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Error Estimation for Eigenvalues of Unbounded Linear Operators and an Application to Energy Levels in Graphene Quantum Dots
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-2681-8965
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
2017 (English)In: Numerical Functional Analysis and Optimization, ISSN 0163-0563, E-ISSN 1532-2467, Vol. 38, no 3, 293-305 p.Article in journal (Refereed) Published
Abstract [en]

The eigenvalue problem for linear differential operators is important since eigenvalues correspond to the possible energy levels of a physical system. It is also important to have good estimates of the error in the computed eigenvalues. In this work, we use spline interpolation to construct approximate eigenfunctions of a linear operator using the corresponding eigenvectors of a discretized approximation of the operator. We show that an error estimate for the approximate eigenvalues can be obtained by evaluating the residual for an approximate eigenpair. The interpolation scheme is selected in such a way that the residual can be evaluated analytically. To demonstrate that the method gives useful error bounds, we apply it to a problem originating from the study of graphene quantum dots where the goal was to investigate the change in the spectrum from incorporating electron–electron interactions in the potential.

Place, publisher, year, edition, pages
Taylor & Francis, 2017. Vol. 38, no 3, 293-305 p.
Keyword [en]
Energy levels, error estimation, graphene, linear operator, quantum dot, spectrum
National Category
Computational Mathematics Mathematical Analysis
Identifiers
URN: urn:nbn:se:liu:diva-135540DOI: 10.1080/01630563.2017.1279176ISI: 000396822500002OAI: oai:DiVA.org:liu-135540DiVA: diva2:1083772
Available from: 2017-03-22 Created: 2017-03-22 Last updated: 2017-04-11
In thesis
1. Quantum scattering and interaction in graphene structures
Open this publication in new window or tab >>Quantum scattering and interaction in graphene structures
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Since its isolation in 2004, that resulted in the Nobel Prize award in 2010, graphene has been the object of an intense interest, due to its novel physics and possible applications in electronic devices. Graphene has many properties that differ it from usual semiconductors, for example its low-energy electrons behave like massless particles. To exploit the full potential of this material, one first needs to investigate its fundamental properties that depend on shape, number of layers, defects and interaction. The goal of this thesis is to perform such an investigation.

In paper I, we study electronic transport in monolayer and bilayer graphene nanoribbons with single and many short-range defects, focusing on the role of the edge termination (zigzag vs armchair). Within the discrete tight-binding model, we perform an-alytical analysis of the scattering on a single defect and combine it with the numerical calculations based on the Recursive Green's Function technique for many defects. We find that conductivity of zigzag nanoribbons is practically insensitive to defects situated close to the edges. In contrast, armchair nanoribbons are strongly affected by such defects, even in small concentration. When the concentration of the defects increases, the difference between different edge terminations disappears. This behaviour is related to the effective boundary condition at the edges, which respectively does not and does couple valleys for zigzag and armchair ribbons. We also study the Fano resonances.

In the second paper we consider electron-electron interaction in graphene quantum dots defined by external electrostatic potential and a high magnetic field. The interaction is introduced on the semi-classical level within the Thomas Fermi approximation and results in compressible strips, visible in the potential profile. We numerically solve the Dirac equation for our quantum dot and demonstrate that compressible strips lead to the appearance of plateaus in the electron energies as a function of the magnetic field. This analysis is complemented by the last paper (VI) covering a general error estimation of eigenvalues for unbounded linear operators, which can be used for the energy spectrum of the quantum dot considered in paper II. We show that an error estimate for the approximate eigenvalues can be obtained by evaluating the residual for an approximate eigenpair. The interpolation scheme is selected in such a way that the residual can be evaluated analytically.

In the papers III, IV and V, we focus on the scattering on ultra-low long-range potentials in graphene nanoribbons. Within the continuous Dirac model, we perform analytical analysis and show that, considering scattering of not only the propagating modes but also a few extended modes, we can predict the appearance of the trapped mode with an energy eigenvalue close to one of the thresholds in the continuous spectrum. We prove that trapped modes do not appear outside the threshold, provided the potential is sufficiently small. The approach to the problem is different for zigzag vs armchair nanoribbons as the related systems are non-elliptic and elliptic respectively; however the resulting condition for the existence of the trapped mode is analogous in both cases.

Abstract [sv]

Sedan isoleringen av grafen 2004, vilket belönades med Nobelpriset 2010, har intresset för grafen varit väldigt stort på grund av dess nya fysikaliska egenskaper med möjliga tillämpningar i elektronisk apparatur. Grafen har många egenskaper som skiljer sig från vanliga halvledare, exempelvis dess lågenergi-elektroner som beter sig som masslösa partiklar. För att kunna utnyttja dess fulla potential måste vi först undersöka vissa grundläggande egenskaper vilka beror på dess form, antal lager, defekter och interaktion. Målet med denna avhandling är att genomföra sådana undersökningar.

I den första artikeln studerar vi elektrontransporter i monolager- och multilagergrafennanoband med en eller flera kortdistansdefekter, och fokuserar på inverkan av randstrukturen (zigzag vs armchair), härefter kallade zigzag-nanomband respektive armchair-nanoband. Vi upptäcker att ledningsförmågan hos zigzag-nanoband är praktiskt taget okänslig för defekter som ligger nära kanten, i skarp kontrast till armchairnanoband som påverkas starkt av sådana defekter även i små koncentrationer. När defektkoncentrationen ökar så försvinner skillnaden mellan de två randstrukturerna. Vi studerar också Fanoresonanser.

I den andra artikeln betraktar vi elektron-elektron interaktion i grafen-kvantprickar som definieras genom en extern elektrostatisk potential med ett starkt magnetfält. Interaktionen visar sig i kompressibla band (compressible strips) i potentialfunktionens profil. Vi visar att kompressibla band manifesteras i uppkomsten av platåer i elektronenergierna som en funktion av det magnetiska fältet. Denna analys kompletteras i den sista artikeln (VI), vilken presenterar en allmän feluppskattning för egenvärden till linjära operatorer, och kan användas för energispektrumav kvantprickar betraktade i artikel II.

I artiklarna III, IV och V fokuserar vi på spridning på ultra-låg långdistanspotential i grafennanoband. Vi utför en teoretisk analys av spridningsproblemet och betraktar de framåtskridande vågor, och dessutom några utökade vågor. Vi visar att analysen låter oss förutsäga förekomsten av fångade tillstånd inom ett specifikt energiintervall förutsatt att potentialen är tillräckligt liten.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2017. 51 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1837
National Category
Condensed Matter Physics Other Physics Topics Computational Mathematics Mathematical Analysis
Identifiers
urn:nbn:se:liu:diva-136093 (URN)10.3384/diss.diva-136093 (DOI)9789176855621 (ISBN)
Public defence
2017-04-28, Visionen, B-huset, Campus Valla, Linköping, 13:15 (English)
Opponent
Supervisors
Available from: 2017-03-27 Created: 2017-03-27 Last updated: 2017-03-31Bibliographically approved

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