Quantum scattering and interaction in graphene structures

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.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

Trapped modes in armchair graphene nanoribbons

We study scattering on an ultra-low potential in armchair graphene nanorib bon. Using the continuous Dirac model and including a couple of articial waves in the scattering process, described by an augumented scattering matrix, we derive a condition for the existence of a trapped mode. We consider the threshold energies, where the the multiplicity of the continuous spectrum changes and show that a trapped mode may appear for energies slightly less than a thresold and its multiplicity does not exceed one. For energies which are higher than a threshold, there are no trapped modes, provided that the potential is suciently small

Error Estimation for Eigenvalues of Unbounded Linear Operators and an Application to Energy Levels in Graphene Quantum Dots

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 electronâ\u80\u93electron interactions in the potential

Effect of zigzag and armchair edges on the electronic transport in single-layer and bilayer graphene nanoribbons with defects

We study electronic transport in monolayer and bilayer graphene with single and many short-range defects focusing on the role of edge termination (zigzag versus armchair). Within the tight-binding approximation, we derive analytical expressions for the transmission amplitude in monolayer graphene nanoribbons with a single short-range defect. The analytical calculations are complemented by exact numerical transport calculations for monolayer and bilayer graphene nanoribbons with a single and many short-range defects and edge disorder. We find that for the case of the zigzag edge termination, both monolayer and bilayer nanoribbons in a single- and few-mode regime remain practically insensitive to defects situated close to the edges. In contrast, the transmission of both armchair monolayer and bilayer nanoribbons is strongly affected by even a small edge defect concentration. This behavior is related to the effective boundary condition at the edges, which, respectively, does not and does couple valleys for zigzag and armchair nanoribbons. In the many-mode regime and for sufficiently high defect concentration, the difference of the transmission between armchair and zigzag nanoribbons diminishes. We also study resonant features (Fano resonances) in monolayer and bilayer nanoribbons in a single-mode regime with a short-range defect. We discuss in detail how an interplay between the defect's position at different sublattices in the ribbons, the defect's distance to the edge, and the structure of the extended states in ribbons with different edge termination influence the width and the energy of Fano resonances.Funding Agencies|Swedish Institute||</p

Electron-electron interactions in graphene field-induced quantum dots in a high magnetic field

We study the effect of electron-electron interaction in graphene quantum dots defined by an external electrostatic potential and a high magnetic field. To account for the electron-electron interaction, we use the Thomas-Fermi approximation and find that electron screening causes the formation of compressible strips in the potential profile and the electron density. We numerically solve the Dirac equations describing the electron dynamics in quantum dots, and we demonstrate that compressible strips lead to the appearance of plateaus in the electron energies as a function of the magnetic field. Finally, we discuss how our predictions can be observed using the Kelvin probe force microscope measurements.Funding Agencies|Danish National Research Foundation [DNRF58]</p

Trapped modes in armchair graphene nanoribbons

We study scattering on an ultra-low potential in armchair graphene nanorib bon. Using the continuous Dirac model and including a couple of articial waves in the scattering process, described by an augumented scattering matrix, we derive a condition for the existence of a trapped mode. We consider the threshold energies, where the the multiplicity of the continuous spectrum changes and show that a trapped mode may appear for energies slightly less than a thresold and its multiplicity does not exceed one. For energies which are higher than a threshold, there are no trapped modes, provided that the potential is suciently small