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Magnetic structure of chiral graphene nanoribbons Pierce, James Kevin 2016

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Magnetic Structure of ChiralGraphene NanoribbonsA thesisbyJames Kevin PierceBS, West Virginia University, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 15 2016c© James Kevin Pierce 2016AbstractWe study the magnetic structure of narrow graphene ribbons with patternededges. Neglecting interactions, a broad class of edge terminations supportzero-energy states localized at the edges of the ribbon. For the simplest(zigzag) ribbon supporting these edge states, electron-electron interactionshave been shown to induce ferromagnetic ordering along the edges of theribbon. We generalize this argument for such a magnetic edge state tocarbon ribbons with more complex chiral edge terminations.iiPrefaceThis thesis is original work by the author Kevin Pierce, created in collabo-ration with research supervisor Dr. Ian Affleck.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Geometry and Properties of Graphene Ribbons . . . . . . 32.1 Zigzag and armchair . . . . . . . . . . . . . . . . . . . . . . . 62.2 Chiral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Electronic and Magnetic Structure of Zigzag Ribbons . . 83.1 Electronic structure of zigzag ribbons . . . . . . . . . . . . . 83.1.1 Zero energy edge states in zigzag ribbons . . . . . . . 93.1.2 Density of states in zigzag . . . . . . . . . . . . . . . 113.2 Magnetic structure of zigzag edge states . . . . . . . . . . . . 123.2.1 Edge-projected Hubbard interactions in zigzag . . . . 123.2.2 Uniqueness of ferromagnetic ground state in zigzag . 153.3 Summary of electronic and magnetic properties of zigzag . . 194 Electronic and Magnetic Structure of Chiral Ribbons . . . 204.1 Electronic structure of chiral ribbons . . . . . . . . . . . . . 204.1.1 Degeneracy of chiral edge mode bands . . . . . . . . . 214.1.2 (2,1) chiral ribbon edge mode structure . . . . . . . . 23ivTable of Contents4.1.3 (s,1) chiral ribbon edge mode structure . . . . . . . . 254.2 Magnetic structure of chiral ribbons . . . . . . . . . . . . . . 274.2.1 (2,1) magnetism . . . . . . . . . . . . . . . . . . . . . 274.2.2 (3,1) magnetism . . . . . . . . . . . . . . . . . . . . . 325 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38AppendicesThe Vandermonde Argument for Zigzag . . . . . . . . . . . . . 40vList of Figures2.1 Graphene decomposes into two triangular lattices. Primitivetranslation vectors are indicated. . . . . . . . . . . . . . . . . 42.2 A collection of possible minimal edge terminations of graphene.Grey–zigzag; yellow– armchair; red– (2,1) chiral; blue– (3,2)chiral; green–(3,1) chiral. . . . . . . . . . . . . . . . . . . . . . 53.1 A schematic spectrum of the zigzag ribbon with edge modeband highlighted in red and bulk bands in grey. . . . . . . . . 93.2 A zigzag ribbon segment is shown with the primitive transla-tion vector indicated. One sublattice is highlighted and ourchoice of unit cell is indicated. Several sites are labeled withtheir wavefunctions to indicate the αn,m notation. n is theindex along the direction of T, while m is the distance awayfrom the upper edge. . . . . . . . . . . . . . . . . . . . . . . . 104.1 From [7]– Schematic band structures of zigzag (1,0), (2,0) andgeneral (S,0) edges after folding the (1,0) zigzag edge band,where S = I + 3M . The shaded areas represent bulk states.Degeneracies of zero-energy bands (M,M + 1) are indicatedin the lower panels. . . . . . . . . . . . . . . . . . . . . . . . 224.2 One division into unit cells and the decomposition of the prim-itive translation vector for the (2, 1) ribbon into zigzag andarmchair components. Several sites are labeled with theirwavefunction to indicate the labeling scheme on the (2, 1)ribbon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23viChapter 1IntroductionAmong its many allotropes, carbon forms one atom thick two-dimensionalhoneycomb ribbons with a wide variety of edge geometries [4]. The edgegeometry of these ribbons is intimately connected to their electronic andmagnetic properties, and for many of the simplest possible ribbon edges,it has been shown that electrons with energies exponentially small in thewidth of the ribbon will localize near the ribbon edge [1] [2]. Signatures ofmagnetic ordering among these zero-energy zigzag edge-localized electronshas been seen experimentally [5], and it has been rigorously proven thatzigzag graphene ribbons will be ferromagnetic at their edges with an edge-projected Hubbard model in the limit U/t 1 [8].The topic of this thesis is a generalization of this rigorous proof of ferro-magnetism in zigzag ribbons to a wider class of edge geometries called chiralribbons. In chapter 2 we define the zigzag, armchair, and chiral geometries.In chapter 3 we will review the electronic and magnetic structure of zigzagribbons, and in chapter 4 we will discuss the electronic structure of chiralribbons in the absense of electron-electron interactions before we includeelectron-electron interactions to lowest order by projecting a Hubbard inter-action onto the localized edge states of the non-interacting spectrum. Thisprocess will indicate a ferromagnetic ordering of localized edge electrons inseveral particular chiral graphene ribbons.Graphene owes its hexagonal structure to the sp2 hybridization of itscarbon atoms. On each atom, the two p-orbitals oriented in the plane of thegraphene sheet mix with the s orbitalOne s-orbital and two p-orbitals hybridize in the plane of the graphenesheet, leading to a trigonal-planar structure bound by σ-bonds in the planebetween neighbouring carbon atoms. The third p-orbital is oriented per-1Chapter 1. Introductionpendicular to the plane and forms covalent pi-bonds with its neighbours,generating a pi-band. This pi-band is half-filled since each pi-orbital has oneelectron.Such a half-filled band has strong tight-binding character, justifying theuse of a local approximation. Graphene exhibits a large intra-site Coulombinteraction, with U/t ≈ 3.5 in such a local approximation [15]. Graphene isa strongly interacting material. Nevertheless, we hope to glean informationabout its electronic and magnetic properties by projecting the interactingproblem onto the non-interacting eigenstates in a small U/t limit. Withinthis approach we will show certain minimal chiral graphene ribbons haveferromagnetic ordering of electrons in their localized edge states.2Chapter 2Geometry and Properties ofGraphene RibbonsThe electronic structure of graphene ribbons is highly dependent on edgegeometry. In this chapter we will discuss the geometry of graphene and de-fine the zigzag, armchair, and minimal chiral edge terminations of grapheneribbons. The chapter will conclude with summaries of the electric and mag-netic properties of these different edge geometries.Graphene is a two-dimensional hexagonal arrangement of carbon atoms.A honeycomb lattice is not a Bravais lattice, but can be constructed froma triangular Bravais lattice with a basis of two atoms per unit cell. Thistriangular lattice is represented by two vectors as indicated in figure 2.1which we represent as R1 = (a, 0), R2 = (a/2,√3a/2), where a ≈ 2.46A˚ isthe triangular lattice constant.When the graphene sheet is terminated to form a ribbon, there are manypossible edge geometries. The two simplest edge geometries are called zigzagand armchair. The next level of complication one can imagine for the edgeterminations are the so-called minimal chiral ribbons, which are the con-centration of the rest of the thesis. These chiral edges are combinations ofarmchair and zigzag edges. Several minimal chiral terminations can be seenin figure 2.2.For any graphene ribbon having a periodic edge, there will be a primitivetranslation vector leaving the wavefunction invariant up to a phase. Wedenote this vector by T, and project it into the basis indicated indicated infigure 2.1:3Chapter 2. Geometry and Properties of Graphene RibbonsFigure 2.1: Graphene decomposes into two triangular lattices. Primitivetranslation vectors are indicated.T = nR1 +mR2, (2.1)where n and m are non-negative integers. For so-called minimal chiral rib-bons, these integers n and m completely determine the ribbon, and theyrepresent the number of respective zigzag and armchair components alongthe edge of the ribbon unit cell. In the remainder of this thesis we willconcentrate on minimal ribbons, using the notation (n,m) to refer to a par-ticular minimal chiral ribbon.The primitive translation vectors of zigzag and armchair ribbons can bewrittenTZ = R1 (2.2)TA = R1 +R2. (2.3)Comparing with equation 2.1, we see that a zigzag ribbon is a (1, 0) ribbon,while an armchair ribbon is a (1, 1) ribbon. Instead of characterising a given4Chapter 2. Geometry and Properties of Graphene RibbonsFigure 2.2: A collection of possible minimal edge terminations of graphene.Grey–zigzag; yellow– armchair; red– (2,1) chiral; blue– (3,2) chiral; green–(3,1) chiral.minimal chiral ribbon by (n,m), we can alternatively use the so-called chiralangle, which is the angle between the translation vector of the (n,m) ribbonand the translation vector of the zigzag ribbon:θ = arccosT ·TZ|T||TZ | = arcsin√34m2n2 + nm+m2. (2.4)This chiral angle ranges between the zigzag limit θ = 0 and the armchairlimit θ = pi/6. We are interested in the intermediate chiralities. The mag-nitude of the (n,m) translation vector isT =√T ·T = a√n2 + nm+m2. (2.5)52.1. Zigzag and armchairWe can project the (n,m) translation vector into the basis of zigzag andarmchair edges:T = (n−m)TZ +mTA. (2.6)This representation will later be useful for describing the electronic structureof chiral ribbons.2.1 Zigzag and armchairThe two simplest and most-studied graphene ribbons are the zigzag and arm-chair ribbons. In 1995 Nakata et al. showed that a semi-infinite graphenesheet with a zigzag edge has a band of states localized to the edge with ex-actly zero energy, while the armchair ribbon has no edge-localized states. [9]Analytic studies of finite-width zigzag graphene ribbons in a non-interactingelectron picture have indicated that these edge states persist with confine-ment, but the two edge states –one at each edge– mix across the ribbon dueto their finite overlap, splitting into bonding and anti-bonding states with aquantum confinement gap exponentially small in ribbon width. [14]This exponential dependence of this gap on zigzag ribbon width is in-consistent with ab initio calculations, and it has been proposed there is anadditional contribution to the energy gap due to magnetic ordering on theedges of the graphene ribbon [5] [12]. Edge-localized ferromagnetic orderingwas later proven in the local approximation in the limit of small U/t byKarimi and Affleck in 2010 [8]. This proof of magnetism by Karimi andAffleck will be reviewed in chapter 3. We will then apply a generalizationof this work to numerically prove edge ferromagnetism on certain chiral rib-bons in chapter 4. Armchair ribbons in contrast to zigzag have no low energylocalized edge states, and the gap at the Dirac point is due only to quantumconfinement in a finite ribbon. There is no edge-localized ferromagnetism.62.2. Chiral2.2 ChiralA minimal chiral ribbon is a ribbon with edges which are a mixture of zigzagand armchair components. Several studies suggest edge states will persist onthe zigzag components of the chiral edge, leading to some density of zigzag-like localized edge states near zero energy. Experimentally, these chiraledge states have been seen in the local density of states obtained throughscanning tunneling experiments [13]. Theoretically, the existence of chiraledge modes has been proven, and their density has been calculated by a2D band-structure projection scheme [1]. Additonally, mean-field theorycalculations suggest magnetic ordering the edges of minimal chiral grapheneribbons [3] [16].We would like to predict this magnetic ordering on chiral ribbons byanother method, using an edge-projected Hubbard interaction which is validin a small U/t limit, generalizing the work of Karimi and Affleck on thezigzag ribbon [8]. Using this method we will prove edge ferromagnetism onseveral particular chiral ribbons.7Chapter 3Electronic and MagneticStructure of Zigzag RibbonsSemi-infinite zigzag ribbons have one localized edge mode per spin on eachedge. These flat bands exist across one-third of the Brillouin zone at exactlyzero energy. In the first section of this chapter, we will show the existence ofthese bands and calculate their density. In the second section of this chapter,following [8], we will prove in an edge-projected Hubbard approximationthat interactions among these localized edge electrons induce ferromagneticordering along the edge in the ground state.3.1 Electronic structure of zigzag ribbonsIn this section we will derive the edge state spectra and wavefunctions onthe zigzag ribbon in the limit of semi-infinite width. We will show the edgemode bands are dispersionless at exactly zero energy. These edge modebands span the Brillouin zone between the so-called Dirac points, where thebulk bands contact the Fermi level with the linear dispersion characteristicof massless Dirac fermions [4] [2].We will not discuss bulk electronic properties in any detail, as they arenot relevant for our present study of edge magnetism. A rough schematic ofthe band structure of a semi-infinite zigzag ribbon is indicated in figure 3.1.83.1. Electronic structure of zigzag ribbonsFigure 3.1: A schematic spectrum of the zigzag ribbon with edge mode bandhighlighted in red and bulk bands in grey.3.1.1 Zero energy edge states in zigzag ribbonsWith nearest neighbour tight-binding hamiltonianH = t∑〈i,j〉c†icj + h.c., (3.1)using the unit cell and labeling scheme indicated in figure 3.2, we obtainSchro¨dinger equationEαn,m = t(βn−1,m + βn,m + βn,m+1) (3.2)Eβn,m = t(αn+1,m + αn,m + αn,m−1). (3.3)Translational invariance over T invites a crystal momentum k ·T = kT withψr+T = exp [ikT ]ψr, and the Bloch equation isEαm(k) = t((1 + e−ika)βm(k) + βm+1(k) (3.4)Eβm(k) = t((1 + eika)αm(k) + αm−1(k). (3.5)We would like to find any modes which may exist at exactly zero energy.Notice the two sublattices decouple at zero energy. This occurs generallyfor more complicated chiral edge terminations as well. Forming the Bloch93.1. Electronic structure of zigzag ribbonsFigure 3.2: A zigzag ribbon segment is shown with the primitive translationvector indicated. One sublattice is highlighted and our choice of unit cellis indicated. Several sites are labeled with their wavefunctions to indicatethe αn,m notation. n is the index along the direction of T, while m is thedistance away from the upper edge.equation at E = 0 and making an exponential ansatzαm(k) ∝ e−ma/Λ(k), (3.6)(and similarly for βm(k)) we find eigenstatesαm(k) = [−2 cos ka/2]meikam/2α0(k) (3.7)βm(k) = [−2 cos ka/2]W−1−me−ikam(W−1−m)/2βW−1(k), (3.8)where the dimensionless width W − 1 is the largest value m takes on. Weobserve that the αm(k) modes decay exponentially away from the m = 0edge, while the βm(k) decay exponentially away from the m = W − 1 edge.Considering normalizability, we see these exponentially decaying eigen-states are only simultaneous eigenstates in the limit of infinite width W →∞, which is the limit in which the localized edge modes do not overlap acrossthe ribbon. Because there are no other modes near zero energy these edgestates can mix with, in a finite system these two modes will overlap and splitinto bonding and anti-bonding pairs, gapping the edge-mode spectrum awayfrom E = 0 with a gap exponentially small in W [14]. In the semi-infinitelimit of large W , the edge modes will be approximately dispersionless zero103.1. Electronic structure of zigzag ribbonsenergy bands. Since these bands have no curvature, their effective mass isinfinite. Electrons in these bands are localized to the edge of the ribbon andare delocalized in the direction along T.From this point onward we concentrate only on semi-infinite ribbons forwhich W is large enough that there is no mixing across the ribbon. We havetwo zero energy states (per spin). One on each edge or equivalently one persublattice. The electron density in these edge modes decays exponentiallyinto the ribbon away from the edge with a characteristic length Λ(k) =−a ln |2 cos ka/2|−1:|αm(k)| =√2aΛ(k)e−ma/Λ(k)|α0(k)| (3.9)|βm(k)| =√2aΛ(k)e−(W−1−m)a/Λ(k)|β0(k)|. (3.10)This penetration length diverges in the limits ka = 2pi/3 and ka = 4pi/3–the location of the Dirac points in the zigzag ribbon spectrum. In the finitesystem, the overlap of the edge modes on opposite edges is largest at valuesof k for which the correlation length diverges, so we expect the largest finite-size splitting at the Dirac points.3.1.2 Density of states in zigzagThe density of localized edge states per length of ribbon and per spin in thesemi-infinite zigzag ribbon is approximately 1/3a:ρ(E) =∑2pi/3<ka<4pi/3δ(E) ≈∫ 4pi/3a2pi/3adk2piδ(E) =13aδ(E). (3.11)We see that there is about one edge-localized electron state per 3 unit cellsin the zigzag ribbon.113.2. Magnetic structure of zigzag edge states3.2 Magnetic structure of zigzag edge statesIn order to understand the low energy magnetic structure of zigzag ribbons,we include local Coulomb repulsion via a Hubbard interaction assumed tobe small relative to the non-interacting hamiltonian. We project this inter-action into the non-interacting eigenbasis. In the semi-infinite ribbon, theedge and bulk modes will be well separated in energy– the dispersionlessedge modes being at exactly zero energy. In the low energy limit thereforeonly the E = 0 bands will be accessible to scattering processes. In this waywe can justify neglecting bulk-to-bulk (bulk-bulk) scattering and scatteringbetween bulk modes and edge modes (bulk-edge) [8]. This is the centralapproximation which allows an analytical understanding of magnetism inzigzag ribbons. This edge-projection scheme supports a rigorous proof ofthe existence of a unique edge-ferromagnetic ground state, accessible in thelimits of U  t and semi-infinite width.3.2.1 Edge-projected Hubbard interactions in zigzagWe seek the ground state of the hamiltonianH = t∑〈i,j〉,σ{c†iσcjσ + c†jσciσ}+HU − µ∑i,σc†iσciσ, (3.12)where HU is the Hubbard hamiltonianHU = U∑ic†i↑ci↑c†i↓ci↓, (3.13)and the last term is included to preserve particle-hole symmetry.In this section we will review an appproximation to this interacting prob-lem created by Karimi and Affleck [8] and based on the work of Schmidtand Loss [11], in which we project the interacting problem onto the non-interacting eigenstates and neglect bulk-bulk and bulk-edge interactions inanticipation of an effective theory with all bulk states at negative energyoccupied and all bulk states at positive energy unoccupied. Our neglectof bulk-edge interactions will be justified in the limit of U/t  1 because123.2. Magnetic structure of zigzag edge statesthe overlap of localized edge states with delocalized bulk states is small.The neglect of bulk-bulk interactions is justified because the effect of bulkinteractions is marginal [10].We diagonalize the hamiltonian H = t∑c†iσcjσ + h.c. and obtain a setof non-interacting eigenstates characterized in Bloch space by creation op-erators c†σ(k). We partition this set of eigenstates into edge states e†σ(k) andbulk states b†σ(k):{c†σ(k)} = {e†σ(k)} ∪ {b†σ(k)}. (3.14)Our central approximation is to work in a low energy limit in which bulk-edge and bulk-bulk interactions are insignificant. Projecting 3.13 into thispartitioned basis 3.14 we obtain different terms. Some describe scatteringprocesses between edge modes, while others describe scattering processesbetween edge and bulk or exclusively between bulk modes:HU =∑k1k2k3k4Γ(k1, k2, k3, k4)e†↑(k1)e↑(k2)e†↓(k3)e↓(k4)+∑k1k2k3k4Γ′(k1, k2, k3, k4)b†↑(k1)b↑(k2)e†↓(k3)e↓(k4) + . . . (3.15)At this point we perform the aforementioned approximations and neglect allbut the first term describing scattering only among edge states. We concen-trate on a semi-infinite ribbon so that the zero-energy edge wavefunctionstake the formαn,m(k) = eika(n+m/2)[−2 cos ka/2]mα0(k) = eiknaαm(k), (3.16)and the only vertex factor Γ we are concerned with in 3.15 becomes:133.2. Magnetic structure of zigzag edge statesΓ(k1, k2, k3, k4) = U∑n,mα∗n,m(k1)αn,m(k2)α∗n,m(k3)αn,m(k4)= LUδk1−k2+k3−k4∑mα∗m(k1)αm(k2)α∗m(k3)αm(k4),(3.17)where L is the length of the ribbon (the number of unit cells).Using this interaction function we obtain our edge-projected Hubbardhamiltonian for the zigzag ribbon:HU =∑kk′qΓ(k, k′, q)e†↑(k + q)e↑(k)e†↓(k′ − q)e↓(k′). (3.18)HereΓ(k, k′, q) = LU∑mα∗m(k + q)αm(k)α∗m(k′ − q)αm(k′)= LU{[1− 4 cos2 (k+q)a2 ][1− 4 cos2 ka2 ][1− 4 cos2 (k′−q)a2 ][1− 4 cos2 k′a2 ]}1/21− 16 cos (k+q)a2 cos ka2 cos (k′−q)a2 cosk′a2.(3.19)The sum over k, k′ and q is restricted to the band in which 2pi/3 < (k +q)a, ka, (k′−q)a, k′a < 4pi/3. Incorporating the energy shift−µ∑k,σ e†σ(k)eσ(k)into the Hubbard hamiltonian to restore particle-hole symmetry, the edge-projected Hubbard hamiltonian takes the formHU =∑kk′qΓ(k, k′, q)[∑σe†σ(k+q)eσ(k)−δq=0][∑σ′e†σ′(k′−q)eσ′(k′)−δq=0].(3.20)Introducing the operators Om(q) and O†m(q) defined byO†m(q) =√LU∑kα∗m(k + q)αm(k)[∑σe†σ(k + q)eσ(k)− δq=0], (3.21)143.2. Magnetic structure of zigzag edge statesthe Hubbard hamiltonian takes positive definite formHU =∑q,mO†m(q)Om(q). (3.22)In this form it is obvious that the ground state of our hamiltonian hasE = 0. One can see that a fully spin-polarized state is a zero energy stateand therefore a ground state. Consider for example a state with spin upelectrons at every momentum k. It is clear that Om(q) annihilates thisstate, because the spin-up terms in Om(q) try to create a spin-up electronwith momentum k + q in an occupied fermionic state, and the spin-downterms in Om(q) try to annihilate a spin-down electron with momentum kin an unoccupied state. It remains to show that this ferromagnetic groundstate is also unique.3.2.2 Uniqueness of ferromagnetic ground state in zigzagWe now argue that the two fully polarized multiplets of total spin S = L/6are the unique ground states of 3.12 in our edge-projected approximation.In order to prove this uniqueness, we need only show that the only groundstates annihilated by the operator Om(q) for all m and q are fully polarized,i.e. they have maximal total spin. This means the ground state wavefunctionis only expressible as a symmetric combination of Fock states with a singleoccupancy at each momentum.Assume that |ψ〉 is annihilated by Om(q) for all m and q. In symbols,0 = Om(q)|ψ〉∝∑kα∗m(k)αm(k + q)[∑σe†σ(k + q)eσ(k)− δq=0]|ψ〉. (3.23)We argue in the Appendix 5 using the Vandermonde theorem that equation3.25 implies the following condition which we will apply repeatedly in orderto prove ferromagnetism in the ground state:153.2. Magnetic structure of zigzag edge states0 =[∑σ{e†σ(k + q)eσ(k) + e†σ(−k)eσ(−k − q)} − 2δq=0]|ψ〉. (3.24)In particular, let us choose this equation with q = 0. Then we have0 =[∑σ{e†σ(k)eσ(k) + e†σ(−k)eσ(−k)} − 2]|ψ〉. (3.25)This equation is satisfied if and only if |ψ〉 is such that[n(k) + n(−k)]|ψ〉 = 2|ψ〉 (3.26)for all k. Here n(k) =∑σ=↑,↓ e†σ(k)eσ(k) is the number operator for theedge state at momentum k. This restriction leaves us two options for theoccupation of momentum states −k and k: Either n(k) = n(−k) = 1, orn(−k) = 0 and n(k) = 2 (or vice-versa). We call the second possibility anexcition. Now we will prove that excitons are not permitted in the groundstate of the hamiltonian 3.22.In general, any state |ψ〉 may be represented as a linear combination ofall possible Fock states∏k,σe†σ(k)|0〉. (3.27)Here the product can run over any subset of the momenta k and spins σsupporting edge modes: 2pi/3 < |ka| < 4pi/3, of which there are approxi-mately L/3. Consider a state |φ〉 satisfying n(k) + n(−k) = 2 for all k withan exciton arbitrarily placed at momentum l. We write this as|φ〉 = | . . . , 0︸︷︷︸−l, . . . , ↓↑︸︷︷︸l, . . .〉. (3.28)We assume this excitonic state |φ〉 may enter the Fock state expansion of ourground state |ψ〉. That is, we assume 〈φ|ψ〉 6= 0. Now seeking a contradictionwe impose condition 3.24 on |ψ〉 with the particular choice of q = −2l and163.2. Magnetic structure of zigzag edge statesk = l. We should then have∑σe†σ(−l)eσ(l)|ψ〉 = 0. (3.29)Since we assume the excitonic state |φ〉 enters the expansion of our groundstate |ψ〉, we are led to consider the action of this operator ∑σ e†σ(−l)eσ(l)on |φ〉:∑σe†σ(−l)eσ(l)| . . . , 0︸︷︷︸−l, . . . , ↓↑︸︷︷︸l, . . .〉= | . . . , ↑︸︷︷︸−l, . . . , ↓︸︷︷︸l, . . .〉 − | . . . , ↓︸︷︷︸−l, . . . , ↑︸︷︷︸l, . . .〉 (3.30)Clearly, this operator acting on |ψ〉 gives a non-zero contribution. There-fore in order to satisfy the condition 3.29 there must be some other states inthe Fock state representation of the ground state |ψ〉 which together cancelthe excitonic state |φ〉’s non-zero contribution in 3.30. Since the operator∑σ e†σ(−l)eσ(l) only connects momenta l and −l, all other momenta are un-changed by the operator and there are only three other states which couldcancel the terms generated from the excitonic state |φ〉 in 3.30:|1〉 = | . . . , ↑↓︸︷︷︸−l, . . . , 0︸︷︷︸l, . . .〉 ∑σ e†σ(−l)eσ(l)|1〉 = 0 (3.31)|2〉 = | . . . , ↑︸︷︷︸−l, . . . , ↓︸︷︷︸l, . . .〉 ∑σ e†σ(−l)eσ(l)|2〉 = |1〉 (3.32)|3〉 = | . . . , ↓︸︷︷︸−l, . . . , ↑︸︷︷︸l, . . .〉 ∑σ e†σ(−l)eσ(l)|3〉 = −|1〉. (3.33)Since∑σ e†σ(−l)eσ(l)|φ〉 = |2〉 − |3〉, and none of the three possible statesmap through∑σ e†σ(−l)eσ(l) into |2〉 or |3〉, the terms generated by theexcitonic state in 3.30 cannot be canceled by any other allowed kets in theFock state expansion of the ground state, and therefore 〈φ|ψ〉 = 0. That is,173.2. Magnetic structure of zigzag edge stateswe’ve contradicted the assumption that there is an excitonic contributionto ground state wavefunction of the edge-projected Hubbard hamiltonian.We’ve now narrowed down the possible states in the Fock state expansion ofthe ground state to states with n(k) = n(−k) = 1 at all momenta k. Thereare no excitions allowed in the ground state.Having shown there are no excitonic states, we now show that the allowedstates with n(k) = n(−k) = 1 for all k in the Fock state expansion of theground state |ψ〉 always enter symmetrically like| . . . , ↑︸︷︷︸k, . . . , ↓︸︷︷︸k′, . . .〉+ | . . . , ↓︸︷︷︸k, . . . , ↑︸︷︷︸k′, . . .〉 (3.34)and never anti-symmetrically like| . . . , ↑︸︷︷︸k, . . . , ↓︸︷︷︸k′, . . .〉 − | . . . , ↓︸︷︷︸k, . . . , ↑︸︷︷︸k′, . . .〉. (3.35)To see this, consider the constraint 3.24 again but with q = k′− k. We have∑σ[e†σ(k′)eσ(k) + e†σ(−k)eσ(−k′)]|ψ〉 = 0. (3.36)Now assume the expansion of |ψ〉 has a state like | . . . , ↑︸︷︷︸k, . . . , ↓︸︷︷︸k′, . . .〉 init. The action of the operator in 3.36 is∑σ[e†σ(k′)eσ(k) + e†σ(−k)eσ(−k′)]| . . . , ↑︸︷︷︸k, . . . , ↓︸︷︷︸k′, . . .〉= | . . . , 0︸︷︷︸k, . . . , ↓↑︸︷︷︸k′, . . .〉. (3.37)183.3. Summary of electronic and magnetic properties of zigzagWe also have∑σ[e†σ(k′)eσ(k) + e†σ(−k)eσ(−k′)]| . . . , ↓︸︷︷︸k, . . . , ↑︸︷︷︸k′, . . .〉= −| . . . , 0︸︷︷︸k, . . . , ↓↑︸︷︷︸k′, . . .〉. (3.38)Since 3.36 must hold, it’s clear that states must enter symmetrically into theFock state expansion of the ground state |ψ〉. Therefore, we have found ourground state |ψ〉 is a symmetric combination of Fock states with a singleoccupancy at each momentum. These allowed ground states are then allSU(2) rotations of the fully polarized states | ↑, ↑, ↑, . . .〉 (and spin-down)with total spin S ≈ L/6. This completes the proof of ferromagnetism in theedge states of the zigzag ribbon in the U/t 1 limit.3.3 Summary of electronic and magneticproperties of zigzagZigzag ribbons have been shown to manifest a density of edge-localized statesat exactly zero energy in the limit of semi-infinite width [4]. Using an edge-projected Hubbard interaction, we have argued following Karimi and Affleckin [8] that a small on-site Coulomb repulsion drives a ferromagnetic orderingof electrons localized at the edge in these dispersionless zero energy states.In the next chapter we will apply a generalization of this argument forferromagnetism to certain chiral ribbons.19Chapter 4Electronic and MagneticStructure of Chiral RibbonsChiral ribbons are known experimentally to manifest edge states from STMand STS experiments [13]. It has been established that there is always someenhancement of the local density of states due to edge-localized states forall carbon ribbon edge terminations except for armchair [1]. Several authorshave studied an appropriate Hubbard model in the mean field approximationand have found ferromagnetic edge state ordering in chiral ribbons [16] [3] [6].We would like to clarify these findings by calculating edge mode spectra andshowing ferromagnetic ordering in the chiral edge state by a means otherthan mean field theory.4.1 Electronic structure of chiral ribbonsThe band structure of any minimal graphene ribbon can be obtained fromprojecting the two-dimensional band structure of graphene onto an appro-priate direction. This projection has been used to derive the density of zeroenergy edge states per length and per spin on semi-infinite chiral ribbons [1].The result isρ(θ) =23acos(θ + pi/3), (4.1)where θ is the chiral angle which ranges between 0 for zigzag and pi/6 forarmchair. We can see the density of edge states decreases as more armchaircomponents are incorporated into the primitive translation vector. In lowchirality ribbons with a large ratio of zigzag to armchair links, the edge mode204.1. Electronic structure of chiral ribbonsbands can become degenerate, with more than one localized edge state at agiven momentum. One group has performed many numerical chiral ribbonband structure calculations, and they have created a scheme to predict thedegeneracy of edge mode bands from a band-folding argument [7]. Thoughthese rules do not appear to have a rigorous foundation, we have found themconsistent with every case we’ve checked. In the next section, we will reviewthis band-folding conjecture.4.1.1 Degeneracy of chiral edge mode bandsThe band folding conjecture of Jasko´lski et al. allows one to predict the de-generacy of chiral edge mode bands without performing calculations. Theirclaim is this: The edge mode spectrum of an (n,m) chiral ribbon is the sameas the edge mode spectrum of an (n −m, 0) zigzag ribbon. This supportsa belief that chiral zero mode properties are determined only by the zigzagsegments in the chiral unit cell, and not by armchair segments.The (n,m) chiral translation vector isTn,m = (n−m)TZ +mTA, (4.2)where the basis is defined in equation 2.2. This decomposition is indicatedfor the particular case of a (2, 1) ribbon in figure 4.2. We see there aren − m zigzag links in the (n,m) chiral unit cell. Jasko´lski et al. reportfrom their numerical calculations that the zero mode spectrum of the (n,m)chiral ribbon is identical to the zero mode spectrum of the zigzag ribbonwith its unit cell enlarged artificially by a factor of S = n−m, i.e. an (S, 0)ribbon. Therefore, let us determine the edge mode spectrum of an (S, 0)ribbon.With translation vector STZ , the first Brillouin zone has extent |k| ≤pi/aS. We know the edge mode spectrum in an extended zone scheme with|k| < pi/a from our previous calculations in chapter 3. We can then fold thebands from higher zones into our first Brillouin zone |k| ≤ pi/aS to obtainthe band structure of the (S, 0) ribbon. This process is indicated in thetop panels of figure 4.1. Depending on S = n −m, the zero energy bands214.1. Electronic structure of chiral ribbonsFigure 4.1: From [7]– Schematic band structures of zigzag (1,0), (2,0) andgeneral (S,0) edges after folding the (1,0) zigzag edge band, where S =I + 3M . The shaded areas represent bulk states. Degeneracies of zero-energy bands (M,M + 1) are indicated in the lower panels.resulting from this folding process may become degenerate, with multiplezero energy states at a given momentum. The two Dirac points of the bulkbands can also become degenerate. Whenever this Dirac degeneracy occurs,zero energy bands will span the entire Brillouin zone. If the Dirac points arenon-degenerate, zero energy bands will span the segments of the Brillouinzone between the Dirac points.The result of folding for a given S is summarized by the following rela-tionship: S = I + 3M , where I = 1, 2, 3 and M = 0, 1, 2, ...; if I = 1 or 2in this decomposition of S, the Dirac points are non-degenerate. If I = 3,they’re degenerate. M then determines the degeneracy of the edge modebands according to the lower panels of figure 4.1.224.1. Electronic structure of chiral ribbonsFigure 4.2: One division into unit cells and the decomposition of the prim-itive translation vector for the (2, 1) ribbon into zigzag and armchair com-ponents. Several sites are labeled with their wavefunction to indicate thelabeling scheme on the (2, 1) ribbon.4.1.2 (2,1) chiral ribbon edge mode structureThe (2, 1) ribbon has translation vector T2,1 = TZ +TA. The magnitude ofthis translation vector is T = a√7. Its chiral angle (eq. 2.4) is θ2,1 = 19.1◦.Since the (2,1) ribbon has one zigzag component per unit cell, following [7]and noting 2−1 = 3(0)+1, we expect a single band of zero energy states (perspin) extending between the Dirac points at kT = 2pi/3 and kT = 4pi/3.We divide the (2, 1) ribbon into unit cells as shown in figure 4.2. Welabel the wavefunction on a given site as αi,n,m. Here n labels the unit cell.m = 0, 1, 2, ... labels the distance away from the upper edge in each unit cell,and i = 1, 2, 3 labels the three distinct sites at each m within the unit cell.At zero energy the two sublattices of the graphene ribbon decouple. Nearestneighbour hopping on the (2, 1) ribbon leads to the following for the latticeSchro¨dinger equation on one of the two sublattices at zero energy. The othersublattice is accessible by inversion.234.1. Electronic structure of chiral ribbons0 = α1,n,m + α2,n,m + α2,n,m+1 (4.3)0 = α2,n,m + α3,n,m + α3,n,m+1 (4.4)0 = α3,n,m + α1,n+1,m + α1,n+1,m−1. (4.5)Translational invariance along T invites the use of Bloch’s theorem αi,n,m =eiknTαi,m(k), where T is the magnitude of the primitive translation vectoron the ribbon. The Bloch equations areα1,m + α2,m + α2,m+1 = 0 (4.6)α2,m + α3,m + α3,m+1 = 0 (4.7)e−ikTα3,m + α1,m + α1,m−1 = 0. (4.8)These equations only hold away from the edge. The boundary condition atthe edge is0 = α1,m=0 + α2,m=1. (4.9)We also require that the wavefunctions must be normalizable:3∑i=1∞∑m=0|αi,m(k)|2 = 1. (4.10)Making an exponential ansatz α1,m(k) ∝ ν(k)m in the Bloch equations 4.6to 4.8 provides a secular equation for the roots ν(k):ν(k)3 + 3ν(k)2 + (3 + e−ikT )ν(k) + 1 = 0. (4.11)Note the appearance of the binomial coefficients 1, 3, 3, 1. This equationhas three complex roots ν1(k), ν2(k), ν3(k) which we can find numerically.Imposing the normalization condition, we find two of the roots are normal-izable for 2pi/3 < |kT | < pi, while the third is not normalizable at any k Letus denote the two normalizable roots ν1(k) and ν2(k).We’ve found our expected single zero energy mode wavefunctions which244.1. Electronic structure of chiral ribbonsare non-zero on 2pi/3 < |kT | < pi (one-third of the Brillouin zone) with theformαi,n,m(k) = eiknT[gi(k)ν1(k)m + li(k)ν2(k)m], (4.12)where i = 1, 2, 3 and the gi(k), li(k) are phase factors fully determinedin terms of the νi(k) by the boundary and normalization conditions. Thesethree wavefunctions at eachm and k describe a density of electrons at exactlyzero energy which decays away from the (2,1) chiral edge exponentially. Thisdensity isρ(θ2,1) ≈∫ 4pi/3T2pi/3Tdk2pi=13a1√7. (4.13)We see the (2, 1) edge mode density is suppressed by a factor of 1/√7 ≈ 0.38from zigzag. This result agrees with that obtained from equation 4.1 fromref. [1] using θ2,1 = 19.1◦. Later in this chapter we demonstrate that theform of these zero energy edge modes supports ferromagnetic ordering atthe edge of the (2, 1) ribbon.4.1.3 (s,1) chiral ribbon edge mode structureExtending the calculation to (s,1) ribbons, where s is an arbitrary integerlarger than 1, since there are s− 1 zigzag links, we expect following [7] thatthe degeneracy and location of the Dirac points are determined by M and Iin s−1 = 3I+M which follows from the band-folding scheme those authorspresent. The translation vector is given by T = (n − 1)TZ + TA and itsmagnitude is T = a√s2 + s+ 1.The s+1 lattice Bloch equations at each choice of m on the upper edge’ssublattice, using an extension of the unit cell and labeling scheme indicatedfor the (2,1) ribbon in figure 4.2, are254.1. Electronic structure of chiral ribbons0 = α1,m + α2,m + α2,m+1 (4.14)0 = α2,m + α3,m + α3,m+1 (4.15)... (4.16)0 = αs,m + αs+1,m + αs+1,m+1 (4.17)0 = e−ikTαs+1,m + α1,m−1 + α1,m. (4.18)The boundary condition isαs−1,0 + αs,1 = 0. (4.19)We again make the ansatz of an exponentially decaying wave-function intothe lattice Bloch equations: α1,m ∝ ν(k)m, which provides a secular equationof s+ 1 order:0 =(s+ 10)ν(k)s+1 +(s+ 11)ν(k)s + . . .+(s+ 1s− 1)ν(k)2+{(s+ 1s)+ (−1)se−ikT }ν(k)1 +(s+ 1s+ 1).(4.20)Solving this equation numerically for many different values of s we find thatthe normalizable solutions are consistent in every case checked with thedegeneracy predicition rules posited in ref. [7]. For example, if s = 16 wehave fifteen zigzag links in the unit cell so that 15 = I + 3M with I = 3 andM = 4. The folding rules of Jasko´lski et al. then indicate we should have a5-fold degenerate zero energy band across the entire Brillouin zone. We findequation 4.20 has 7 solutions with |ν(k)| < 1 across the entire Brillouin zone.The wavefunctions then have seven undetermined coefficients. Since we haveone boundary condition and one normalization condition, we are left withfive undetermined coefficients in the wavefunction, which is consistent witha degeneracy of five across the whole Brillouin zone. Other cases of (s, 1)ribbons check out against the folding rules of Jasko´lski et al. similarly. We264.2. Magnetic structure of chiral ribbonsalso find the density of zero modes for all (s, 1) ribbons we have investigatedto be in agreement with the Akhmerov et al. result 4.1 with chiral anglecalculated by equation 2.4.4.2 Magnetic structure of chiral ribbonsWe have found a general proof of magnetism on chiral ribbons inaccessiblethus far. Though minimal chiral ribbons certainly do support localizededge modes of known density and degeneracy, their wavefunctions cannotbe known in closed form due to our inability to solve secular equations suchas eq. 4.20 exactly. The degeneracy of chiral edge bands is an additionaldifficulty. This degeneracy introduces arbitrary coefficients into the zeroenergy wavefunctions. Because we need to invert matrices of arbitary sizeinvolving products of these wavefunctions in order to prove magnetism, thesearbitrary coefficients lead to a generally intractable problem.We have however numerically proven edge ferromagnetism on severalribbons with non-degenerate edge modes by generalizing the edge-projectedHubbard scheme developed on zigzag. These numerical calculations showmagnetism on ribbons of a fixed length. We have calculated ferromagneticedge ordering for many diferent ribbon lengths, and this suggests this or-dering is a general property independent of length.4.2.1 (2,1) magnetismThe (2, 1) ribbon has a single edge state per spin extending across one-thirdof its Brillouin zone, exactly like the zigzag ribbon. Our method of showingedge ferromagnetism proceeds in exact analogy with the zigzag ribbon, ex-cept one step of the mathematics is more complicated and cannot be doneanalytically as before. The process is this: First, we project the Hubbardinteraction into the non-interacting eigenbasis and drop bulk-bulk and edge-bulk terms on the basis of U  t. Second, we argue the unique ground statesof this edge-projected Hubbard hamiltonian are fully polarized.274.2. Magnetic structure of chiral ribbonsThe real space wavefunctions areαi,n,m(k) = eiknT[gi(k)ν1(k)m + li(k)ν2(k)m]= eiknTαi,m(k),(4.21)where T = a√7 is the magnitude of the primitive translation vector andthe gi(k) and li(k) are phase factors fully determined by boundary andnormalization conditions in terms of the two normalizable roots ν1(k) andν2(k) of the secular equation 4.11. Here there are four distinct sites per unitcell at a given distance m away from the edge, so i = 1, 2, 3, 4.In the limit U/t  1 we can make the approximate transformationeσ(k) ≈∑i,n,m αi,n,m(k)c(i,n,m)σ which corresponds to neglecting bulk-bulkand edge-bulk interactions, so thatHU = U∑i,n,mc†(i,n,m)↑c(i,n,m)↑c†(i,n,m)↓c(i,n,m)↓≈ U∑i,n,m∑k1k2k3k4αi,n,m(k1)α∗i,n,m(k2)αi,n,m(k3)α∗i,n,m(k4)× e†↑(k1)e↑(k2)e†↓(k3)e↓(k4). (4.22)Noting that∑nαi,n,m(k1)α∗i,n,m(k2)αi,n,m(k3)α∗i,n,m(k4)= Lδk1−k2+k3−k4=0αi,m(k1)α∗i,m(k2)αi,m(k3)α∗i,m(k4)(4.23)and renaming the momenta, equation 4.22 takes a form exactly analogousto the Hubbard hamiltonian from the zigzag case equation 3.18:HU ≈∑kk′qΓ(k, k′, q)e†↑(k + q)e↑(k)e†↓(k′ − q)e↓(k′), (4.24)where Γ(k, k′, q) is defined by284.2. Magnetic structure of chiral ribbonsΓ(k, k′, q) = LU∑i,mαi,m(k + q)α∗i,m(k)αi,m(k′ − q)α∗i,m(k′). (4.25)In imitation of the zigzag case we now define the operatorsO†i,m(q) =√LU∑kα∗i,m(k + q)αi,m(k)[∑σe†σ(k + q)eσ(k)− δq=0], (4.26)so that the Hubbard hamiltonian takes the positive definite formHU =∑i,m,qO†i,m(q)Oi,m(q). (4.27)Here i = 1, 2, 3 is a sum over inequivalent sites at a distance m away fromthe edge.We see again that a fully polarized state is a zero energy ground state.From here, in order to show this ferromagnetic ground state is the onlypossibility, we must show as in the zigzag case that the only states annihi-lated by Oi,m(q) at every q, m, and i are fully polarized. Suppose Oi,m(q)annihilates a ground state |ψ〉. We have0 =∑kα∗i,m(k + q)αi,m(k)[∑σe†σ(k + q)eσ(k)− δq=0]|ψ〉. (4.28)Again we shift the momenta in the sum by pi and choose q ≤ 0, putting kin the convenient range −pi/3− qT < kT < pi/3, and we note that α∗i,m(k+q)αi,m(k) = α∗i,m(−k)αi,m(−k− q). We split the sum in 4.28 into two equalhalves −pi/3− qT < kT ≤ −qT/2 and −qT/2 ≤ kT < pi/3 and restrict oursum to the upper half at the expense of writing two terms:294.2. Magnetic structure of chiral ribbons0 =∑−qT/2≤kT<pi/3α∗i,m(k + q)αi,m(k)× [∑σ{e†σ(k + q)eσ(k) + e†σ(−k)eσ(−k − q)} − 2δq=0]|ψ〉. (4.29)Now we argue that the summand vanishes term by term at every k. Hadwe the resulting condition[∑σ{e†σ(k + q)eσ(k) + e†σ(−k)eσ(−k − q)} − 2δq=0]|ψ〉 = 0, (4.30)exactly analogous with the zigzag case’s equation 3.24, the ferromagnetismproof would follow immediately.In the zigzag case we argued that the summand vanished at each k bydemonstrating the invertibility of a matrix with the Vandermonde theorem,equation 6 in the Appendix. This hinged upon the zigzag wavefunctionshaving an exponentially decaying form with m. An additional complicationwith the (2, 1) ribbon is that the wavefunctions decay bi-exponentially withm. This removes our ability to use the Vandermonde theorem because thematrix which we need to invert becomes a sum of four Vandermonde matri-ces, and the determinant of a sum of matrices does not necessarily relate tothe determinants of the individual matrices in the sum. However, we stillneed to show 4.30 holds to prove ferromagnetism of the edge state.To this end, we fix the length of the ribbon to be L unit cells, therebyfixing the number D + 1 of momenta on the range −qT/2 ≤ kT < pi/3 tobe at most L/6 + 1, and we construct a matrix by collating equation 4.29at the first D + 1 values of m:304.2. Magnetic structure of chiral ribbonsM i0,k0(q) Mi0,k1(q) . . . M i0,kD(q)M i1,k0(q) Mi1,k1(q) . . . M i1,kD(q).... . .. . ....M iD,k0(q) MiD,k1(q) . . . M iD,kD(q)|ψqk0〉|ψqk1〉...|ψqkD〉 =00...0 . (4.31)Here the matrix elements are defined byM im,k(q) = α∗i,m(k + q)αi,m(k), (4.32)and the kets by|ψqk〉 =[∑σ{e†σ(k + q)eσ(k) + e†σ(−k)eσ(−k − q)} − 2δq=0]|ψ〉. (4.33)In order to derive the condition 4.30 from which ferromagnetism follows,we need to show this matrix is invertible. Generally, there is no clear wayto do this. However, we have calculated the determinant of this matrix atevery q numerically for each choice of i = 1, 2, 3 and for system sizes up toL = 192– making at most a 32 × 32 matrix. The determinant is non-zeroand therefore we have proven the matrix in question is invertible in everycase checked. Therefore it holds for L ≤ 192 that[∑σ{e†σ(k + q)eσ(k) + e†σ(−k)eσ(−k − q)} − 2δq=0]|ψ〉 = 0. (4.34)From this relationship it follows by exact recapitulation of the steps outlinedbetween equations 3.25 to 3.38 that the ground state of the (2, 1) ribbon isferromagnetically ordered at the edge. We have thus numerically provenferromagnetism of the edge state for L ≤ 192. We expect this magnetism isalso plausible for L > 192.314.2. Magnetic structure of chiral ribbons4.2.2 (3,1) magnetismOur analysis of magnetism in the (3, 1) chiral ribbon is closely analogousto that used on the (2, 1)-chiral and (1, 0)-zigzag ribbons. Following theJasko´lski et al. band-folding argument, noting 3 − 1 = 2 = I + 3M withI = 2 and M = 0, we expect (see figure 4.1) a single non-degenerate bandof zero energy edge modes extending along −2pi/3 < kT < 2pi/3, whereT = a√13. This is two-thirds of the Brillouin zone.The lattice Bloch equations at zero energy areα1,m + α2,m + α2,m+1 = 0 (4.35)α2,m + α3,m + α3,m+1 = 0 (4.36)α3,m + α4,m + α4,m+1 = 0 (4.37)e−ikTα4,m + α1,m−1 + α1,m = 0, (4.38)and with the ansatz α1,m(k) ∝ ν(k)m they yield the secular equation (com-pare with equation 4.20)ν(k)4 + 4ν(k)3 + 6ν(k)2 + {4− e−ikT }ν(k) + 1 = 0. (4.39)This equation has four complex roots which can be found numerically. Twoof the four are normalizable with |νi(k)| < 1 along the range −2pi/3 < kT <2pi/3 in agreement with the Jasko´lski band-folding prediction. The densityof zero modes is thenρ(θ3,1) ≈∫ 2pi/3T−2pi/3Tdk2pi=13a2√13(4.40)in agreement with the Akhmerov et al. result 4.1 using θ3,1 = 13.9◦. Thedensity of zero modes is reduced by a factor of 2/√13 ≈ .55 from zigzag.The real-space zero energy edge mode wavefunctions have the formαi,n,m = eiknT {gi(k)ν1(k)m + li(k)ν2(k)m} = eiknTαi,m(k), (4.41)324.2. Magnetic structure of chiral ribbonsfor i = 1, 2, 3, 4 where the gi(k) and li(k) are phase factors fully determinedin terms of the two normalizable roots ν1(k)andν2(k) from the normalizationand boundary conditions. We again edge-project the Hubbard interaction.We take the interaction in the site basis HU = U∑i c†i↑ci↑c†i↓ci↓ and projectit into the non-interacting eigenbasis. We neglect bulk-bulk and bulk-edgecorrelations, sum over the transverse index n, and rename the momenta,obtainingHU ≈∑kk′qΓ(k, k′, q)e†↑(k + q)e↑(k)e†↓(k′ − q)e↓(k′), (4.42)valid for U/t  1, where the vertex factor Γ(k, k′, q) is defined in terms ofthe four wavefunctions αi,m(k) byΓ(k, k′, q) = LU∑i,mαi,m(k + q)α∗i,m(k)αi,m(k′ − q)α∗i,m(k′). (4.43)Again we define the operatorsO†i,m(q) =√LU∑kα∗i,m(k + q)αi,m(k)[∑σe†σ(k + q)eσ(k)− δq=0], (4.44)which bring the hamiltonian into positive definite form, exactly as in the(2, 1) case but with one additional term in the sum i = 1, 2, 3, 4 over distinctsites in the unit cell at a chosen m:HU =∑i,m,qO†i,m(q)Oi,m(q). (4.45)At this point it is again clear from the definition of Oi,m(q) that a fullypolarized state is annihilated by HU and is therefore a ground state. Wemust show these ferromagnetic ground states are the unique ground states.To this end we again consider the action of Oi,m(q) on a ground state|ψ〉, setting q ≤ 0 to obtain the condition334.2. Magnetic structure of chiral ribbons0 =∑−2pi/3−qT<kT<2pi/3α∗i,m(k + q)αi,m(k)[∑σe†σ(k + q)eσ(k)− δq=0]|ψ〉.(4.46)We note α∗i,m(k + q)αi,m(k) = α∗i,m(−k)αi,m(−k − q) and we break the suminto two equal halves −2pi/3−qT < kT ≤ −qT/2 and −qT/2 ≤ kT < 2pi/3.We restrict the sum to the upper half of the range and write two terms:0 =∑−qT/2≤kT<2pi/3α∗i,m(k + q)αi,m(k)× [∑σ{e†σ(k + q)eσ(k) + e†σ(−k)eσ(−k − q)} − 2δq=0]|ψ〉. (4.47)Again, we would like to argue that the summand vanishes at every k. Inthe zigzag case we were able to argue this rigorously with the Vandermondetheorem. For the (2, 1) case the best we can do is show this invertibilitynumerically for different choices of L or equivalently different choices of thenumber of independent momenta.Fixing the number of unit cells to be L and defining matrix elementsand kets viaM im,k(q) = α∗i,m(k + q)αi,m(k), (4.48)|ψqk〉 =[∑σ{e†σ(k + q)eσ(k) + e†σ(−k)eσ(−k − q)} − 2δq=0]|ψ〉, (4.49)we can again construct an analogue of the Vandermonde matrix from thezigzag case:344.2. Magnetic structure of chiral ribbonsM i0,k0(q) Mi0,k1(q) . . . M i0,kD(q)M i1,k0(q) Mi1,k1(q) . . . M i1,kD(q).... . .. . ....M iD,k0(q) MiD,k1(q) . . . M iD,kD(q)|ψqk0〉|ψqk1〉...|ψqkD〉 =00...0 . (4.50)We note this matrix is not of Vandermonde form, but is instead a sumof four Vandermonde matrices. Therefore the approach we used to isolateparticular momentum channels for the zigzag ferromagnetism proof doesnot work here. Instead we have done this numerically for particular cases.We have calculated the determinant of these four matrices (one for each i)to be non-zero at every q for system sizes up to L = 54 unit cells, whichcorresponds to at most a 19× 19 matrix. For cases of L ≤ 54 we obtain ourcondition[∑σ{e†σ(k + q)eσ(k) + e†σ(−k)eσ(−k − q)} − 2δq=0]|ψ〉 = 0, (4.51)from which the proof of edge ferromagnetism on the (3, 1) minimal chiralribbon follows by exact recapitulation of the steps around equations 3.25 to3.38 in the zigzag proof. We then have a rigorous proof of magnetism forL ≤ 54. It is plausible that this proof can be done for any L.35Chapter 5ConclusionWe have reviewed literature relevant to an understanding of confinement-driven magnetism in carbon ribbons with patterned edges. For the simplestcase of a zigzag edged ribbon, edge ferromagnetism has been rigorouslyproven in the small U/t limit by previous authors by projecting a Hubbardinteraction onto the localized edge modes and neglecting certain marginalinteractions. We have built upon this previous work by numerically demon-strating edge ferromagnetism on minimal chiral ribbons of the (2, 1) and(3, 1) variety within this edge-projected Hubbard approximation.Our numerical proof of magnetism within this edge-projected hubbardinteraction becomes more difficult for lower chirality ribbons, since we needto concentrate on individual momentum channels in order to rigorously provemagnetism. To isolate individual momentum channels, we must invert amatrix with elements which are products of the zero-energy edge modewavefunctions. In general, these chiral edge mode wavefunctions cannotbe exactly known, and they may also have undetermined coefficients associ-ated with edge-mode degeneracy. These two complications prevent us frominverting these matrices and isolating momentum channels in the generalcase.The best we have been able to achieve is show the invertibility of thismatrix numerically for many cases, which provides a strong suggestion ofedge ferromagnetism on (2, 1) and (3, 1) chiral ribbons of any length, anda numerical proof of edge ferromagnetism on (2, 1) ribbons with L ≤ 192and (3, 1) ribbons with L ≤ 54. We expect our methodology is sufficient tonumerically demonstrate edge ferromagnetism in any minimal chiral ribbonof a given length supporting non-degenerate edge modes, given we havesufficient computational power to calculate the determinant of the resulting36Chapter 5. Conclusionmatrices analogous to those in equation 4.50.37Bibliography[1] A. R. Akhmerov and C. W. J. Beenakker. Boundary conditions for diracfermions on a terminated honeycomb lattice. Phys. Rev. B, 77:085423,Feb 2008.[2] L. Brey and H. A. Fertig. Electronic states of graphene nanoribbonsstudied with the dirac equation. Phys. Rev. B, 73:235411, Jun 2006.[3] A. R. Carvalho, J. H. Warnes, and C. H. Lewenkopf. Edge magnetiza-tion and local density of states in chiral graphene nanoribbons. Phys.Rev. B, 89:245444, Jun 2014.[4] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, andA. K. Geim. The electronic properties of graphene. Rev. Mod. Phys.,81:109–162, Jan 2009.[5] Imre Hagymasi Peter Vanscso-Zoltan Osvath Peter Nemes-InczeChanyong Hwang Laszlo P. Biro Levente Tapaszto Gabor Zsolt Magda,Xiaozhan Jin. Room-temperature magnetic order on zigzag edges ofnarrow graphene nanoribbons. Nature, 514:608, Sept 2014.[6] Michael Golor, Thomas C. Lang, and Stefan Wessel. Quantum montecarlo studies of edge magnetism in chiral graphene nanoribbons. Phys.Rev. B, 87:155441, Apr 2013.[7] W. Jasko´lski, A. Ayuela, M. Pelc, H. Santos, and L. Chico. Edgestates and flat bands in graphene nanoribbons with arbitrary geome-tries. Phys. Rev. B, 83:235424, Jun 2011.38[8] Hamed Karimi and Ian Affleck. Towards a rigorous proof of magnetismon the edges of graphene nanoribbons. Phys. Rev. B, 86:115446, Sep2012.[9] Kyoko Nakada, Mitsutaka Fujita, Gene Dresselhaus, and Mildred S.Dresselhaus. Edge state in graphene ribbons: Nanometer size effectand edge shape dependence. Phys. Rev. B, 54:17954–17961, Dec 1996.[10] Manuel J. Schmidt, Michael Golor, Thomas C. Lang, and Stefan Wes-sel. Effective models for strong correlations and edge magnetism ingraphene. Phys. Rev. B, 87:245431, Jun 2013.[11] Manuel J. Schmidt and Daniel Loss. Tunable edge magnetism atgraphene/graphane interfaces. Phys. Rev. B, 82:085422, Aug 2010.[12] Young-Woo Son, Marvin L. Cohen, and Steven G. Louie. Energy gapsin graphene nanoribbons. Phys. Rev. Lett., 97:216803, Nov 2006.[13] Liying; Yazyev Oleg V.; Chen Yen-Chia; Feng Juanjuan; Zhang-Xiaowei; Capaz Rodrigo B.; Tour James M.; Zettl Alex; Louie StevenG.; Dai Hongjie; Tao, Chenggang; Jiao and Michael F. Crommie. Spa-tially resolving edge states of chiral graphene nanoribbons. Nat Phys,7:616–620, Aug 2011.[14] Katsunori Wakabayashi, Yositake Takane, Masayuki Yamamoto, andManfred Sigrist. Electronic transport properties of graphene nanorib-bons. New Journal of Physics, 11(9):095016, 2009.[15] T. O. Wehling, E. S¸as¸ıog˘lu, C. Friedrich, A. I. Lichtenstein, M. I. Kat-snelson, and S. Blu¨gel. Strength of effective coulomb interactions ingraphene and graphite. Phys. Rev. Lett., 106:236805, Jun 2011.[16] Oleg V. Yazyev, Rodrigo B. Capaz, and Steven G. Louie. Theory ofmagnetic edge states in chiral graphene nanoribbons. Phys. Rev. B,84:115406, Sep 2011.39The Vandermonde Argumentfor ZigzagHere we will show equation 3.24 holds given equation 3.23.We begin from equation 3.23, and we set q ≤ 0 so the range of k in thesum can be written 2pi/3− qa < ka < 4pi/3:0 =∑2pi/3−qa<ka<4pi/3α∗m(k)αm(k + q)× [∑σe†σ(k + q)eσ(k)− δq=0]|ψ〉. (1)Now we perform a series of manipulations on this equation. First, we shift allka by pi so the range of allowed momenta becomes −pi/3− qa < ka < pi/3–this is just for convenience. We haveα∗m(k)αm(k + q) ∝ sin {ka/2} sin {(k + q)a/2}=12{cos(qa/2)− cos(ka+ qa/2)} (2)by double-angle formulas. We can see two values of k in the range −pi/3−qa < ka < pi/3 will give equal values of α∗m(k)αm(k + q). We determinethese values k, k′ by requiringcos(ka+ qa/2) = cos(k′a+ qa/2) (3)40The Vandermonde Argument for Zigzagwhich has the solutionk′ = −k − q. (4)We would like to manipulate equation 1 into a form whereby each termin the sum over k must vanish independently. It is convenient to break thesum into two equal halves −pi/3 − qa < ka ≤ −qa/2 and −qa/2 ≤ ka <pi/3. We see that −qa/2 ≤ ka < pi/3 places k′ = −k − q into the range−pi/3− qa < k′a ≤ −qa/2. This allows us to restrict the sum in equation 1to the upper half at the expense of writing two terms:0 =∑−qa/2≤ka<pi/3α∗m(k)αm(k + q)× [∑σ{e†σ(k + q)eσ(k) + e†σ(−k)eσ(−k − q)} − 2δq=0]|ψ〉. (5)In writing this, we have used the property α∗m(k)αm(k + q) = α∗m(−k −q)αm(−k) of the wavefunctions which can be seen from their definition 3.16.Now we will show that each term in this sum vanishes independently.Our argument hinges upon the Vandermonde theorem concerning the deter-minant of a particular (said to be of Vandermonde form) square matrix:∣∣∣∣∣∣∣∣∣∣x00 x01 x02 . . . x0nx10 x11 x12 . . . x1n... . . .. . . . . ....xn0 xn1 xn2 . . . xnn∣∣∣∣∣∣∣∣∣∣∝∏i<j(xi − xj). (6)Notice if every xi is distinct, the Vandermonde determinant is necessarilynon-vanishing, meaning the matrix is invertible. Now let us use the explicitform of the wavefunctions αm(k) from equation 3.7, remembering we haveshifted ka by pi, and denoting|ψqk〉 = eiqam/2α∗0(k)α0(k + q)× [∑σ{e†σ(k + q)eσ(k) + e†σ(−k)eσ(−k − q)} − 2δq=0]|ψ〉 (7)41The Vandermonde Argument for ZigzagandMk(q) = 4 sin[ka2]sin[(k + q)a2]. (8)Equation 5 takes the simple-looking form:0 =∑−qa/2≤ka<pi/3Mk(q)m|ψqk〉. (9)Consider a zigzag ribbon of finite length L. There are at most L/3 + 1allowed momenta k in the range −pi/3 − qa < ka < pi/3, and at mostL/6 + 1 independent momenta to sum over in 9. Denote the set of momentain this range k0, k1, . . .,kD.Since equation 9 holds for every m, the index describing distance awayfrom the ribbon edge, which ranges from 0 to∞ in integer steps, we can writedown the equation for the first D+1 values of m and form a (D+1)×(D+1)square matrix:Mk0(q)0 Mk1(q)0 . . . MkD(q)0Mk0(q)1 Mk1(q)1 . . . MkD(q)1.... . .. . ....Mk0(q)D Mk1(q)D . . . MkD(q)D|ψqk0〉|ψqk1〉...|ψqkD〉 =00...0 . (10)We note this matrix is of Vandermonde form. We have purposefully manipu-lated the range of k in the sum in equation 9 to ensure Mk(q) is single-valuedand non-zero with k on this range for any q ≤ 0. Therefore, the Vander-monde determinant is non-zero, and we can invert the Vandermonde matrixto obtain0 =[∑σ{e†σ(k + q)eσ(k) + e†σ(−k)eσ(−k − q)} − 2δq=0]|ψ〉 (11)as claimed in equation 3.24.42

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