a quaternary fragment of a discrete system of coupled nonlinear oscillators with modulated frequency parameters is investigated, and the stability of its gap and out-gap soliton-like excitations is studied.

National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-14654 (URN)10.1063/1.2737564 (DOI)
Available from: 2007-09-20 Created: 2007-09-20 Last updated: 2017-12-13
6. The appearance of gap solitons in a nonlinear Schrödinger lattice
Open this publication in new window or tab >>The appearance of gap solitons in a nonlinear Schrödinger lattice
Manuscript (Other academic)
Identifiers
urn:nbn:se:liu:diva-14655 (URN)
Available from: 2007-09-20 Created: 2007-09-20 Last updated: 2010-01-13

Open Access in DiVA

cover(43 kB)38 downloads
File information
File name COVER01.pdfFile size 43 kBChecksum MD5
7aa93ed6280291259f37be0eed160846a8293263fa25b6f0727e4e32a35244053be5ecc9
Type coverMimetype application/pdf
fulltext(966 kB)1188 downloads
File information
File name FULLTEXT01.pdfFile size 966 kBChecksum MD5
fd93cc92a2d5fe18166a75d8ee683e046fccc91e2951b349b3847996f78c58bc22bb250c
Type fulltextMimetype application/pdf

Authority records BETA

Kroon, Lars

Search in DiVA

By author/editor
Kroon, Lars
By organisation
Theoretical Physics The Institute of Technology
Condensed Matter Physics

Search outside of DiVA

GoogleGoogle Scholar
Total: 1188 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

isbn
urn-nbn

Altmetric score

isbn
urn-nbn
Total: 2172 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf