Ph. D. Thesis - The University of Texas at Austin

CopyrightbyYafis Barlas2008

Role of electron-electron interactions in Chiral 2DEGsbyYafis Barlas, B.S.; M.A.DISSERTATIONPresented to the Faculty of the Graduate School ofThe University of Texas at Austinin Partial Fulfillmentof the Requirementsfor the Degree ofDOCTOR OF PHILOSOPHYTHE UNIVERSITY OF TEXAS AT AUSTINAugust 2008

Dedicated to Ammi, Abbu and Amma.

AcknowledgmentsI have been fortunate to work with Prof. Allan H MacDonald, who hasbeen an excellent teacher, mentor and friend. He taught me physics that cannotbe learnt from textbooks, and his intuition and deep insight in condensedmatter theory has been a source of inspiration to me. I am specially indebtedto his extraordinary degree of patience, kindness and sense of humor. He isnot only a great physicist but also one of the best people I have known.I would also like to thank my collaborators: Tami Pareg-Berena, Marco Polini,Reza Asgari, Rene Cote and especially Kentaro Nomura, without whom thiswork would not have been possibile. I would also like to acknowledge discussionsand support from our group members: Paul Haney, Dagim Tilahun,Hongki Min, Ion Garate, Wei-Cheng Lee, Zhang Fan, Alvaro Nunez, JasonHill, Murat Tas and Sergey Maslennikov. I would especially like to thankBecky Drake for patience, kindness and immense help she provided during mywork with the group. I would also like to thank Prof. Alex Demkov, MaximTsoi, Jim Chelikowsky and Qian Niu for taking the time to be on my dissertationcommittee.I would like to thank my Jibroni friends and friends from the climbing wallfor friendship and support. The road trips we took over weekends and breakswere essential to my morale. I could not have asked for better company ontop of the Colorado fourteeners and New Mexico’s peaks.Finally I would like to thank my family especially my parents for their unconv

ditional love and support throught out the course of my life. Your financialand emotional sacrifices especially during my study in the States will never beforgotten. You have always been close to my heart, mere words cannot conveymy immense gratitude and love.vi

Role of electron-electron interactions in Chiral 2DEGsPublication No.Yafis Barlas, Ph.D.The University of Texas at Austin, 2008Supervisor: Allan H MacDonaldIn this thesis we study the effect of electron-electron interactions onChiral two-dimensional electron gas (C2DEGs). C2DEGs are a very good descriptionof the low-energy electronic properties of single layer and multilayergraphene systems. The low-energy properties of single layer and multilayergraphene are described by Chiral Hamiltoninans whose band eigenstates havedefinite chirality. In this thesis we focus on the effect of electron-electron interactionson two of these systems: monolayer and bilayer graphene.In the first half of this thesis we use the massless Dirac Fermion model andrandom-phase-approximation to study the effect of interactions in graphenesheets. The interplay of graphene’s single particle chiral eigenstates alongwith electron-electron interactions lead to a peculiar supression of spin susseptibilityand compressibility, and also to an unusual velocity renormalization.We also report on a theoretical study of the influence of electron-electron interactionson ARPES spectra in graphene. We find that level repulsion betweenquasiparticle and plasmaron resonances gives rise to a gap-like feature nearvii

the Dirac point.In the second half we anticipate interaction driven integer quantum Hall effectsin bilayer graphene because of the near-degeneracy of the eight Landaulevels which appear near the neutral system Fermi level. We predict that anintra-Landau-level cyclotron resonance signal will appear at some odd-integerfilling factors, accompanied by collective modes which are nearly gapless andhave approximate q 3/2 dispersion. We speculate on the possibility of unusuallocalization physics associated with these modes.viii

Table of ContentsAcknowledgmentsAbstractList of TablesList of FiguresvviixixiiChapter 1. Introduction 11.1 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . 3Chapter 2. Two dimensional Chiral Hamiltonians 62.1 Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.1 Low energy theory of graphene . . . . . . . . . . . . . . 122.1.2 Cyclotron Mass and Density of States . . . . . . . . . . 172.2 Bilayer Graphene . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Chiral Fermions in a magnetic field . . . . . . . . . . . . . . . 232.3.1 Quantum Hall Effect of Chiral Fermions . . . . . . . . . 25Chapter 3. Chirality and Correlations in Graphene 293.1 RPA Theory of Graphene . . . . . . . . . . . . . . . . . . . . 353.2 Charge and Spin Susceptibilities . . . . . . . . . . . . . . . . . 403.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Chapter 4. Graphene: A Pseudochiral Fermi Liquid 464.1 Random Phase Approximation of Self-energy . . . . . . . . . . 504.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55ix

Chapter 5. Plasmons and The Spectral Function of Graphene 625.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.2 Doped Dirac Sea Charge Fluctuations . . . . . . . . . . . . . . 645.3 Dirac Quasiparticle Decay . . . . . . . . . . . . . . . . . . . . 665.4 Spectral function . . . . . . . . . . . . . . . . . . . . . . . . . 69Chapter 6. Quantum Hall Ferromagnets 766.1 Review of Quantum Hall Effect . . . . . . . . . . . . . . . . . 776.2 Quantum Hall Ferromagnets . . . . . . . . . . . . . . . . . . . 80Chapter 7. Octet Quantum Hall Ferromagnets in Bilayer Graphene 877.1 Graphene Bilayer Landau Levels . . . . . . . . . . . . . . . . . 887.2 Octet Hunds Rules . . . . . . . . . . . . . . . . . . . . . . . . 907.3 Landau-Level Pseudospin Dipoles . . . . . . . . . . . . . . . . 937.4 Intra-Landau-Level Cyclotron Resonance . . . . . . . . . . . . 97Chapter 8. Conclusion 100Appendices 103Appendix A. Graphene’s Lindhard Function 104A.1 Half-Filling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105A.2 Pauli-blocking effects . . . . . . . . . . . . . . . . . . . . . . . 107Appendix B. Correlation Self-energy of a quasiparticle in graphene111Bibliography 114Index 124Vita 125x

List of Tablesxi

List of Figures2.1 This figure shows an atomic layer of graphene identified usingAtomic Force Microscopy . . . . . . . . . . . . . . . . . . . . . 102.2 (Adapted from [22]) Energy gap vs ribbon width. The insetshows energy gap vs relative angle for the device sets. Dashedlines in the inset represent the value of energy gap as predictedby empirical scaling of energy gap vs ribbon width. . . . . . . 122.3 Left: Lattice structure of graphene composed from two triangularlattices (⃗a and ⃗ b are the lattice vectors), with sublatticesA(Blue) and B(Red). Right: Shows the corresponding Brillouinzone. The Dirac cones are located at K and K ′ . . . . . . . . . 132.4 (Adapted from [10]) Left: Numerically calculated energy spectrumof graphene (in units if t). Right: Magnified linear energydispersion near the Dirac point. . . . . . . . . . . . . . . . . . 152.5 (Adapted from [10]) Cyclotron mass of quasiparticles in grapheneas a function of their concentration(n), positive and negative ncorrespond to electron and holes respectively. Experimentaldata extracted from SdH oscillations. . . . . . . . . . . . . . . 182.6 Lattice structure of bilayer graphene with a honeycomb latticeconstant a = 2.46A and interlayer separation d = 3.35A. . . . 192.7 (Adapted from [26]) This figure shows the evolution of gap closingand reopening by changing the doping level by potassiumadsorption. Experimental and theoretical bands (solid lines)(A) for an as-prepared graphene bilayer and (B and C) withprogressive adsorption of potassium are shown. The number ofdoping electrons per unit cell, estimated from the relative sizeof the Fermi surface, is indicated at the top of each panel . . 232.8 (Adapted from [3] Hall conductivity σ xy and longitudinal ρ xx ofgraphene as a function of the concentration at B = 14T. Theinset show bilayer graphene’s hall conductivity as a function ofconcentration. . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.1 Random-phase-approximation in terms of Feynman diagrams. 303.2 (Adapted from [10])Particle-hole continuum and collective modesof:(a) 2DEG ;(b) neutral graphene; (c) doped graphene . . . . 32xii

3.3 Dirac cone for n-doped graphene, the yellow arrows representsthe chirality of the bands s = +(−) for clockwise and anticlockwise,and the red arrows represent particle-hole transitions. 333.4 (Color online) Cut-off dependence of the regularized exchangeenergy δε x in units of the Fermi energy ε F . . . . . . . . . . . . 373.5 (Color online) Cut-off dependence of the regularized correlationenergy δε RPAc in units of the Fermi energy ε F . . . . . . . . . . 393.6 (Color online) Cut off Λ and coupling constant f dependence ofκ/κ 0 . The color coding is as in Figs. 3.4-3.5. . . . . . . . . . . 413.7 (Color online) Cut-off Λ and coupling constant f dependence ofthe spin susceptibility χ S . The color coding is as in Figs. 3.4-3.5. 424.1 Honeycomb lattice of a single layer graphite flake with one sublatticein yellow and the other sublattice in blue. In the continuumlimit the sublattice degree of freedom may be regardedas a pseudospin. When momentum k is measured away fromthe Dirac points at the K and K ′ Brillouin zone corners, bandeigenstates have definite projection of pseudospin in the k direction,i.e. definite pseudochirality. The angle φ k above denotesthe momentum-dependent phase difference between wavefunctionamplitudes on the two sublattices. For spin-1/2 quantumparticles this angle is the azimuthal orientation of a pseudospincoherent state in the equatorial plane. . . . . . . . . . . . . . . 584.2 In a weakly doped material, graphene’s energy bands can be describedby a massless Dirac equation in which the role of spin isplayed by pseudospin. Like an ordinary 2DES, doped graphenehas a circular Fermi surface. The Fermi liquid properties ofgraphene are a consequence of both exchange interactions betweenquasiparticles near the Fermi surface and states in thepositive and negative energy Fermi seas and of interactions withboth intra-band (short red vertical arrow) and inter-band (longred vertical arrow) virtual fluctuations of the electronic system.The yellow arrows in this figure indicate the pseudospinchirality of band eigenstates. Because of the difference in chiralitybetween positive and negative energy bands, the velocityof graphene quasiparticles is enhanced by inter-band exchangeinteractions, tending to protect the system from magnetic andother instabilities, and reducing both charge and spin responsefunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59xiii

4.3 “Lindhard” function χ (0) (q, iΩ) of a C2DES, in units of the noninteractingdensity-of-states at the Fermi surface ν = gk F /(2πv),as a function of q/k F and Ω/µ on the imaginary frequencyaxis. k F = (4πn/g) 1/2 is the Fermi wavenumber, µ = vk F theFermi energy, n the electron density and the flavor multiplicityg = g s g v = 4 for graphene because of its two-fold valley degeneracy.Because of interband fluctuations χ (0) diverges linearlywith q for q → ∞ and decays only like Ω −1 for Ω → ∞ in theC2DES, in contrast to the q −2 and Ω −2 behaviors of the ordinary2DES. In the static Ω = 0 limit χ (0) (q, 0) = −ν for allq ≤ 2k F for both chiral and ordinary 2DESs. . . . . . . . . . 604.4 Density and coupling constant f dependence of some C2DESFermi-liquid parameters. The density is specified by Λ ≡ q c /k F .The density range studied most extensively in experiment, n ∼10 11 cm −2 to n ∼ 10 13 cm −2 , corresponds to Λ = 100 to Λ = 10.In all panels the black solid line corresponds to the highest valueof the cut-off parameter we have considered, Λ = 2.7 × 10 5 .The red dashed line illustrates the RPA Fermi-liquid parametersof an ordinary non-chiral 2DES with parabolic bands. In thiscase the f = √ 2 r s [see Eq. (4.11)], where r s = (πna 2 B )−1/2 isthe usual Wigner-Seitz density parameter and a B = ǫ 2 /(m b e 2 )the effective Bohr radius. From the left the three panels show:(a) the quasiparticle renormalization factor Z evaluated fromEq. (4.7); (b) the velocity renormalization factor evaluated fromEq. (4.8); and (c) the l = 0 dimensionless Landau parameter F a 0which characterizes spin-dependent quasiparticle interactions.The color coding for Λ is the same in all panels. . . . . . . . . 615.1 Spectral function A(k, ω) of an n-doped graphene sheet as afunction of k (in units of Fermi wavevector k F ) and ω (in unitsof and measured from the Fermi energy vk F where v is theFermi velocity). These results are for coupling constant α gr =ge 2 /(ǫv) = 2 (here g = 4 is a spin-valley degeneracy factorand the dielectric constant ǫ depends on the material whichsurrounds the graphene layer). For each k ARPES detects theportion of the spectral function with ω < 0. The k-dependenceis represented in this figure by results for twenty discrete k ∈[0.0, 0.95]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.2 Left panel: −Im[ε −1 (q, ω)] as a function of q/k F and ω/ε Ffor α gr = 2. The solid line is the RPA plasmon dispersionrelation. The dashed lines are the boundaries of the electronholecontinuum. Right panel: −v q Im[χ (0) (q, ω)] as a functionof q/k F and ω/ε F . The left and right panels become identicalin the non-interacting α gr → 0 limit. . . . . . . . . . . . . . . 73xiv

5.3 (Color online) Top left panel: ω ⋆ pl (solid line) and Γ ⋆ (filledsquares) as functions of α gr . Other panels: The absolute value|Im[Σ s (k, ω)]| of the imaginary part of the RPA quasiparticleself-energy (in units of ε F ) of an n-doped system as a functionof energy ω for k = 0, 0.25, and 0.75 and α gr = 2. . . . . . . . 745.4 (Color online) Re[Σ + (k, ω)], Im[Σ + (k, ω)], and spectral functionA + (k, ω) for k = 0.25 and k = 0.75. The band energyand Re[Σ + ] are measured from the band ε F and interaction[Σ + (k F , ω = 0)] contributions to the chemical potential. . . . 756.1 This figure shows hall plateaus at integer and fractional fillingas a function of the magnetic field strength B. . . . . . . . . . 776.2 The direction of the local magnetization is ⃗m = (sin θ cosφ, sin θ sin θ, cosθ) 816.3 Phase diagram of a double layer quantum hall system. Thephase boundary indicates the collapse of hall plateau as a functionof layer separation and tunnelling. . . . . . . . . . . . . . 857.1 (Color online) Filling factor dependence of the integer filling factorHF theory occupied state ( spectrum of the bilayer grapheneoctet at ∆ V = 0). Energies of occupied (red - solid lines) andunoccupied (blue - dashed lines) are in units of (π/2) 1/2 e 2 /εl B .The Zeeman field ∆ Z value in these units is 0.023 at a magneticfield of 20T. Octet space fractional pseudospin polarizationsoffset for clarity: spin(red boxes), valley(green circles) and LLpseudospin(blue triangles). . . . . . . . . . . . . . . . . . . . 927.2 Collective mode ω q of the Landau-level pseudospin polarizedstate in units of interaction strength e 2 /ǫl B = 11.2 √ B[Tesla]meV as a function of ql B at different values of the external potentialdifference ∆ V at a magnetic field of 20 T. The black(solid)line indicates the ql B → ∞ asymptote for ∆ B = 0. . . . . . . . 97A.1 The surface plot shows graphene’s Lindhard function for ˜Ω =Ω/µ and ˜q = q/v . . . . . . . . . . . . . . . . . . . . . . . . . 110xv

Chapter 1IntroductionThe discovery of quantum mechanics in the early twentieth centuryled to a number of unintuitive and fascinating insights into matter at atomiclength scales. It only seems natural to extend the laws of quantum mechanicsto assemblies of many atoms and molecules in matter. However a bruteforce calculation of Schrödinger’s equation for just a few particles becomesintractable even on the world’s most powerful computers. Condensed matterphysics involves study of systems with huge degrees of freedom (of order 10 23 )with various couplings between these degrees of freedom (usually due to interactions).For example in a solid the atoms or molecules are close to each otherand normally crystallize or form other novel exotic states in order to reducethe interaction energy between each other. It seems we are at an impasse;to understand the nature of these systems we must understand the effect ofinteractions with a microscopic number of degrees of freedom.Fortunately, however, we are generally interested in the long distanceor low energy physics of these systems and hence only the degrees of freedomthat are involved in the low energy excitations of such systems. For examplein metals the core electrons are strongly bound to the nuclei and require ahigh energy for excitation, whereas the conduction electrons which are free to1

move are essentially important in low-energy processes such as transport. It isthen prudent to ignore the core electrons. In fact when considering a physicalprocess there are always certain low-energy degrees of freedom that are moreimportant than others. Condensed matter systems can then be described bya ”low-energy effective theory” by writing an effective Lagrangian involvingonly the relevant low-energy degrees of freedom. The effect of the high energyphysics is to ”renormalize” the bare parameters of the low-energy effectivetheory [1].Landau’s Fermi liquid theory is a perfect example of a low-energy effectivetheory. The interacting electrons act as weakly interacting ”quasiparticles”with renormalized mass and a finite lifetime. In the renormalizationgroup language Landau’s Fermi liquid theory is a fixed-point theory parameterizedby marginal couplings m ∗ (effective mass) with some effective interactionbetween the quasiparticles. The effect of the high energy physics is encodedin the effective mass and effective interactions. The other manifestation oflow-energy effective theory is the ”emergence” of a collective coordinate, oftencalled an order parameter. In this case the microscopic degrees of freedomcondense into some symmetry broken collective state where the phase can bedescribed by this collective variable or order parameter. Notable examplesinclude superconductivity and ferromagnetism. Here we can also write an effectivetheory in terms of the order parameter. This leads to the notion of”emergent phenomena” in condensed matter systems[2].Condensed matter systems also exhibit a wealth of strongly correlated2

systems where the notion of ”quasiparticle” and ”order parameter” breaksdown. Common examples of such systems include high T c superconductors,heavy fermions, Luttinger liquids and Quantum Hall Systems. For exampleQuantum Hall Systems have incompressible ground states that support collectiveexcitations with fractional charge and statistics which do not conformto a quasiparticle picture or with a broken symmetry phase. In these systemsthe kinetic energy is quenched by the magnetic field and electron-electron interactionsplay a dominant role in determining the low-energy physics. Theirlow-energy description is then taken into account by model hamiltonians oreffective theories in some ”dual” sector 1 . Strongly correlated systems are generallyrelated to low dimensionality i.e. they exist in one or two dimensions.This is related to the interesting topological properties of low dimensionalspace; it seems there is more room for exotic behavior in lower dimensions.1.1 Outline of the thesisIn this thesis we study the effect of electron-electron interactions ingraphene sheets. We also study the effect of electron-electron interactions onbilayer graphene in the Quantum Hall regime. These systems can be classifiedunder a general class of systems which will be referred to as Chiral twodimensional electron gas (C2DEGs). Below we describe the contents of eachchapter.1 Jain’s composite fermion picture maps Fractional Hall Effect of electrons to Integer HallEffect of composite fermions using the method of flux attachment, thereby mapping stronglyinteracting electrons to non-interacting quasiparticles.3

Ch.(2) introduces the concept of two-dimensional Chiral Fermions andChiral two-dimensional electron gas (C2DEGs), focusing on the single particleproperties of two-dimensional Chiral Fermions. In particular we show thatgraphene and bilayer graphene fall within this family of C2DEGs with chiralityindex J = 1 and J = 2 respectively. Towards the end of this chapter wealso study two-dimensional Chiral Fermions in a magnetic field.Ch.(3) investigates the influence of electron-electron interactions indoped graphene sheets based on the random-phase-approximation. We showthat the tendency of Coulomb interactions in lightly doped graphene to favorstates with larger net chirality leads to suppressed spin and charge susceptibilities.Our conclusions are based on an evaluation of graphene’s exchange andrandom-phase-approximation (RPA) correlation energies. This suppression isa consequence of the quasiparticle chirality switch which enhances quasiparticlevelocities near the Dirac point.Ch.(4) addresses graphene’s Fermi liquid properties quantitatively usinga microscopic random-phase-approximation theory. We find a weak dopingdependence on the renormalized velocity and quasiparticle spectral weight.We also comment on the importance of using exchange-correlation potentialsbased on the properties of a chiral two-dimensional electron gas in densityfunctional-theoryapplications to graphene nanostructures.Ch.(5) reports on a theoretical study of the influence of electronelectroninteractions on ARPES spectra in graphene that is based on therandom-phase-approximation and on graphene’s massless Dirac equation con-4

tinuum model. We find that level repulsion between quasiparticle and plasmaronresonances gives rise to a gap-like feature at small k. ARPES spectraare sensitive to the electron-electron interaction coupling strength α gr andmight enable an experimental determination of this material parameter.Ch.(6) reviews the exotic and novel properties of single layer and bilayerQuantum Hall Ferromagnets. We study the the ground state and theneutral low energy collective excitations of Quantum Hall Ferromagnets. Wealso comment on the topologically charged excitations within these systems.Ch.(7) reports on a study of interaction driven integer quantum Halleffects in bilayer graphene. These systems are of interest due to the neardegeneracyof the eight Landau levels which appear near the neutral systemFermi level. We predict that an intra-Landau-level cyclotron resonance signalwill appear at some odd-integer filling factors, accompanied by collectivemodes which are nearly gapless and have approximate q 3/2 dispersion. Wespeculate on the possibility of unusual localization physics associated withthese modes.Parts of this thesis have been or will be published separately.5

Chapter 2Two dimensional Chiral HamiltoniansIn this chapter we discuss a particular class of single particle hamiltoniansfrom here on referred to as Chiral Hamiltonians restricting ourselvesto two dimensions. We refer to the quasiparticles described by Chiral Hamiltonianas Chiral Fermions. The main theme of this dissertation is to studythe effect of electron-electron interactions in systems where the single particleproperties are determined by Chiral hamiltonians in two dimensions. We referto this collective system as a Chiral two dimensional electron gas (C2DEG).The single particle Chiral Hamiltonian with chirality index J can bewritten asH J (⃗q) ∝ ξ J q J [cos(Jφ ⃗q )σ x + sin(Jφ ⃗q )σ y ], (2.1)where σ α is a Pauli matrix acting on a pseudospin doublet, ⃗q is an envelop functionmomentum measured from some nodal points in the Brillouin-zone(BZ),ξ = ± accounts for the presence of two nodal points K and K ′ in the BZ generallyreferred to as valley degree of freedom, q = |⃗q| and φ ⃗q = tan −1 (q y /q x ).The pseudospin doublets in their respective valleys K(ξ = +) and K ′ (ξ = −)are defined as Φ † ξ=+1 = (φ† ↑ , φ† ↓ ) and Φ† ξ=−1 = (φ† ↓ , φ† ↑ ). In H J J is the chiralityindex of the pseudospin doublet. It can be easily seen that the energy6

dispersion of 2.1 is ǫ J (⃗q) ∝ ±q J/2 with chiral band eigenstates:|±,⃗q〉 = √ 1 ( )12 ±e iJφ , (2.2)⃗qwhere the sign s = ± of the eigenstate is called the chirality. With our inverteddefinition of the pseudospin components of the wavefunction, quasiparticles indifferent valleys have opposite chirality.In semiconducting language these systems are called zero-gap semiconductorswith the positive(negative) energies identified with the conduction(valance)bands. The energy bands exhibit degeneracy points K and K ′in momentum space where the conduction and valence bands meet. For aneutral structure (i.e undoped with holes or electrons) the Fermi energy liesat the degeneracy points. As we see later time reversal symmetry requires thepresence of the two valleys K and K ′ .The quasiparticles described by H J acquire a Berry phase of Jπ uponan adiabatic propagation along a closed orbit. This has unusual consequenceson the single particle properties of chiral systems most notable of which isthe presence of anomalous Half-integer Quantum Hall effect of odd J, whichhas been measured in some of these systems [3–5]. We leave discussion of theproperties of Chiral fermions in a magnetic field towards of this chapter.Apart from the single particle properties the existence of chiral eigenstateshas interesting consequences on the many-body properties of the system[6–8]. The effects of disorder and electron-electron interactions have recentlybeen an area of intense theoretical and experimental study [9,10]. Aswe see through out this thesis the presence of chiral eigenstates has interesting7

consequences on the electronic and thermodynamic properties of C2DEGs.The existence of two valleys is crucial for time reveral symmetry ofChiral hamiltonians. Time reversal symmetry for chiral hamiltonian can bedescribed by (Π ⊗ σ x )HJ ∗(⃗q)(Π ⊗ σ x) = H J (−⃗q) where Π swaps ξ = +1 andξ = −1 in valley space. Chiral hamiltonians also satisfy spatial inversion symmetrygiven by (Π ⊗ σ 0 )H J (⃗q)(Π ⊗ σ 0 ) = H J (−⃗q). Note that vanishing of theenergy at the nodal points is essential for the existence of Chiral Hamiltonians.Such a structure can also appear due to Fermi surface nesting properties:for example d-wave superconductors can be described by J = 1 chiral systems[11].This family of chiral hamiltonians is a good description of the lowenergyelectronic properties of graphene multilayers, where the stacking sequenceof an N-layer graphene system determines the exact decomposition ofthe chiral pseudospin doublets and the chirality index [12]. A special caseof this is monolayer graphene (J = 1 chiral system) [10] and Bernal stackedbilayer graphene (J = 2 chiral system) [13] and remains the focus of this dissertation.Below we show how to derive the low energy properties of singlelayer and bilayer graphene in the process indicating that they belong to a muchwider class of chiral hamiltonians with chirality index J = 1, 2 respectively.2.1 GrapheneGraphene is a two-dimensional array of carbon atoms stacked on ahoneycomb lattice that is isolated from its parent compound Graphite by me-8

chanical exfoliation[9,10] . Graphite is a well known and extensively studiedthree dimensional allotrope of carbon most commonly used in pencils and lubricants.Graphite is composed of stacks of graphene layers weakly coupledtogether due to van der Waals forces. It was originally assumed that graphenewas unstable could not exist in a free state. However the real fact is thateven though it is deposited every time one writes with a pencil it is hard todetect as no experimental tool existed to detect a one-atom-thick flake. It wasnot until the seminal work of researches at University of Manchaster [14] thatgraphene was eventually spotted it due to the subtle optical effect it createson SiO 2 substrate.P. R. Wallace [15] was the first person to show that the band structureof graphene exhibits unusual semimetallic behavior. The low energy theory ofgraphene can be described by massless Dirac Fermions, resembling the physicsof quantum electrodynamics(QED) of massless particles with a material specificspeed of light v F (approximately 300 times smaller than the speed of lightin vacuum) [16–18]. Graphene is a condensed matter realization of a relativisticsystem where many of the unusual properties of QED can show up ingraphene at much smaller speeds. Dirac Fermions also behave in unusual ways,compared to ordinary electrons, when subjected to magnetic fields, leading toan anomalous half-integer quantum hall effect(IQHE) [3,4].A particularly interesting feature of Dirac fermions is their insensitivityto external electrostatic potentials due to the Klein paradox which statesthat Dirac fermions can be transmitted with a nonzero probability through a9

Figure 2.1: This figure shows an atomic layer of graphene identified usingAtomic Force Microscopyclassically forbidden region [19,20]. Due to this fact Dirac fermions behave inan unusual way in the presence of a confining electrostatic potential leadingto the phemomenon of zitterbewegung, or jittery motion of the wavefunction.Such an electrostatic potential can be generated by disorder, and recent effortshave focused on trying to understand how disorder can effect these masslessDirac quasiparticles and transport properties of graphene [10].The role of electron-electron interactions is particularly interesting ingraphene. As mentioned earlier relativistic quasiparticles in graphene have a10

Fermi velocity much smaller than the speed of light, and therefore the electronelectroninteractions are non-relativistic long-range Coulomb interactions. Thestrength of electron-electron interactions in graphene is given by the dimensionlessconstant α gr = e 2 /(ǫv F ) very different from the density dependentparameter r s which determines the strength of interactions in 2DEGs. Thisinterplay of relativistic quasiparticles interacting through long-range Coulombinteraction has interesting consequences most striking of which is the absenceof screening in neutral graphene and a peculiar renormalization of Fermi velocityin graphene [8,21]. We discuss the role of electron-electron interactionsin graphene extensively in the up coming chapters.Apart from the theoretical interest in massless Dirac quasiparticles,graphene might have possible applications in the semiconductor industry. Itexhibits very high mobility (µ > 10 4 cm 2 /V s), an order of magnitude higherthan modern Si transistors [4]. This fact makes graphene a strong candidatefor future device applications. The mobility remains high even at the highestelectric-field induced concentrations ensuring ballistic transport at submicrometerscale even at room temperature. However the fact that graphene remainsmetallic at the neutrality point is a hindrance to future device application.Recent work on graphene nanoribbons indicates that a gap can induced byspatial confinement leading to the possibility of future graphene based transistors[22].11

Figure 2.2: (Adapted from [22]) Energy gap vs ribbon width. The inset showsenergy gap vs relative angle for the device sets. Dashed lines in the insetrepresent the value of energy gap as predicted by empirical scaling of energygap vs ribbon width.2.1.1 Low energy theory of grapheneGraphene is made out of carbon atoms arranged in a hexagonal structureas shown in Figure 2.3. A honeycomb lattice is a textbook example of anon- Bravias lattice usually referred to as a lattice with a basis of two atomsper unit cell [23]. In this section starting from a nearest neighbor tight-bindingdescription of electrons on a planar honeycomb lattice we show how the low-12

G 2KG 1baK'Figure 2.3: Left: Lattice structure of graphene composed from two triangularlattices (⃗a and ⃗ b are the lattice vectors), with sublattices A(Blue) and B(Red).Right: Shows the corresponding Brillouin zone. The Dirac cones are locatedat K and K ′ .energy theory of graphene is a condensed-matter analog of (2+1)-dimensionalQED. The basis vectors of a honeycomb lattice can be written as√⃗a = (1, 0)a, ⃗1 3 b = (−2 , )a, (2.3)2where a/ √ 3 = 1.42A is the carbon-carbon atom distance. With a judiciouschoice of lattice vectors graphene’s hamiltonian in the nearest neighbor tightbindingapproximation can be written as(H TB ( ⃗ 0 −tγ(k) =⃗ k)−tγ ∗ ( ⃗ k) 0), (2.4)13

where t(∼ 2.8eV) is the nearest neighbor hopping energy (hopping betweendifferent sublattices A and B) and γ( ⃗ k) is defined asγ( ⃗ k) = 1 + e i⃗ k·⃗a + e i⃗ k·( ⃗ b+⃗a)(2.5)The energy dispersion derived from this hamiltonian can be written as E( ⃗ k) =±t|γ( ⃗ k)|. The reciprocal lattice vectors in the hexagonal BZ are given by⃗G 1 = 2π a (1, 1 √3 ),⃗ G2 = 4π √3a(0, 1). (2.6)It is easy to see that γ( ⃗ k) and hence the energy E( ⃗ k) vanishes at two pointsK and K ′ at the corners of the Brillouin zone, these points are called Diracpoints for reasons that will become clear towards the end of this section. TheDirac points are given by⃗K = 2π a ( 1 √3,1√ ), K′ ⃗ = − 2π 3 a ( √ 1 1, √ ). (2.7)3 3Expanding γ( ⃗ k) around the Dirac point K with ⃗p = ⃗ K + ⃗q gives:γ( ⃗ k) = ⃗q · ∂γ∂p | ⃗p= ⃗K = √32 ta(q x + iq y ). (2.8)The effective hamiltonian around the K point can therefore be written asH(⃗q) = v F ⃗σ · ⃗q, (2.9)where σ i are Pauli matrices, ⃗q is momentum measured relative to the Diracpoint K and v F = ( √ 3/2)ta ≈ 1×10 6 m/s is graphene’s material specific speedof light. A similar expression can be obtained around the other Dirac point14

Figure 2.4: (Adapted from [10]) Left: Numerically calculated energy spectrumof graphene (in units if t). Right: Magnified linear energy dispersion near theDirac point.K ′1 . The energy dispersion near the Dirac points K and K ′ , ǫ(⃗q) = ±v F |⃗q| islinear. It is now clear that the low energy effective theory for graphene can bedescribed by massless Dirac Fermions with a material specific speed of light.Figure 2.4 shows a numerical calculation of graphene’s spectrum, the lineardispersion around the Dirac points K and K ′ is apparent [10].1 With our definition of pseudospin components graphene’s low-energy effective hamiltonianis just H(⃗q) = ξv F ⃗σ · ⃗q in valley K ′ (ξ = −1).15

When the Fermi energy lies at the Dirac points, this system is referredto as neutral graphene, raising the Fermi energy above(below) the Diracpoint gives hole(electron) doped graphene. The most striking difference due tographene’s unusual linear band structure is that unlike the energy dispersionof a regular 2DEG ǫ(⃗q) = q 2 /2m graphene’s Fermi velocity is not energy ormomentum dependent. This has important consequences on the Fermi liquidproperties as we see in the upcoming chapters.For J = 1 chiral systems the eigenstates around valley K can be easilywritten from 2.2:|±,⃗q〉 K = 1 √2(1±e iφ ⃗q)(2.10)where ± corresponds to the eigenenergies ǫ q = ±v F q which in tight bindinglanguage correspond to the π and π ∗ band respectively. The eigenstatesaround K and K ′ are related by time reversal symmetry leading to oppositechirality in the two valleys. The eigenstates also exhibits a Berry’s phase ofπ (i.e. the wavefunction changes sign if the phase φ is rotated by 2π), with asimilar property characteristic of other chiral systems.For J = 1 chiral systems, chirality can be defined in another waycommonly referred to as helicity. Helicity is defined as the projection of thepseudo(spin) operator in the direction of momentum [24]:ĥ =⃗σ · ⃗q|⃗q| , (2.11)from the definition of the helicity operator it is easy to see that it commuteswith the hamiltonian H(⃗q) = v F ⃗σ · ⃗q, and that |±,⃗q〉 K and |±,⃗q〉 K ′are also16

eigenstates of the helicity operator. Electrons(holes) have positive(negative)chirality or helicity, stating that only states close to the Dirac point have a welldefinedhelicity. It is only good a quantum number as long as hamiltonian 2.9is valid, and therefore holds only as an asymptotic property, well defined closeto the Dirac points.2.1.2 Cyclotron Mass and Density of StatesAn immediate consequence of this massless Dirac-like dispersion is acyclotron mass that depends on the electronic density as its square root. Thecyclotron mass within the semiclassical approximation can be expressed asm ∗ = 1 [∂A(ǫ)], (2.12)2π ∂ǫǫ=ǫ Fwith A(ǫ) the area enclosed by an orbit in momentum space given by:giving usA(ǫ) = πq(ǫ) = π ǫ2, (2.13)vF2m ∗ = k √F π √= n, (2.14)v F v Fwhere the electronic density n is related to the Fermi momentum k F via n =kF 2 /π (here we have already accounted for the valley degeneracy). Fittingthe above expression to the experimental data provides an estimate for theFermi velocity and the hopping parameter (v F∼ 10 6 m/s and t ∼ 2.8eVrespectively). This experimental observation of the √ n dependence providesevidence for the existence of massless Dirac quasiparticles in graphene [3,4].17

Figure 2.5: (Adapted from [10]) Cyclotron mass of quasiparticles in grapheneas a function of their concentration(n), positive and negative n correspondto electron and holes respectively. Experimental data extracted from SdHoscillations.Close to the Dirac point the density of states per unit area of a unitcell for spin polarized Dirac fermions per valley is given by the expressionD(ǫ) =ǫ2πv 2 F(2.15)from 2.15 we can see that the density of states in graphene vanishes close tothe Dirac point. From this density of states we can see that graphene exhibitsa semimetallic behavior.18

Figure 2.6: Lattice structure of bilayer graphene with a honeycomb latticeconstant a = 2.46A and interlayer separation d = 3.35A.2.2 Bilayer GrapheneIn this section we describe another intriguing chiral system closelyrelated to graphene. Soon after the experimental discovery of graphene itwas recognized that a graphite bilayer’s low energy effective hamiltonian fallswithin the family of Chiral hamiltonians discussed in this chapter, with chiralityindex J = 2 [13]. This has important consequences for Quantum HallEffect as we see later in this chapter and towards the end of the dissertation.Starting from the continuum low-energy effective theory of graphene, in thissection we show that bilayer graphene is a J = 2 chiral system.A graphite bilayer has two graphene layers with one stacked on top of19

the other in a particular arrangement as shown in Fig 2.6. commonly referredto as Bernal stacking. We refer to the two sublattice degree of freedom inbilayer graphene as A(Ã) and B( ˜B) in the bottom(top) layers respectively.As we have already seen single layer graphene’s honeycomb lattice supportsa degeneracy point at the two inequivalent corners of a hexagonal BrillouinZone K and K ′ which coincide with the Fermi point in a neutral structure anddetermine the centers of the two valleys of a gapless spectrum. The direct interlayerhopping Ã−B γ Ã−B ≡ γ 1 ≈ 0.4eV forms a dimer state from the Ã−Borbitals thus leading to the formation of high energy bands. There also existsweak intralayer hopping from A to ˜B via the dimer state γ A− ˜B≡ γ 3

Identifying m α v 2 = |γ 1 cos( απ )| the above energy resembles the relativistic3energy spectrum for a particle with momentum ⃗q and mass m. The low-energyspectrum given by:ǫ(⃗q) =q 22mα = 1− q22m α = 2, (2.18)resembles the quadratic dispersion of a massive particle with mass m = γ 1 /2v 2 ≈0.054m e , not too different from the effective mass in GaAs(m ∗ = 0.064m e ).The range of validity of 2.18 can be determined from 2.17 giving the condition|ǫ|

with a parabolic dispersion quite distinct from graphene and normal 2DEGs.This effective hamiltonian also belongs to the class of Chiral hamiltonians withchirality index J = 2.Including the remote interlayer hopping γ 3 and allowing the possibilityof a onsite energy difference between the layers ∆ V give additional contributionsto the effective hamiltonian [13] H eff → H eff + h w + h ∆ where:( ) ( 1 0 πh w + h ∆ = v 3π † + ∆ − 12 2mγ 1π † )π 00 V0 − 1 + 12 2mγ 1ππ †(2.21)The external potential difference ∆ V opens a gap in the spectrum modifyingthe parabolic dispersion to a mexican hat type potential [25] with the gap sizeincreasing with increasing electric potential. The ability to control this gapusing gate controlled external potential makes bilayer graphene more interestingfor technological applications [26,27].The role of electron-electron interactions in bilayer graphene is quiteinteresting. The Fermi liquid properties of bilayer graphene 2 are still notunderstood as no sophisticated treatment of electron-electron interactions inbilayer graphene exists in the literature. There are number of interesting propertiesthat have been highlighted by other authors that are of particular interestdue to the presence of chiral bands: for examples it has been predicted thatbilayer graphene exhibits spontaneous pseuodspin polarization [28] (i.e. chargeis spontaneously transferred to one layer). It has also been predicted that neutralbilayer graphene is unstable towards Wigner crystallization [29] and also2 An interesting question yet unanswered is whether bilayer graphene is a Fermi liquid atthe neutrality point.22

Figure 2.7: (Adapted from [26]) This figure shows the evolution of gap closingand reopening by changing the doping level by potassium adsorption. Experimentaland theoretical bands (solid lines) (A) for an as-prepared graphenebilayer and (B and C) with progressive adsorption of potassium are shown.The number of doping electrons per unit cell, estimated from the relative sizeof the Fermi surface, is indicated at the top of each panelferromagnetism [30]. Towards the end of this thesis we focus on yet anotherinteresting question of electron-electron interactions in bilayer graphene in theQuantum Hall regime.2.3 Chiral Fermions in a magnetic fieldIn the presence of a uniform magnetic field B ⃗ = Bẑ applied in a directionperpendicular to plane of the C2DEG, Hamiltonian 2.2 is modifiedby ⃗p → ⃗π = ⃗p + (e/c) A ⃗ where A ⃗ is the vector potential with B ⃗ = ∇ × A.23

Defining the usual raising and lowering Landau level operator a † and a witha † = (l B / √ 2)π, where l B = (c/eB) 1/2 = 25.6/ √ (B[Tesla])nm is the magneticlength, the hamiltonian 2.2 in ξ = +1 becomes :H J ∝(√ ) J ( 2 0 aJl B (a † ) J 0). (2.22)One immediate consequence is the appearance of zero-energy eigenstateswhich can be identified by a J φ n = 0 for 2D orbitals with Landau levelindex n = 0, ..., J (here φ n are the well known Landau level wavefunctions).This yields a 4J-fold degenerate (including valley and spin) zero state, whichleads to the presence of anomalous Half-integer Quantum hall effect for J oddin these chiral systems. The 4J-fold degeneracy is already evident in quantumhall measurements performed on monolayer graphene (J = 1)[3,4] and bilayergraphene (J = 2) [5] chiral systems, we discuss this in the next section. Thezero-energy eigenstates are localized on the ↑ (↓) pseudospin in the K(K ′ )valley, which for graphene corresponds to localization on sublattice A(B) andtop(bottom) layer in the case of bilayer graphene.The energy of the other Landau levels is given by:(√ ) J 2 √ǫ n,J ∝ ± n(n − 1)...(n − J + 1), (2.23)l Bwith chiral eigenstates in a magnetic field associated with the s = ± energies:( )1 sφn−J+1√ . (2.24)2 φ n24

The energy spectrum of a chiral system in a magnetic field is remarkablydifferent from that of a normal 2DEG, where the Landau levels are equallyspaced with the energy between adjacent levels equal to ω (ω = eB/mc) andthere are no zero energy eigenstates. In a J chiral system the energy scales asB J/2 also very different from the standard 2DEG for J ≠ 2.The existence of the zero-energy 4J-fold degenerate state for sufficientlyclean chiral systems is very interesting from the point of view of electronelectroninteractions [31–33] . Since the kinetic energy is quenched in a Landaulevel electron-electron interactions play a dominant role in determining theground state of the system, leading to broken symmetry states at integer fillingfactors and also to the possibility of novel exotic strongly correlated states atfractional filling factors. Broken symmetry states were predicted to exist inmonolayer graphene [31] have already been seen in exprimental studies onmonolayer graphene. In the last chapter we investigate yet another and moreinteresting set of broken symmetry states that we believe should appear inbilayer graphene samples with high mobility at sufficiently strong magneticfields. In the next section we specifically discuss the presence of anomaloushall effect in monolayer and bilayer graphene.2.3.1 Quantum Hall Effect of Chiral FermionsQuantum Hall Effect(QHE) is one of the most remarkable phenomenonin condensed matter physics [34–37]. Its discovery in 1980s was one of thewatershed moments ranking it among the most important discoveries within25

the last three decades. The basic experimental fact characterizing QHE isthe vanishing diagonal conductivity σ xx → 0 and the quantization of the offdiagonalconductivity σ xy = νe 2 /h where ν is an integer for Integer QHE or afraction for Fractional QHE.IQHE in normal 2DEGs is different from that of chiral systems, thepresence 4J-fold degenerate zero energy state leads to an unusual Half-IntegerQuantum Hall Effect for J odd with the off-diagonal conductivity:σ xy = ±4 e2h (n + J ), n = 0, 1, 2, · · · . (2.25)2To understand this unusual quantization condition let us first review IQHE innormal 2DEGs. The quantization condition for a 2DEG can be understoodfrom the dispersion. The dispersion of a 2DEG is given by ǫ n = (n + 1/2)ω cwith the LL equally spaced by the cyclotron energy ω c . In the presence ofdisorder LLs get broadened with the current carrying states localized whenthe chemical potential lies between the LLs. So when the Fermi energy liesin a gap between LLs, electrons can not move to new states and there is noscattering. Thus the transport is dissipationless and the resistance falls tozero. However there is a delocalized state at the position of each LL with thenumber of current carrying states at each LL given by eB/h. When n LLsare filled below the chemical potential there are neB/h current carrying statesgiving σ xy = ne 2 /h. As the chemical potential crosses a LL there is an extracontribution to the current from the delocalized state at the center of the LLand hence the off-diagonal conductivity jumps by an integer.The unusual QHE unique to Chiral Fermions and can be understood26

Figure 2.8: (Adapted from [3] Hall conductivity σ xy and longitudinal ρ xx ofgraphene as a function of the concentration at B = 14T. The inset showbilayer graphene’s hall conductivity as a function of concentration.from the unusual spectrum of Landau levels and the presence of zero-energyeigenstates. We can identify the spectrum for J Chiral Fermions as electronlike(n ≥ J) and hole-like (n ≤ −J) along with the 4J-fold degenerate zeroenergystate where electrons and holes are degenerate. Due to the presencevalley and spin each LL has a degeneracy 4. With the given spectrum ofLLs for Chiral Fermions the Hall conductance σ xy exhibits QH plateau when(|n| ≥ J) are fully occupied and jumps by an amount 4e 2 /h when the chem-27

ical potential crosses a (|n| ≥ J) LL. The unusual half-integer is due to thepresence of a zero-energy state, the first plateau for the electrons (n ≤ J) andholes (n ≤ −J) are situated at ±2Je 2 /h. As the chemical potential crosses thenext electron(hole) LL the conductivity increases(decreases) by and amountof 4e 2 /h which gives us the quantization condition 2.25.This unusual Hall Effect has already been measured in Hall conductivityexperiments in single layer and bilayer graphene. In fact broken symmetrystates have also been measured in graphene, giving rise plateaus at all intermediateinteger fillings. The appearance of QH plateaus at these intermediateintegers is due to the effect of electron-electron interactions. At present thereis no experimental verification of broken symmetry states in bilayer graphene.Furthermore the cyclotron energy in 2DEG is measured to be about 10 Kat a magnetic field strength of 10 T, where as in graphene at the same fieldstrength cyclotron gap could in principle be as large as 1300 K. This opens thepossibility of observing integer quantum hall effect at room temperature, andrecent observation of room temperature Hall effect in graphene has alreadybeen reported [38].28

Chapter 3Chirality and Correlations in GrapheneThe study of electron-electron interactions is an important and fundamentalpursuit of condensed matter physics [40]. The study of the effect ofinteractions is quite complex as it involves understanding the behavior of amicroscopic number of variables, and hence physicists have to rely on a numberof approximations to study interacting systems.The simplest approach generally used as a starting point towards amany-body problem is the Hartree-Fock (HF) approximation. The basic ideabehind the HF approximation is an attempt to approximate the ground-stateof the interacting system by that of an effective hamiltonian which is quadraticin the electron creation and destruction operator thereby resembling the formof a single-particle hamiltonian that can be easily diagonalized. The most commonversion of this procedure is to approximate the true ground-state by aN-electron single Slater determinant wavefunction in an attempt to figure outthe best independent-electron approximation to the interacting system. TheHF approximation is quite unreliable as it has a tendency to over estimate thepresence of broken symmetry states. This approximation does not include thequantum fluctuations which in some cases tend to restore the full symmetry,therefore HF approximation is only a guide towards broken symmetry states29

Figure 3.1: Random-phase-approximation in terms of Feynman diagrams.and HF results should not be accepted without validation of more accuratestudies. The approximate nature of the HF ground-state leads to the conceptof correlations. Correlation effects are, by definition, effects that stem fromthe fact that the true interacting ground state wavefunction is not a singleSlater determinant.The simplest approximation involving correlations is the RandomPhase Approximation (RPA). This is generally the most popular and significantattempt to go beyond the HF approximation. RPA accounts for quantumfluctuations of the interaction by including virtual particle-hole pairs. Itcan be best understood in terms of Feynman diagrams where a subgroup ofall possible diagrams can be formally be summed upto infinite order. Fig 3shows RPA summation in terms of Feynman diagrams, the double wavy linerepresents the renormalized interaction, single wavy line represents the bareinteraction, and the bubble are virtual particle-hole pairs. The diagrams canbe formally written as a geometric series 1 . The physics captured in this sum-1 Strictly speaking ∑ x n = 11−xrequires |x| < 1, however this approximation yieldsqualitatively correct results when compared to experiments even outside this range.30

mation in the screening of Coulomb potential at large distances. In electrongas RPA is generally a good approximation in the high density limit therebybecoming exact as r s ∼ 1/ √ n(n is the density) becomes small. In this chapterwe employ RPA to study thermodynamic properties of doped graphene sheets.However, before we study doped graphene sheets let us understand the effectof electron-electon interactions in neutral or undoped graphene sheets.Since graphene is truly a two-dimensional system lets us compare it tothe more standard 2DEG which has been studied extensively since the developmentof heterostructures and the discovery of quantum hall effect. At thesimplest level metallic systems have two main kinds of excitations: Particleholepairs and collective modes (such as plasmons). Particle-hole pairs areincoherent excitations of the Fermi sea and a direct result of Pauli’s exclusionprinciple. An electron inside the Fermi sea at momentum ⃗ k can only be excitedoutside the Fermi Sea to a new state with momentum ⃗ k + ⃗q leaving behind ahole. The energy associated to such an excitation is simply: ω = ǫ ⃗k+⃗q −ǫ ⃗k , andfor states close to the Fermi sea scales like ω q ∼ v F q. In 2DEGs the electronhole continuum is made out of intra-band transitions only, and exists evenat zero energy since it is always possible to produce electron-hole pairs witharbitrary low-energy close to the Fermi surface. 2DEGs also contain collectiveexcitations such as plasmons with dispersion: ω plasmon ∝ √ q that exist outsidethe particle-hole continuum at sufficiently long wavelengths.In undoped graphene where Fermi energy lies at the Dirac point, theFermi surface shrinks to a point and hence intra-band excitations disappear31

Figure 3.2: (Adapted from [10])Particle-hole continuum and collective modesof:(a) 2DEG ;(b) neutral graphene; (c) doped grapheneand only inter-band transitions between the lower and upper Dirac cones areallowed. In neutral graphene there are no particle-hole excitations at lowenergyand each particle-hole pair costs energy, the particle-hole continuumoccupies the upper triangle as shown in Fig 3.2. Plasmons are also suppressedand no coherent collective modes can exit. This can be seen from neutralgraphene’s Lindhard function calculated in Appendix A:q 216 √ (3.1)v 2 q 2 − Ω2, 32

Figure 3.3: Dirac cone for n-doped graphene, the yellow arrows represents thechirality of the bands s = +(−) for clockwise and anti-clockwise, and the redarrows represent particle-hole transitions.which is imaginary for Ω > v F q indicating a damping of the particle-hole pairs.The static polarization function (Ω = 0) vanishes linearly with q, also indicatinga lack of screening in neutral graphene.In doped graphene sheets where Fermi energy is moved away from theDirac point, the situation becomes a little different, intra-band excitations arerestored and inter-band excitations have to account for Pauli blocking effectof the filled Fermi sea in the upper cone as shown in the Fig 3.3. The plasmon33

modes are now allowed as can be seen from Fig 3.2, however the situation isstill very different from 2DEGs as intra-band transition along with chiralityplay an important role on the thermodynamic properties of doped graphenesheets as we show in this chapter. Due the presence of intra-band transitionsand Pauli blocking effect of the filled Fermi sea in the upper cone these propertieshave a dependence on the doping: as we shift the Fermi energy inter-bandtransitions begin to dominate more and more and the system starts to resemblea standard 2DEG.In particular we show that quasiparticle chirality in weakly dopedgraphene layers also leads to a peculiar suppression of the charge and spinsusceptibilities. We predict that both quantities are suppressed by approximately15% in current samples and that the suppression will be larger ifuniform samples with much lower densities can be realized. At a qualitativelevel, these effects arise from an interaction energy preference for MDF stateswith larger chiral polarization. Our conclusions are based on an evaluationof the exchange and RPA correlation energies of uniform spin-polarized MDFsystems with Coulomb interactions. We first describe this calculation, payingcareful attention to the MDF model’s ultraviolet cutoff, present its predictionsfor the charge and spin susceptibilities of graphene, and finally discussthe origin of this unusual physics.34

3.1 RPA Theory of GrapheneWe study the following MDF model Hamiltonian,Ĥ = v ∑ ˆψ † k [σ3 ⊗ I ⊗ (⃗σ · ⃗k)] ˆψ k + 1 ∑v qˆn qˆn −q , (3.2)2Skq≠0where σ 3 acts on the two-degenerate (K and K’) valleys, ⃗ k is two-dimensionalvector measured from the K and K’ points, σ 1 and σ 2 are Pauli matrices thatact on graphene’s pseudospin degrees of freedom, I is the 2 ×2 identity matrixthat acts on the physical spin, S is the sample area, ˆn q = ∑ k ˆψ † k−q ˆψ k isthe total density operator, and v q = 2πe 2 /(ǫq) is the 2D Fourier transformof the Coulomb interaction potential e 2 /(ǫr). In Eq. (3.2) the field operatorˆψ k is a eight-component spinor that encompasses valley, spin and pseudospindegrees of freedom 2 . In this chapter chirality (s, s ′= ±) is given by theexpectation value of ⃗σ · ⃗k/|k|, which is the same as the definition of chiralitystated in chapter 1. The model (3.2) requires an ultraviolet cutoff, as we discussbelow. To evaluate the interaction energy we follow a familiar strategy [40]by combining a coupling constant integration expression for the interactionenergy valid for uniform continuum models,E int = N ∫ 1∫ d 2 qdλ2 (2π) v 2 q [S (λ) (q) − 1] , (3.3)0with a fluctuation-dissipation-theorem (FDT) expression [40] for the staticstructure factor,S (λ) (q) = − 1πn∫ +∞0dΩ χ ρρ (λ) (q, iΩ) , (3.4)2 The MDF model can only describe low-energy processes, therefore we neglect intervalleyscatters35

where n is the total electron density. This form of the FDT theorem takes advantageof the smooth behavior of the density-density response function alongthe imaginary axis χ (λ)ρρ (q, iΩ). The RPA approximation for the interactionenergy then follows from the RPA approximation for χ:χ (λ)ρρ (q, iΩ) =χ (0) (q, iΩ)1 − λv q χ (0) (q, iΩ)(3.5)where χ (0) (q, iΩ) is the non-interacting density-density response-function. We [41]have derived the following compact expression for the χ (0) contribution for anindividual MDF model channel, the details of the calculation are presented inAppendix A.q 2χ MDF (q, iΩ) = −16 √ Ω 2 + v 2 q − εc F(3.6)2 2πv 2⎡( ) ( ) √ ⎤( )q 22ε+8π √ Ω 2 + v 2 q Re ⎣sin −1 cF + iΩ 2εc+ F + iΩ 2εc 21 − F+ iΩ⎦ .2 vq vqvqIn Eq. (3.6) ε c F = vkc F where kc F is the channel Fermi momentum. χ(0) inEq. (4.6) is constructed by summing the channel response function (χ MDF ) overvalley and spin with appropriate ε c F values. For a spin- and valley-unpolarizedsystem χ (0) = gχ MDF [with k c F → k F = (4πn/g) 1/2 ] where g = g s g v = 4accounts for spin and valley degeneracy.The energy constructed by combining Eqs. (3.3)-(3.6) is clearly divergentsince χ (0) increases with q at large q and falls only like Ω −1 at large Ω 3 .The divergence is expected since the energy calculated in this way includes3 These asymptotic expressions are calculated in Appendix A36

δεx/εF2.11.751.41.050.7Λ = 10Λ = 10 2Λ = 10 3Λ = 10 4Λ = 10 50.3500 1 2 3 4 5Figure 3.4: (Color online) Cut-off dependence of the regularized exchangeenergy δε x in units of the Fermi energy ε F .fthe interaction energy of the model’s infinite sea of negative energy particles.The MDF model can be expected to describe only changes in energy withdensity and spin-density at small ε c Ftotal energy of undoped graphene (ε c Fvalues. For definiteness we choose the= 0 in all channels) as our zero of energy.For pedagogical and numerical reasons it is also helpful to separate thecontribution that is first order in e 2 , the exchange energy, from the higherorder contributions conventionally referred to in electron gas theory as thecorrelation energy. Using Eqs. (3.3)-(3.6) we find that for unpolarized doped37

graphene the excess exchange energy per excess electron isδε x = − 1 ∫ d 2 ∫q +∞ [2πn (2π) v 2 q dΩ χ (0) (q, iΩ) − χ (0) (q, iΩ) ∣ ]εF(3.7)=00≡ − 1 ∫ d 2 ∫q +∞2πn (2π) v 2 q dΩ δχ (0) (q, iΩ) ,0and that the corresponding correlation energy isδε RPAc = 1 ∫ d 2 ∫ {[]}q +∞dΩ v2πn (2π) 2 q δχ (0) 1 − v q χ (0) (q, iΩ)(q, iΩ) + ln.01 − v q χ (0) (q, iΩ)| εF =0(3.8)With this regularization the Ω integrals are finite and the q integralshave logarithmic ultraviolet divergences. The remaining divergences are physicaland follow from the interaction between electrons near the Fermi energyand electrons very far from the Fermi energy as we discuss at length later.The best we can do in using the MDF model to make predictions relevant tographene sheets is to introduce an ultraviolet cutoff for the wavevector integrals,k c . k c should be assigned a value corresponding to the wavevector rangeover which the MDF model describes graphene. Based on this criterion [42]we estimate that k c ∼ 1/a where a ∼ 0.246 nm is graphene’s lattice constant.The MDF is useful when k c is much larger than k F in all channels.With this regularization the properties of graphene’s MDF model dependon the dimensionless coupling constantf ≡ g e2ǫv = g cα, (3.9)ǫ vand on Λ = k c /k F . In Eq. (4.11), c ∼ 300v is the speed of light, α ≈ 1/137 isthe fine structure constant, and ǫ depends on the dielectric environment of the38

0-0.35δε RPAc /εF-0.7-1.05Λ = 10Λ = 10 2Λ = 10 3Λ = 10 4Λ = 10 5-1.40 1 2 3 4 5Figure 3.5: (Color online) Cut-off dependence of the regularized correlationenergy δε RPAc in units of the Fermi energy ε F .fgraphene layer. In typical circumstances f ∼ 2. Λ is ∼ 10 in the most heavilydoped samples studied experimentally and can in principle be arbitrarily largein lightly doped systems. We expect, however, that many of the electronicproperties of graphene layers will be dominated by disorder when the dopingis extremely light.In Fig. 3.4 and Fig. 3.5 we plot the exchange and correlation energiesof the graphene MDF model as a function of f for a range of Λ values. Notethat both δε x and δε RPAchave the same density dependence as ε F ∝ n 1/2 apartfrom the weak dependence on Λ. The exchange energy is positive because our39

egularization procedure implicitly selects the chemical potential of undopedgraphene as the zero of energy; doping either occupies quasiparticle states withpositive energies or empties quasiparticles with negative energies. Note thatincluding the RPA correlation energy weakens the Λ dependence so that theexchange energy per electron scales more accurately with ε F . It is possible toanalytically extract the asymptotic behavior of the exchange and correlationenergies at large Λ by Laurent expanding the integrands of Eqs. (3.7)-(3.8) inq and retaining only the 1/q terms:andwhereδε x = 16g fε F ln (Λ) + regular terms, (3.10)δε RPAc = − 16g f2 ξ(f)ε F ln (Λ) + regular terms (3.11)∫ +∞dxξ(f) = 40 (1 + x 2 ) 2 (8 √ 1 + x 2 + fπ) . (3.12)[Note that ξ(f = 0) = 1/3 so that the exchange and correlation energies arecomparable in size for typical f values.]3.2 Charge and Spin SusceptibilitiesIn an electron gas, the physical observables most directly related to theenergy are the Ω → 0, q → 0 charge and spin-susceptibilities, normally discussedin terms of dimensionless ratios between non-interacting and interactingsystem values. The charge susceptibility is the inverse of the thermodynamic40

10.9κ/κ00.80.70.60.50 1 2 3 4 5fFigure 3.6: (Color online) Cut off Λ and coupling constant f dependence ofκ/κ 0 . The color coding is as in Figs. 3.4-3.5.compressibility κ of the system up to a factor of n 2 . For the MDF model ofdoped grapheneandκ 0κ = 2nε F∂ 2 (nδε tot )∂n 2 , (3.13)χ 0= 2 ∣∂ 2 [δε tot (ζ)] ∣∣∣ζ=0, (3.14)χ S ε F ∂ζ 2where δε tot includes band, exchange, and correlation contributions. In Eq. (3.14)ζ ≡ (n ↑ − n ↓ )/(n ↑ + n ↓ ), χ −1Smeasuring the stiffness of the system againstchanges in the density of electrons with spin ↑ and spin ↓. In a 2D elec-41

10.9χS/χ00.80.70.60.50 1 2 3 4 5Figure 3.7: (Color online) Cut-off Λ and coupling constant f dependence ofthe spin susceptibility χ S . The color coding is as in Figs. 3.4-3.5.ftron systems the compressibility can be measured [43] capacitively. We notethat this type of measurement is less difficult when ∂µ/∂n is large as it isin weakly-doped graphene. In bulk electronic systems, the spin-susceptibilitycan usually be extracted successfully from total magnetic susceptibility measurements,but these are likely to be challenging in the case of single-layergraphene. In two-dimensional electron systems, however, information aboutthe spin-susceptibility can often [44] be extracted from weak-field magnetotransportexperiments using a tilted magnetic field to distinguish spin and42

orbital response.Our results for the charge and spin-susceptibilities are summarized inFig. 3.6 and Fig. 3.7. For experiments performed over the density range overwhich properties appear to be intrinsic in current samples (Λ between ∼ 10 and∼ 40), these results predict compressibility and susceptibility suppression (apparentquasiparticle velocity enhancement) by approximately 15%. Both thesign of the interaction effect and the similarity of κ and χ are in remarkablecontrast with familiar electron gas behavior. In 3D and 2D non-relativisticelectron gases both are [40,45] strongly enhanced by interactions, with thecharge response diverging at intermediate coupling and the spin response divergingat very strong coupling.3.3 DiscussionThe qualitative physics of ferromagnetism in metals is most transparentat the Hartree-Fock (HF) level. Similarly, the mechanism responsible for theunusual interaction physics of weakly doped graphene becomes clear whenthe exchange energy is expressed in terms of HF theory quasiparticle selfenergies.Correlations do however play an essential quantitative role. Fordoped graphene the contribution of an individual channel to the HF theoryinteraction energy isδε x = − 12nS 2 ∑+ 1nS∑k,ss,s ′∑k,k ′ V s,s ′(k,k ′ ) δn ks δn k ′ s ′Σ (0)k,s δn ks (3.15)43

where s, s ′ = ± are the chirality indices of the MDF bands (i.e. the eigenvaluesof the chirality operator defined above),Σ (0)k,s = − 1 ∑V ss ′(k,k ′ )n (0)s(k ′ ) (3.16)S′ k ′ ,s ′is the HF self-energy of the undoped MDF model,[ ]V s,s ′(k,k ′ ) =2πe2 1 + ss ′ cos(θ k,k ′)|k − k ′ | 2is the exchange matrix elements between band states s ′ ,k ′(3.17)and s,k, andcos(θ k,k ′) is the angle between k and k ′ . For Coulomb interactions, the factorin square brackets on the right hand side of Eq. (3.17) tends to be larger betweenstates in the same band, i.e. states with the same chirality.The first term on the right-hand side of Eq. (3.15) is similar to theexchange energy of an ordinary two-dimensional electron system. Because itis negative and increases with density, its contribution to the exchange energyis lowered when spins are unequally populated. If this was the only exchangeenergy contribution, the spin-susceptibility and inverse compressibility wouldbe enhanced by interactions as usual. The unusual behavior comes from thesecond term. For weakly doped graphene it is sufficient to expand Σ (0)k,sto firstorder in k; Σ (0)k=0,sis a physically irrelevant constant that is included in thechemical potential chosen as the zero of energy by our renormalization procedure.Expanding to first order in k gives the leading interaction contributionto the velocity renormalization [7,8] of undoped graphene. In agreement withprevious work we find that for large Λv → v[1 + f ]4g ln(Λ) . (3.18)44

The physical origin of the velocity increase is the loss in exchange energy oncrossing the Dirac point from states that have the same chirality as the occupiednegative energy sea to states that have the opposite chirality. It is easyto verify that this velocity renormalization is responsible for the leading ln(Λ)terms in the exchange energy and in the exchange contributions to κ −1 andχ −1S . The conventional exchange energy contributes negatively to κ−1 and χ −1Sand competes with the Dirac point velocity renormalization.When correlations are included, the leading ln(Λ) contributions to theinteraction energy and to κ −1 and χ −1S, still follow from the (now altered)undoped system quasiparticle velocity renormalization. The enhanced quasiparticlevelocity is tied to the Dirac point, i.e. to the switch in chirality,and results in an interaction energy that tends to be lower when the chemicalpotential is close to the Dirac point in all channels. Fig. 3.6 and Fig. 3.7illustrate RPA theory predictions for experimentally observable consequencesof the competition between this interband effect and conventional intrabandcorrelations.45

Chapter 4Graphene: A Pseudochiral Fermi LiquidFermi liquid theory has been one of the seminal concepts in condensedmatter physics. It is not only describes the low-energy behavior of interactingelectron in most metal but also the low-energy behavior of Fermi systems suchas 3 He and nuclear matter irrespective of the complex details of these systems.Since the very first studies of the physical properties of metals it has beenapparent that inspite of their mutual attraction, electrons in a metal behaveas noninteracting independent particles. This is not easily justified as in mostmetals the Coulomb interaction energy for typical values of density can beclose to the Fermi energy, which at first glance invalidates the use of standardperturbation theory. This issue was resolved by Landau who provided a theoreticalframework for understanding the low-energy properties of interactingFermi systems. Before we talk about graphene let us review some of the basictenents of Landau’s Fermi liquid theory.Landau’s basic idea is that the low-energy excitations of a system of interactingfermions with repulsive interaction can be constructed from the low-energyexcitations of a non-interacting system by adiabatically switching on the interactionbetween the particles. This establishes a one-to-one correpondencebetween the eigenstates of an ideal system and a set of approximate eigen-46

states of the interacting system. Since the eigenstates of the noninteractingsystem are specified by a set of occupations numbers, say {N ⃗k,σ } of singleparticle eigenstates, the low-energy excitation of the interacting systems canbe studied by the same set of eigenstates. Landau argued that for excitationsclose to the Fermi surface these occupation numbers change very very slowlyeven for strong interactions thereby retaining their identity as approximatequantum numbers. The low-energy properties can be described by an additionor removal of quasiparticles from a filled Fermi sea with momentum k F :for example the ideal state of a particle with momentum k > k F outside theFermi sea evolves into an excited state of the interacting system containingone quasiparticle with the same momentum outside a slightly modified Fermisea.The physical basis of Laudau’s Fermi liquid is in the ineffectiveness ofthe interaction’s influence on the momentum distribution of the particles. Itcan be shown that in the limit of k → k F the quasiparticle lifetime τ ⃗k approachesinfinity. Thus on a time scale short compared to τ ⃗k the occupationquantum number {N ⃗k,σ } can be regarded as a good quantum number.The main properties of a quasiparticle excitation are included in theeffective mass m ∗and the Landau interaction function f ⃗k ′ σ, ⃗ k ′ σ ′ . The effectivemass modifies the bare mass m due to interactions and determines theenergy of the quasiparticles ⃗ε ⃗k = k 2 /2m ∗ . The Landau interaction functionintroduces an effective interaction between the quasiparticles. The beautyLandau’s Fermi liquid theory is that the low-energy behavior of Fermi systems47

can be described in terms of a few Landau parameters which can be simplyrelated to f ⃗k ′ σ, ⃗ k ′ σ ′ . A more rigorous derivation of the concepts discussed aboveis provided by Shankar [1], in the Renormalization Group(RG) sense m ∗ andf ⃗k ′ σ, ⃗ k ′ σ ′are fixed-points describing Landau’s Fermi liquid theory and the excitationsabout this fixed points are the Landau quasiparticles.As mentioned in the last chapter neutral graphene is a semi-metal with a pointfor a Fermi surface. It is prudent of ask if a Fermi liquid description survivesfor neutral graphene. The interaction hamiltonian for graphene involves arelativistic kinetic term and a non-relativistic interaction term:∫∫H = v F d 2 ⃗rψ † ⃗σ · i∇ψ + e2d 2 ⃗rd 2 1⃗r2ǫ′ |⃗r − ⃗r ′ | ρ(⃗r)ρ( ⃗r ′ ).. (4.1)The parameters in the theory v F and e 2 /ǫ remain invariant under the dimensionalscaling ⃗r → λ⃗r, ψ → λ −1 ψ. As the strength of the Coulomb interactiondoes not change relative to the change in the kinetic terms, in the RG sensethe Coulomb interaction is considered marginal. As shown towards the end ofthe last chapter the self energy in the Hartree-Fock approximation for dopedgraphene (similar calculation can be performed for neutral graphene) has alogarithmic renormalization:Σ HF ( ⃗ k) = f 4 k log ( Λk), (4.2)where Λ is a high-energy momentum cutoff. This logrithimic behavior survivesfor higher orders in perturbation theory as obtained in the random phaseapproximation [8] and also in the large N approximation [7]. This impliesthat the Fermi velocity is renormalized to higher and higher values and the48

Coulomb interaction is renormalized towards lower values.This can also be seen from the RG point of view: we can evaluate theeffect of reducing the cut-off Λ on the coupling constant f. From 4.2 it canbe shown that within the Hartree-Fock approximation the coupling constantf obeys the equation:Λ ∂f∂Λ = −f 4(4.3)implying that Coulomb interaction become marginally irrelevant. Due to thisrenormalization of the Fermi velocity to higher values and Coulomb interactionto lower values neutral graphene is commonly referred to as a marginalFermi liquid.When Coulombic electron-electron interactions are included, dopedgraphene represents a new type of many-electron problem, distinct from bothan ordinary 2DES and from quantum electrodynamics. The Hamiltonian inEq. (3.2) differs from a Schrödinger equation in two crucial respects: i) its spectrumis not bounded from below and ii) its eigenstates have definite projectionof pseudospin along the direction of momentum, i.e. definite pseudochirality,rather than definite pseudospin. We refer to the graphene 2DES as the chiral2DES (C2DES). In this chapter we explain why the C2DES and the ordinary2DES have distinctly different Fermi liquid properties. In the absence of afield, ordinary 2DESs are normal Fermi liquids [1,46] in which interactionsalter the Fermi velocity v → v ⋆ , introduce marginally irrelevant effective interactionsbetween quasiparticles on the circular Fermi surface, and diminishthe fraction Z of the spectral weight in the one-particle Green’s function as-49

sociated with its quasiparticle peak. The Fermi liquid phenomenologies of aC2DES and an ordinary 2DES have the same structure, since both systems areisotropic and have a single circular Fermi surface as illustrated in Fig. 2. Thestrength of interaction effects in an ordinary 2DES increases with decreasingcarrier density. At low densities, the quasiparticle weight Z is small, the velocityis suppressed, the charge compressibility changes sign from positive tonegative, and the spin-susceptibility is strongly enhanced. These effects, describedwith reasonable consistency [47–51] by theory and experiment, emergefrom an interplay between exchange interactions and quantum fluctuations ofcharge and spin in the 2DES. In the C2DES we find that interaction effectsalso strengthen with decreasing density, although more slowly, that the quasiparticleweight Z tends to larger values, that the velocity is enhanced ratherthan suppressed, and that the influence of interactions on the compressibilityand the spin-susceptibility changes sign. These qualitative differences are dueto exchange interactions between electrons near the Fermi surface and electronsin the negative energy sea, to quasiparticle chirality, and to interbandcontributions to C2DES charge and spin fluctuations. The interband excitationsare closely analogous to virtual particle-antiparticle excitations of a trulyrelativistic electron gas.4.1 Random Phase Approximation of Self-energyThe technical calculation [52] on which our conclusions are based is anevaluation of the electron self-energy Σ of the C2DES near the quasiparticle-50

pole. Σ describes the interaction of a single electron near the 2DES Fermisurface with all states inside the Fermi sea, and with virtual particle-holeand collective excitations of the entire Fermi sea, as illustrated in Fig. 2. Aswe discuss more explicitly below, a direct expansion of electron self-energyin powers of the Coulomb interaction is never possible in a 2DES because ofthe long-range of the Coulomb interaction. Our results for the C2DES arebased on the random phase approximation (RPA) in which the self-energy isexpanded to first order in the dynamically screened Coulomb interaction W(setting = 1):Σ s (k, iω n ) = − 1 ∑∫ d 2 qβ (2π) 2s ′+∞∑m=−∞[ ]1 + ss ′ cos (θ k,k+q )W(q, iΩ m )G (0)s(k+q, iω2′ n +iΩ m ) ,(4.4)where s = + for electron-doped systems and s = − for hole-doped systems,β = 1/(k B T),χ ρρ (q, iΩ) =W(q, iΩ) = v q + v 2 qχ ρρ (q, iΩ) , (4.5)χ (0) (q, iΩ)1 − v q χ (0) (q, iΩ) ≡ χ(0) (q, iΩ)ε(q, iΩ)(4.6)is the RPA density-density response function, χ (0) is its non-interacting limit [6,41], and ε(q, iΩ) is the RPA dielectric function. For definiteness, we limit ourdiscussion to an electron-doped system with positive chemical potential µ: theFermi liquid properties at negative doping are identical because of the C2DESmodel’s particle-hole symmetry (see also Fig. 2).In Eq. (??) ω n = (2n + 1)π/β is a fermionic Matsubara frequency, thesum runs over all the bosonic Matsubara frequencies Ω m = 2mπ/β while in51

Eqs. (4.5) and (4.6), v q is the bare unscreened Coulomb interaction in 2D,v q = 2πe 2 /(ǫq) where ǫ is an effective dielectric constant. The first and secondterms in Eq. (4.5) are responsible respectively for the exchange interaction withthe occupied Fermi sea (including the negative energy component), and for theinteraction with particle-hole and collective virtual fluctuations. The factor insquare brackets in Eq. (4.4), which depends on the angle θ k,k+q between k andk+q, captures the dependence of Coulomb scattering on the relative chiralityss ′ of the interacting electrons. The Green’s function G (0)s (k, iω) = 1/[iω −ξ s (k)] describes the free propogation of states with wavevector k, Dirac energyξ s (k) = svk −µ (relative to the chemical potential) and chirality s = ±. Aftercontinuation from imaginary to real frequencies, iω → ω+iη, the quasi-particleweight factor Z and the renormalized Fermi velocity can be expressed [52] interms of the wavevector and frequency derivatives of the retarted self-energyΣ ret+ (k, ω) evaluated at the Fermi surface (k = k F) and at the quasiparticlepole ω = ξ + (k):andv ⋆Z =1, (4.7)1 − ∂ ω ReΣ ret+ (k, ω)| k=kF ,ω=0v = 1 + (v)−1 ∂ k ReΣ ret+ (k, ω) ∣ k=kF ,ω=0. (4.8)1 − ∂ ω ReΣ ret+ (k, ω)| k=kF ,ω=0Following some standard manipulations [52] the self-energy can be expressedin a form convenient for numerical evaluation, as the sum of a contributionfrom the interaction of quasiparticles at the Fermi energy, the residue contributionΣ res , and a contribution from interactions with quasiparticles far fromthe Fermi energy and via both exchange and virtual fluctuations, the line52

contribution Σ line . In the zero-temperature limitΣ res+ (k, ω) = ∑ ∫ [ ]d 2 q v q 1 + s ′ cos (θ k,k+q )(2π) 2 ε(q, ω − ξs ′ s ′(k + q)) 2andΣ line+ (k, ω) = −∑ s ′ ∫ d 2 q(2π) 2v q× [Θ(ω − ξ s ′(k + q)) − Θ(−ξ s ′(k + q))] (4.9)[ ] 1 + s ′ ∫cos (θ k,k+q ) +∞2−∞dΩ2π1 ω − ξ s ′(k + q)ε(q, iΩ) [ω − ξ s ′(k + q)] 2 + Ω . 2(4.10)Note that at the Fermi energy ∂ k Σ res+ (k, ω) vanishes, and ∂ ω Σ res+ (k, ω) involvesan integral over interactions on the Fermi surface that are statically screened.These expressions differ from the corresponding 2DES expressions because ofthe relative chirality dependence of the Coulomb matrix elements, becauseof the linear dispersion of the bare quasiparticle energies, and most importantlybecause of the fast short-wavelength density fluctuations produced bythe interband contribution to χ (0) (q, iΩ) illustrated in Fig. 3.4.2 ResultsOur results for Z and v ⋆ /v are summarized in Fig. 4 as a function ofthe C2DES dimensionless coupling constant (restoring )f ≡ ν 2πe2ǫk F= g e2ǫv . (4.11)The appropriate value of f for a particular graphene sheet is dependent on itsdielectric environment; for graphene on SiO 2 f ∼ 2. As we discuss at greaterlength below, graphene’s Fermi liquid properties depend only weakly on the53

carrier density which is expressed in these figures in terms of the cut-off parameterΛ. The trends exhibited in Fig. 4 can be understood by consideringthe limits of small f and the limit of large q at all values of f. In the formerlimit screening is weak except at extremely small q. In ∂ ω Σ res+ (k, ω), for example,the integral over q diverges logarithmically at small q when ε(q, ω = 0)is set equal to one, i.e. when screening is neglected. Screening cuts off thislogarithmic divergence at a wavevector proportional to f so that ∂ ω Σ res+ (k, ω)has a contribution proportional to f ln(f) at small f. Because ε(q, ω = 0)happens to be independent of q for transitions between Fermi surface points,it is possible to evaluate ∂ ω Σ res+ (k, ω) analytically. We find that∣ [ (∂∂ω ReΣres + (k, ω) ∣∣∣k=kF= f √4 2 + √ ) ]4 − f − f2 ln 2− 1,ω=02πgf 2 (4 − fπ) (4.12).Similar small q f ln(f) contributions appear in the other elements which contributeto Z and v ⋆ . All this behavior is very familiar from the case of thenormal 2DES; the new differences present in the chiral C2DEG are ones ofdetail. At large q, on the other hand, interband charge fluctuations dominateε(q, ω) − 1, which approaches its simple undoped system form. It becomesespecially clear when ω is expressed in units of vq that the typical value ofε(q, ω) at large q is ∼ 1 with a non-trivial dependence on f. The q integralsall vary as q −1 , requiring that the C2DES model be accompanied by an ultravioletcut-off which for the case of graphene should be [6] q c ∼ 1/a where ais the graphene lattice constant. Since the crossover between intraband andinterband screening occurs for q ∼ k F , it follows that both ∂ k Σ line and ∂ ω Σ line54

have contributions that are analytic in f and vary as ln(Λ) where Λ = q c /k Fwhen Λ is large. To leading order in ln(Λ) we find thatZ −1 − 1 = fλ(f)6gln (Λ) (4.13)and that 1v ⋆v− 1 =f[1 − fξ(f)]4gln(Λ) (4.14)whereandλ(f) = 48πξ(f) = 4∫ +∞0∫ +∞01dx8 √ x 2 − 1(4.15)1 + x 2 + fπ (1 + x 2 ) 3/21dx8 √ 11 + x 2 + fπ (1 + x 2 ) . (4.16)2Note that λ(f) = f/4 − 3π 2 f 2 /256 + ... and ξ(f) = 1/3 − 3π 2 f/256 + ...are analytic functions of f because interband polarization screening does notessentially alter the Coulomb interaction at large q.4.3 DiscussionsThe asymptotic expressions (4.13) and (4.14) approximately capturethe contribution to the corresponding Fermi liquid parameters from interactionsover the wavevector range from ∼ k F to ∼ q c . As the density decreasesand k F → 0 this contribution dominates. The Fermi wavelength then acts likea cut-off on the renormalization group flows which appear in the theory [53] ofinteraction effects in undoped graphene. The fact that the velocity increases1 Eq 4.14 is obtained after performing a weak-coupling expansion on Eq 4.855

in this regime can be understood qualitatively using Hartree-Fock theory [6],which is accurate at small f when Λ is large. In Hartree-Fock theory theenhanced velocity is due to the reduced exchange energy in a right-handedband when the negative energy sea is left-handed. In Fig. 4 we have alsoshown cut-off and coupling constant dependence of the antisymmetric Landauparameter F0 a , which is defined in terms for the antisymmetric Landauinteraction function f a (cos(ϕ)) as [52]F a l = ν⋆ ∫ 2π0dϕ2π f a(cos(ϕ)) cos(lϕ), (4.17)where ν ⋆ = gk F /(2πv ⋆ ) is the density-of-states of the interacting system atthe Fermi surface [f a (cos(ϕ)) is obtained from the Fermi surface dependenceof the self-energy [54]]. Physically, F a 0 determines the spin susceptibility χ S =(v/v ⋆ )/(1 + F0 a ). From our results in panels (b) and (c) of Fig. 4 we predicta rather large suppression of the spin susceptibility which could be measuredin weak-field Shubnikov-de Haas magnetotransport experiments using a tiltedmagnetic field to distinguish spin and orbital response [48].Our findings have important implications for density-functional-theory(DFT) and tight-binding modeling of ribbons [55,56], quantum dots [57,58],and other nanostructures made from graphene. Because of the pseudo-chiralproperties of bulk quasiparticles, states tend to have a lower energy when theyhave the majority chirality. This interaction effect depends specifically on intersitecoherence and is completely missing in the local-density-approximationand in other approximations for exchange and correlation potentials commonly56

used in DFT. The accuracy of graphene nanostructure electronic structure calculationswould be improved if they used exchange-correlation potentials basedon the properties of the C2DES rather than on the properties of the ordinary2DES.57

Figure 4.1: Honeycomb lattice of a single layer graphite flake with one sublatticein yellow and the other sublattice in blue. In the continuum limit thesublattice degree of freedom may be regarded as a pseudospin. When momentumk is measured away from the Dirac points at the K and K ′ Brillouinzone corners, band eigenstates have definite projection of pseudospin in thek direction, i.e. definite pseudochirality. The angle φ k above denotes themomentum-dependent phase difference between wavefunction amplitudes onthe two sublattices. For spin-1/2 quantum particles this angle is the azimuthalorientation of a pseudospin coherent state in the equatorial plane.58

Figure 4.2: In a weakly doped material, graphene’s energy bands can be describedby a massless Dirac equation in which the role of spin is played bypseudospin. Like an ordinary 2DES, doped graphene has a circular Fermisurface. The Fermi liquid properties of graphene are a consequence of bothexchange interactions between quasiparticles near the Fermi surface and statesin the positive and negative energy Fermi seas and of interactions with bothintra-band (short red vertical arrow) and inter-band (long red vertical arrow)virtual fluctuations of the electronic system. The yellow arrows in this figureindicate the pseudospin chirality of band eigenstates. Because of the differencein chirality between positive and negative energy bands, the velocity ofgraphene quasiparticles is enhanced by inter-band exchange interactions, tendingto protect the system from magnetic and other instabilities, and reducingboth charge and spin response functions.59

Figure 4.3: “Lindhard” function χ (0) (q, iΩ) of a C2DES, in units of the noninteractingdensity-of-states at the Fermi surface ν = gk F /(2πv), as a functionof q/k F and Ω/µ on the imaginary frequency axis. k F = (4πn/g) 1/2 is theFermi wavenumber, µ = vk F the Fermi energy, n the electron density and theflavor multiplicity g = g s g v = 4 for graphene because of its two-fold valleydegeneracy. Because of interband fluctuations χ (0) diverges linearly with q forq → ∞ and decays only like Ω −1 for Ω → ∞ in the C2DES, in contrast tothe q −2 and Ω −2 behaviors of the ordinary 2DES. In the static Ω = 0 limitχ (0) (q, 0) = −ν for all q ≤ 2k F for both chiral and ordinary 2DESs.60

Z10.80.60.4(a)Λ = 10 1Λ = 10 2Λ = 10 3Λ = 10 4Λ = 10 52DES0 1 2 3 4 5fv ⋆ /v1.8(b)1.61.41.210.80 1 2 3 4 5fF a 00-0.125-0.25-0.375(c)-0.50 1 2 3 4 5fFigure 4.4: Density and coupling constant f dependence of some C2DESFermi-liquid parameters. The density is specified by Λ ≡ q c /k F . The densityrange studied most extensively in experiment, n ∼ 10 11 cm −2 to n ∼ 10 13 cm −2 ,corresponds to Λ = 100 to Λ = 10. In all panels the black solid line correspondsto the highest value of the cut-off parameter we have considered, Λ = 2.7×10 5 .The red dashed line illustrates the RPA Fermi-liquid parameters of an ordinarynon-chiral 2DES with parabolic bands. In this case the f = √ 2 r s [seeEq. (4.11)], where r s = (πna 2 B )−1/2 is the usual Wigner-Seitz density parameterand a B = ǫ 2 /(m b e 2 ) the effective Bohr radius. From the left the three panelsshow: (a) the quasiparticle renormalization factor Z evaluated from Eq. (4.7);(b) the velocity renormalization factor evaluated from Eq. (4.8); and (c) thel = 0 dimensionless Landau parameter F0 a which characterizes spin-dependentquasiparticle interactions. The color coding for Λ is the same in all panels.61

Chapter 5Plasmons and The Spectral Function ofGraphene5.1 IntroductionThe single-particle spectral function [40] A(k, ω) captures the influenceof Coulomb and phonon-mediated interactions on the energy band propertiesof crystals. In this chapte we report on a random-phase-approximation (RPA)theory of A(k, ω) in two-dimensional (2D) honeycomb-lattice carbon crystalsdescribed by their Dirac equation continuum model [14]. Graphene sheetshave attracted [9,59,60] attention recently because of unusual properties thatfollow from chiral band states, notably unusual quantum Hall effects [3,4],and because of their potential for technological applications. We find thatstates near the Dirac point (k = 0) of a graphene sheet interact strongly withplasmons with a characteristic frequency ωpl ⋆ that scales with the sheet’s Fermienergy and depends on its interaction coupling constant α gr , producing plasmonicspectral function satellites. The resulting spectral functions, illustratedin Fig. 5.1, have a broad energy spread near the Dirac point and a gap betweenthe extrapolations of right-handed and left-handed bands to k = 0. Weexplain below why the Dirac point is special, even when it is not at the Fermienergy.62

Angle-resolved photoemission spectroscopy (ARPES) is a powerful probeof A(k, ω) in 2D crystals because it achieves momentum k resolution [61].Two recent experiments [62,63] have reported ARPES spectra for singlelayergraphene samples prepared by graphitizing the surface of Silicon Carbide(SiC) [64]. Although the data in Refs. [62,63] are similar, the physical interpretationsof the experimental findings are very different. Ref. [62] discussesthe ARPES spectra in terms of electron-phonon [65] and electron-plasmon interactions,while Ref. [63] focuses mainly on the apparent band-gap opening atthe Dirac point. A gap at the Dirac point can be explained without electronelectroninteractions by assuming strong inversion symmetry breaking in thegraphene layer due to coupling with the SiC substrate. Our theoretical resultsappear to allow an intrinsic interpretation for this feature, although it isclear that present experimental data is still partially obscured by incompletelycontrolled interactions with the substrate and by sample inhomogeneity whichproduces momentum space broadening.The self-energy in a system of fermions can be separated into an exchangecontribution due to interactions with occupied states in the static Fermisea, and a correlation contribution due to the sea’s quantum fluctuations [40].Graphene differs [21] from the widely studied 2D systems in semiconductorquantum wells because its quasiparticles are chiral and because it is gaplessand therefore has interband quantum fluctuations on the Fermi energy scale.In graphene, band eigenstate chirality endows exchange interactions with anew source of momentum dependence which renormalizes the quasiparticle63

velocity and strongly influences the compressibility and the spin susceptibility[6,66,67].5.2 Doped Dirac Sea Charge FluctuationsThe massless Dirac band Hamiltonian of graphene can be written as [9]( = 1) H = vτ (σ 1 p 1 + σ 2 p 2 ), where τ = ±1 for the inequivalent K andK ′ valleys at which π and π ∗ bands touch, p i is an envelope function momentumoperator, and σ i is a Pauli matrix which acts on the sublattice pseudospindegree-of-freedom. The low-energy valence band states have pseudospinaligned with momentum, while the high energy conduction band states, splitby 2v|p|, are anti-aligned. In Fig. 5.2 we compare the particle-hole excitationspectra of non-interacting and interacting 2D doped Dirac systems. The noninteractingparticle-hole continuum is represented here by the imaginary part ofgraphene’s Lindhard function [6,41], Im[χ (0) (q, ω)], which weighs transitionsby the strength of the density fluctuation to which they give rise. Transitionsbetween states with opposite pseudospin orientation therefore have zeroweight. More generally the band-chirality related density-fluctuation weightingfactor (called the chirality factor below), which plays a key role in thephysics of the spectral function, is [1 ± cos(θ k,k+q )]/2 with the plus sign applyingfor intraband transitions and the minus sign applying for interbandtransitions, and θ k,k+q equal to the angle between the initial state (k) andfinal state (k+q) momenta. The weight is therefore high for intraband (interband)transitions when k and k + q are in the same (opposite) direction. The64

most important features in Fig. 4.2 are (i) the 1/ √ vq − ω divergence whichoccurs near the upper limit of the q < k F intraband particle-hole continuumand (ii) the relatively weak weight at the lower limit of the q < k F inter-bandparticle-hole continuum. The divergence at the intraband particle-hole spectrumcontrasts with the singular but finite √ ω max − ω behavior at the upperend of the particle-hole continuum in an ordinary electron gas. The differencefollows from the linear quasiparticle dispersion which places the maximumintraband particle-hole excitation energy at vq for all k in the Dirac-modelcase.In the RPA, quasiparticles interact with Coulomb-coupled particleholeexcitations. Because the bare particle-hole excitations are more sharplybunched in energy, Coulomb coupling leads to plasmon excitations that aresharply defined out to larger wavevectors than in the ordinary electron gasand steal more spectral weight from the particle-hole continuum. As seenin Fig. 5.2, the plasmon excitation ω pl (q) of the Dirac sea remains remarkablywell defined even when it enters the interband particle-hole continuum.The persistence occurs because transitions near the bottom of the interbandparticle-hole continuum have nearly parallel k and k + q and therefore littlecharge-fluctuation weight. Interactions between quasiparticles and plasmonsare stronger in the 2D massless Dirac system than in an ordinary parabolicband2D system.65

5.3 Dirac Quasiparticle DecayIn Fig. 4.3 we plot the imaginary part of the RPA theory [40] selfenergy:Im[Σ s (k, ω)] = ∑ s ′ ∫ d 2 q(2π) 2 v q Im[ε −1 (q, ω − ξ s ′(k + q))[Θ(ω − ξ s ′(k + q)) − Θ(−ξ s ′(k + q))][ 1 + ss ′ cos (θ k,k+q )2](5.1)where s, s ′ = ±1 are band (chiral) indices, v q = 2πe 2 /(ǫq) is the 2D Coulombinteraction, ε(q, ω) = 1 −v q χ (0) (q, ω) is the RPA dielectric function, and Θ(x)is the Heaviside step function. Im[Σ] measures the band-quasiparticle decayrate. The two factors in square brackets on the right-hand-side of Eq. (5.1)express respectively the influence of chirality and Fermi statistics on the decayprocess. Note that Σ s depends on the band-index s only through thechirality factor. For ω > 0 and fixed q, the RPA decay process representsscattering of an electron from momentum k and energy ω to k + q andξ s ′(k + q), with all energies in Eq. (5.1) measured from the Fermi energy.Since the Pauli exclusion principle requires that the final state is unoccupied,it must lie in the conduction band, i.e. s ′= +1. Furthermore sincethe Fermi sea is initially in its ground state, the quasiparticle must lowerits energy, i.e. ξ s ′< ω – electrons decay by going down in energy. Becauseinteraction and band energies in graphene’s Dirac model both scaleinversely with length, Im[Σ s (k, ω)] = vk F F(ω/vk F , k/k F ).For large |x|,66

F(x, y) → −παgrl(α 2 gr )|x|/(64g), where l(0) = 4/3 and l(2) ≃ 0.655124 1 . Thisimplies that for |ω| ≫ vk F , the decay rate in a doped system (Im[Σ s (k, ω)])approaches that of an undoped system. As we will see, however, doped systemproperties are quite different from those of an undoped system up to energiesseveral times larger than the Fermi energy, particularly so near the Dirac(k = 0) point. The Fermi energy ε F = vk F is used as the energy unit andk F as the unit of wavevector in all plots and in the remaining sections of thischapter.In explaining the spectra plotted in Fig. 5.3 we start with the Diracpointcase for which the self-energy is band independent. For k = 0, the finalstate energy ξ s ′(q) = s ′ q − 1 is independent of the direction of q. Becausemost charge fluctuation spectral weight is transferred from the particle-holecontinua to plasmonic excitations of the Dirac sea, Im[Σ s (0, ω)] tends to bedominated by plasmon emission contributions. For ω > 0 the final state mustbe unoccupied so that s ′= +1; q is restricted to those values larger than1 for which the the Dirac sea excitation energy Ω(q) = ω + 1 − q is positive.Comparing with Fig. 4.2 we see that Im[Σ + (0, ω)] vanishes like ω 2 forω → 0, a universal property of normal Fermi liquids [46]. The sharp increasein Im[Σ + (0, ω)] which occurs at ω ∼ 1.2 reflects the onset of plasmon emission.For ω < 0 both conduction and valence band final states occur andtransitions are allowed if the transition energy Ω(q) = |ω| − 1 + s ′ q is positive1 l(α gr ) has an analytical expression that is rather cumbersome and will be presentedelsewhere67

and the final hole state is occupied. Given ω, plasmon emission contributionsoccur when Ω(q) = ω pl (q) and are proportional to the plasmon spectral weightand to the density-of-states factor |s ′ − dω pl /dq| −1 . The density-of-states factoris large for s ′= + and diverges when Ω(q) is tangent to ω pl (q). Theplasmon emission features in Im[Σ + (0, ω)] are more prominent for holes thanfor electrons because this factor cannot diverge in the latter case. We findthat Im[Σ s (0, ω)] = −C Θ(ω + 1 + ω ⋆ pl ) ω⋆3/2 pl/ √ ω + 1 + ω ⋆ pl(with C ∼ 0.8)near the decay peak. If we approximate this peak by a δ-function, settingIm[Σ s (0, ω)] ∼ −πΓ ⋆2 δ(ω + 1 + ωpl ⋆ ) and choosing the electron-plasmon couplingconstant Γ ⋆2 to reproduce the integrated strength of the feature over a ω ⋆ plenergy interval, we obtain a simple model in which a single band state hole withenergy −1 interacts with a plasmon with energy ω ⋆ pl . Because Γ⋆ is comparableto ω ⋆ pl for all values of α gr (see top left panel in Fig. 5.3) a significant part of theDirac point spectral weight is always transferred to a plasmaron [68] satellite√separated from the Dirac point band energy by ωpl ⋆2 + 4Γ ⋆2 . This plasmaronsatellite could be responsible for the broad photoemission spectrum [62,63] atthe Dirac point in epitaxial graphene samples if the sharper features presentin Fig. 5.1 are obscured in current data by disorder-induced momentum spacebroadening. Away from the Dirac point, the conduction and valence bandIm[Σ s (k, ω)] peaks broaden because of the dependence on scattering angle ofξ s ′(k + q), weakening any satellite features and the plasmaron satellite fades.The s = + and s = − peaks in Im[Σ s ] in Fig. 5.2 separate at finite k becauseof chirality factors which emphasize k and q in nearly parallel directions for68

conduction band states and k and q in nearly opposite directions for valenceband states. The conduction band plasmon emission peak moves up in energyapproximately as vk and the valence band peak moves down as seen in thebottom panels of Fig. 5.3.5.4 Spectral functionARPES measures the wavevector dependent quasiparticle spectral function[40]. Near the Fermi energy the spectral function consists of a narrowLorentzian centered at the energy E which solves the Dyson equation for thes = + quasiparticle energy, E = ξ + (k) + Re[Σ + (k, E)]. Near the Dirac point,the s = + band spectrum separates into a quasiparticle peak shifted to lowerenergies as explained above. In Fig. 5.4 we see explicitly that for k = 0.25there are already two solutions to the Dyson equation, although the largestpart of the spectral weight still belongs to the quasiparticle peak. We alsonote in Fig. 5.4 that Re[Σ s ] has a negative contribution which is present atthe Fermi energy and persists over a wide regime of energy. This contributionis due to exchange and correlation interactions of quasiparticles near theFermi energy with the negative energy sea. As explained 2[6,21] previously,2 ReΣ s is weakly dependent on the massless Dirac model’s ultraviolet cutoff Λ [6,21]. Forthis reason the spectral function A s (k, ω) is not exactly a function of only k/k F and ω/ε F .This caveat is of little practical signifigance however since deviations are very small over thetwo to three orders of magnitude of graphene sheet charge density which lies the windowbetween the low density cutoff ∼ 10 11 cm −2 below which disorder plays a strong role andthe high density cutoff ∼ 10 14 cm −2 imposed by fundamental gating and doping limitations.The numerical results shown in this work have been calculated with Λ = 10 2 .69

this effect produces a nearly rigid shift in the band energies which is increasinglynegative further below the Fermi energy, increasing the band dispersionand the quasiparticle velocity. Fig. 5.1 was constructed by combining resultsfor A s (k, ω) at twenty different values of |k| (A = ∑ s A s). The plasmaronsatellite in the s = + band spectral function emerges gradually as |k| → 0.The s = − band spectral function is identical to the s = + band function at|k| = 0, but is substantially broader at larger |k| because of the large phasespace for decay via particle-hole excitation further below the Fermi energy.The plasmaron satellite and the quasiparticle peak in the s = − band tendto merge into one broad peak as |k| increases. The wavevector-dependent exchangeand correlation energy shifts discussed above also influence how thespectral function broadens at the lowest energies. It is abundantly clear thespectral function of a doped system is similar to that of an undoped graphenesystem only for k ≫ k F .Graphene ARPES spectra are influenced by disorder, coupling to thesubstrate, and by electron-phonon interactions, in addition to the electronelectroninteraction effects considered here. Because interactions effects scalewith vk F energy scale, while phonon effects are fixed at optical phonon energyscales, these two contributions can be separated experimentally by varyingcarrier density. Our RPA theory demonstrates that broad quasiparticle peaksand apparent energy gaps near the Dirac point are expected even withoutsubstrate coupling. We expect that the present RPA theory results, combinedwith progress in the preparation of samples suitable for ARPES or for 2D to70

2D tunneling spectroscopy [69], will enable further progress.71

Figure 5.1: Spectral function A(k, ω) of an n-doped graphene sheet as a functionof k (in units of Fermi wavevector k F ) and ω (in units of and measuredfrom the Fermi energy vk F where v is the Fermi velocity). These resultsare for coupling constant α gr = ge 2 /(ǫv) = 2 (here g = 4 is a spin-valleydegeneracy factor and the dielectric constant ǫ depends on the material whichsurrounds the graphene layer). For each k ARPES detects the portion of thespectral function with ω < 0. The k-dependence is represented in this figureby results for twenty discrete k ∈ [0.0, 0.95].72

4433ΩεF2ΩεF210.000.07∞ 0.37 0.170 1 2 3 4qk F100.000.08∞ 0.36 0.190 1 2 3 4qk FFigure 5.2: Left panel: −Im[ε −1 (q, ω)] as a function of q/k F and ω/ε Ffor α gr = 2. The solid line is the RPA plasmon dispersion relation. Thedashed lines are the boundaries of the electron-hole continuum. Right panel:−v q Im[χ (0) (q, ω)] as a function of q/k F and ω/ε F . The left and right panelsbecome identical in the non-interacting α gr → 0 limit.73

0.50.450.40.350.30.250.20.150.10.050ωpl⋆Γ ⋆ 0.100 0.5 1 1.5 2 2.5 3 3.5 4α gr0.50.40.30.2s = ±1Undopedk = 0-5 -4 -3 -2 -1 0 1 2 3 4 5ω0.50.4s = +1s = −10.50.4s = +1s = −10.3k = 0.250.3k = 0.750.20.20.10.10-5 -4 -3 -2 -1 0 1 2 3 4 5ω0-5 -4 -3 -2 -1 0 1 2 3 4 5ωFigure 5.3: (Color online) Top left panel: ω ⋆ pl (solid line) and Γ⋆ (filled squares)as functions of α gr . Other panels: The absolute value |Im[Σ s (k, ω)]| of theimaginary part of the RPA quasiparticle self-energy (in units of ε F ) of an n-doped system as a function of energy ω for k = 0, 0.25, and 0.75 and α gr = 2.74

21.510.5ImΣ +ReΣ +A +ω−k+1k=0.2521.51ImΣ +ReΣ +A +ω−k+1k=0.7500.5-0.50-1-3 -2.5 -2 -1.5 -1 -0.5 0ω-0.5-2 -1.5 -1 -0.5 0 0.5ωFigure 5.4: (Color online) Re[Σ + (k, ω)], Im[Σ + (k, ω)], and spectral functionA + (k, ω) for k = 0.25 and k = 0.75. The band energy and Re[Σ + ] are measuredfrom the band ε F and interaction [Σ + (k F , ω = 0)] contributions to thechemical potential.75

Chapter 6Quantum Hall Ferromagnets2DESs have been a long been a fertile source of surprising new physicsfor more than four decades, most notably in the presence of a strong magneticfield when they exhibit the integer and fractional quantum Hall effects. As wehave seen in chapter 2 due to the unusual properties of chiral fermions in amagnetic field, systems like graphene and bilayer graphene have opened a newarena in quantum hall physics. A particularly interesting aspect involves thepossibility of novel and exotic strongly correlated states that can appear dueto the 4J-fold degeneracy of the lowest Landau level. Even though researchin graphene is still in its infancy, an optimistic extrapolation of the trendtowards improved mobility [10] in current samples indicates that experimentalinvestigation of strongly correlated quantum hall physics in chiral systems isnot too far in the future. This might lead to interesting new surprises thatadd to the richness of quantum hall physics. In the next chapter we perform atheoretical study of Quantum Hall Ferromagnetism in bilayer graphene’s chiralsystem at a strong magnetic field. Before we do this let us we review salientfeatures of the rich phenomenology associated with quantum hall physics in2DEGs with a special focus on Quantum Hall Ferromagnetism.76

Figure 6.1: This figure shows hall plateaus at integer and fractional filling asa function of the magnetic field strength B.6.1 Review of Quantum Hall EffectInteger and Fractional quantum hall effect was discovered almost unexpectedlyin the early 1980s [34–37]. Since then the physics community haswitnessed tremendous progress in this area [70]. It is now understood that a2DEG in the presence of a magnetic field is incompressible at certain Landaulevel filling factors [70,71]:1κ ∼ dµ → ∞; (6.1)dn77

the discontinuity of the chemical potential as a function of the density indicatesthat it costs a finite amount of energy introduce a single charge in thesystem. Whenever there is an incompressibility the energy to add or removea particle differ in the thermodynamic limit, it follows that it costs finite energyto create a particle-hole pair that are not bound to each other and cancarry a current, this is referred to as ’charge gap’. This incompressibility orequivalently charge gap is responsible for the plateaus in the off-diagonal Hallresistance ρ xy and the vanishing longitudinal resistance ρ xx at low temperatures.It can also be shown that it is this incompressibility that leads to thequantization of Hall conductance.The origin of this incompressibility is different for the integer and fractionalquantum hall effects. In the integer Hall effect the charge gap arises dueto the quantization of the energy spectrum into Landau levels or due to theZeeman gap between spin states, this is essentially a single particle effect andfairly easy to understand. In contrast Fractional hall effect requires partialfilling of a Landau level, and the charge gap arises due to electron-electroninteractions. The Coulomb interaction energy scale is approximately e 2 /ǫl B(which scales like √ B) much lower than the Landau level gap (which scaleslike B ) in a high magnetic field. It is then useful to think of electrons confinedto the lowest Landau level where the kinetic energy is a constant and can beabsorbed in the zero of energy. Interactions then dominate the physics leadingto many body states that are highly correlated incompressible quantum fluids.Soon after the experimental discovery of Fractional Hall Effect [36]78

Laughlin wrote a variational wavefunction to describe the incompressible quantumhall fluid at filling factor ν = 1/m (where m is odd) [37]:Ψ m (z 1 , ..., z N ) = ∏ i

at a number of excellent reviews on this subject [70,75,76].6.2 Quantum Hall FerromagnetsIn the previous section we have assumed that the electrons in the lowestLandau levels are spin polarized due to the presence of an external magneticfield. This can be justified, as discussed earlier, by arguing that the interactionenergy scales like √ B where as Zeeman energy is proportional to B. Howeverdue to the low effective mass and spin-orbit coupling the Zeeman gap is only1/60 of the Landau level gap in GaAs comparable to the interaction energy atexperimentally attainable magnetic fields. It is therefore important to considerthe spin of the electron. Another interesting class of systems are double layerquantum hall systems where two layers of 2DEGs are brought in close proximityto each other, such that they are coupled by interlayer Coulomb interactionand interlayer tunnelling along with intralayer Coulomb interaction. In suchsystems the which layer degree of freedom introduces a new (pseudo)spin alongwith the physical spin already present in these systems.Any two level system can be viewed as a ’(pseudo)spin’ system, i.e. thehamiltonian and all other operators can be expressed in terms of Pauli matrices.This mapping allows us to relate to the more familiar problem of spinsystems. Although the physical origin for the two systems could in principlebe quite different, the mathematical description and physical consequences areidentical to real spin systems. We can define the pseudospin on a Bloch sphere80

Figure 6.2: The direction of the local magnetization is ⃗m =(sin θ cosφ, sin θ sin φ, cosθ)parameterizing the direction of spin by Euler angles θ and φ,( cos(θ) )2sin( θ . (6.3)2 )eiφIn the layer language θ ≠ 0 would then correspond to the electron being alinear combination of up and down layer i.e. the electron is in both layers.In single layer quantum hall system the kinetic energy is quenched bythe external magnetic field and there is good exchange energy between thespins, i.e. the spins in the systems choose to spontaneously polarize even in81

the limit of vanishing Zeeman energy. This happens for integer and fractionalfilling even though the orbital wavefunction is different at all filling factors.In the presence of Coulomb interaction Hund’s rule suggests that the systemlower its interaction energy by maximizing its total spin, since states withmaximum total spin are symmetric and the orbital wavefunction 6.2 is antisymmetricwith respect to particle interchange, this gives a total antisymmetricwavefunction. This represents 100% spin polarization and is a perfect itinerantferromagnet as there is no competition with kinetic energy.In the limit of zero Zeeman energy single layer quantum hall ferromagnets haveSU(2) invariance in the spin degree of freedom which is spontaneously brokenwhen the system exhibits maximum spin polarization. This simply resemblesa Heisenberg ferromagnet with low energy spin wave excitations ω q ∼ q 2 in thelong wavelength limit. The low energy effective hamiltonian has the familiarNLσM form [84]:∫12 ρ sd 2 ⃗r(∇m µ ) · (∇m µ ), (6.4)here the order parameter ⃗m represents the direction of the local magnetization,and ρ s is the spin stiffness to spatial variations of ⃗m. The spin stiffnessdepends on the nature of Coulomb interactions and the underlying orbitalground state wavefunction. There is a charge gap in this system as the additionor removal of an electron would cause a loss in the good exchange energy.The charge gap here is purely associated to Coulomb interactions. Similar toHeisenberg ferromagnets single layer quantum hall ferromagnets also exhibitspin textures called ’skyrmions’. In the case of quantum hall ferromagnets82

these topological excitation carry charge and are the lowest charged excitationsof this system [77,84].This is very different from the quantum hall effect in double layer systems1 , for the rest of the discussion we neglect real spin in these systems. Unlikethe single layer system the Coulomb interaction in a double layer system islayer(pseudospin) dependent. In the absence of tunneling between the layersthe double layer system can be viewed as an easy-plane quantum itinerant ferromagnet[84] with pseudospin polarized in the XY plane with a spontaneouslybroken U(1) symmetry 2 . This correspond to an electron being in a coherentsuperposition of both layers. Neglecting layer thickness the Coulomb potentialis pseudospin dependent: electron in the same layer interact via intralayerCoulomb potential v S (q) = 2πe 2 /q where as electrons different layer interactvia the interlayer Coulomb potential v D (q) = v S (q)e −qd where d is the distancebetween the layers. The hamiltonian can be separated into a pseudospin independentand pseudospin dependent part. It is the pseudospin dependent partthat reduces the symmetry of the hamiltonian from SU(2) to U(1).The hamiltonian for a double layer system can be written as H =H + + H −H = 1 2∫d 2 ∫q(2π) 2v +(⃗q)ˆρ(−⃗q)ˆρ(⃗q) + 2d 2 q(2π) 2v −(⃗q)Ŝz (−⃗q)Ŝz (⃗q) (6.5)1 Double layer system exhibit a wide variety of nontrivial collective states at differentfilling factors. Here we only focus on total filling factor ν = 1 (i.e. 1/2 in each layer).2 Tunneling introduces peusdospin anisotropy in the XY plane, forcing the pseudospin topoint along the x-axis.83

where ˆρ(⃗q) is the fourier transform of the electronic spin density summed overlayers, Ŝz α = 1 2 (ˆρt α−ˆρ b α) is the z-component pseudospin density operator, and v ±are the symmetric and antisymmetric combinations of interaction potentialsfor electrons in the same(different) layer. Since v S > v D , v − is positive andproduces an easy plane, as opposed to Ising, pseudospin anisotropy. Thisterm prefers that the pseudospin lie in the XY plane: as when the pseudospinorientation moves out of the XY plane (〈S z 〉 ≠ 0) and the energy increases.The physical origin of this energy cost is the charging energy of the capacitordue to unbalanced densities in the two layers, since S z measures the chargedifference between the two layers, the pseudospin lies in the XY plane.This term also increases the effect of quantum fluctuations since itdoes not commute with the order parameter [78]:[H − , S µ ] ≠ 0, (6.6)where µ = x, y. The total spin is no longer a sharp quantum number. Theeffect of quantum fluctuations become important for large layer separationand produce a phase transition to two uncoupled layers. When the layers arewidely separated there will be no correlations between the two layers and noappearance of a quantum hall plateau since each layer has ν = 1/2 [79]. Theresulting phase diagram shown in Fig 6.3 has been verified experimentally.To address the issue of collective modes of a double layer quantumhall system it is convenient to assume the pseudospin is polarized in the ˆxdirection.The energy change associated with small oscillation of the spins84

Figure 6.3: Phase diagram of a double layer quantum hall system. The phaseboundary indicates the collapse of hall plateau as a function of layer separationand tunnelling.from the ˆx-direction, in the long wavelength limit can be written as [84]:∫E[⃗m] ≈ d 2 qβ(m z ) 2 + ρ A2 q2 |m z | 2 + ρ E2 q2 (|m x | 2 + |m y | 2 ). (6.7)We can see from the above expression that rotations out of the xy plane arecostly due to the local mass term associated with m z . This is exactly due tothe capacitive energy as interaction favor equal population of densities in bothlayers. The collective mode dispersion is gapless and is the Goldstone modeassociated with the broken U(1) symmetry. The collective mode dispersion in85

the long wavelength limits is given by ω q ≈ |q|, linear rather than quadraticdue to the presence of the massive field m z .The linear gapless dispersion discussed above and the absence of gaplesscharged excitations suggests that double layer quantum hall systems exhibitsuperfluid behavior. This has interesting consequences on transportexperiments in these systems [80]. These systems also exhibit four flavorsof topologically charged excitations called ’merons’ which are essentially halfskyrmions [84]. The unbinding of these topological defects also leads to theKosterlitz-Thouless phase transition even at zero temperature, due to thequantum fluctuations, if the layer separation exceeds some critical value [81].In the next chapter we study Quantum Hall Ferromagnetism in bilayergraphene which shares some characteristics with double layer quantum hallsystems. This system is interesting due to the presence of a 8-fold degeneracyin neutral bilayer graphene in a magnetic field. This adds an additional orbitalpseudospin degree of freedom and new physics to the already rich structurediscussed in this chapter.86

Chapter 7Octet Quantum Hall Ferromagnets in BilayerGrapheneBecause the Zeeman spin-splitting in most two-dimensional electronsystems (2DES’s) is much smaller than the Landau level separation, the magneticband spectrum usually consists of narrowly-spaced doublets.Whenone of these doublets is half-filled and disorder is weak, Coulomb interactionphysics leads to ferromagnetism i.e. to spontaneous spin polarization in theabsence of a Zeeman field [82–84]. In some circumstances [85] other approximateLandau level degeneracies occur, often associated with layer degrees offreedom. These can also lead to broken symmetries which induce quasiparticlegaps and hence interaction driven integer quantum Hall effects. The caseof bilayer 2DES’s is particularly interesting because the which layer degreeof freedom doubles Landau level degeneracies and leads to exciton condensation[86,87] at odd filling factors and to canted anti-ferromagnetic states[88] at even filling factors. In this Chapter, we address the still richer case ofgraphene bilayer 2DES’s in which chiral bands lead to an additional degeneracydoubling [13] at the Fermi energy of a neutral system. Bilayer graphene’sLandau level octet is already apparent in present experiments [5] through the8 × (e 2 /h) Hall conductivity jump between well formed plateaus at Landau87

level filling factors ν = −4 and ν = +4. We anticipate that when externalmagnetic fields are strong enough or disorder is weak enough [89], interactionswill drive quantum Hall effects at the octet’s seven intermediate integer fillingfactors. We predict that these quantum Hall ferromagnets (QHFs) willexhibit unusual intra-Landau-level cyclotron modes at odd filling factors, andthat the collective mode excitations at these filling factors are nearly gaplesseven when there is no continuous symmetry breaking. Because the conductivityhas Drude weight centered near zero-energy, we speculate that localizationphysics and quantum-Hall related transport phenomena will also be anomalous.7.1 Graphene Bilayer Landau LevelsWhen trigonal warping [90] and Zeeman coupling are neglected, thelow energy properties of Bernal stacked unbalanced bilayer graphene are determinedby electron-electron interactions and by a band Hamiltonian [13]H = H 0 + H ext whereH 0 = 1 ( 0 π†22m π 2 0), (7.1)and the influence of an external potential difference ∆ V between the layers iscaptured byH ext = ξ∆ V[ 12( 1 00 −1)− v2γ 2 1( π † π 00 −ππ † )]. (7.2)In Eqs. (7.1)-(7.2), ⃗π = ⃗p+(e/c) ⃗ A is the 2D kinetic momentum, π = π x +iπ y ,the 2 × 2 matrices act on the pseudospin degree of freedom associated with88

the two low energy sites [13] (the top and bottom layer sites without a nearneighborin the opposite layer), v is the single-layer Dirac velocity, γ 1 ∼ 0.4eVis the inter-layer hopping amplitude, and the effective mass m = γ 1 /2v 2 ≈0.054m e . H describes both K (ξ = 1) and K’ (ξ = −1) valleys provided thatwe choose the pseudospin representation (A, ˜B) for K and ( ˜B, A) for K’.Defining the usual raising and lowering Landau level ladder operatorsa † , a with a † = (l B / √ 2)π, where l B = (c/eB) 1/2 = 25.6/ √ (B[Tesla])nm isthe magnetic length, zero-energy eigenstates of H 0 can be identified using theproperty that a 2 φ n = 0 for 2D orbitals with Landau level index n = 0, 1. Inbilayer graphene the n = 0 and n = 1 orbital Landau levels are members ofthe same octet. This peculiarity is behind most of the physics explored in thischapter. Neutral bilayer graphene’s Landau-level octet is the direct productof three S = 1/2 doublets: real spin and which-layer [91] pseudospins (as in anormal bilayer), and the Landau-level pseudospin n = 0, 1 degree of freedomwhich is responsible for new physics. Zeeman coupling produces real spinsplitting∆ Z while ∆ V gives rise to layer-splitting as in normal bilayers, but alsoto a small splitting of the Landau-level pseudospin which plays a central role inthe physics: ∆ LL = ∆ V ω/γ 1 ≡ ω LL where ω = 2 2 v 2 /lB 2 γ 1 = 2.14 B[Tesla]meV.The interaction contribution to the graphene bilayer Hamiltonian islayer-dependent:H int = ∑ αβ∫12d 2 ∫q(2π) 2v +(⃗q)ˆρ α (−⃗q)ˆρ β (⃗q) + 2d 2 q(2π) 2v−(⃗q)Ŝz α (−⃗q)Ŝz β (⃗q) (7.3)89

In Eq. (7.3), ˆρ α (⃗q) is the α-component of the electronic spin density summedover layers, Ŝz α = 1 2 (ˆρt α − ˆρ b α) is the z-component of the corresponding pseudospindensity operator,and v ± are the symmetric and antisymmetric combinationsof interaction potentials for electrons in the same(different) layerv s = 2πe 2 /εq(v d = v s e −qd ). In graphene bilayers, the layer separationd = 0.334nm so that, in contrast to what is typical in the semiconductorbilayer case, d/l B ≪ 1 and hence v − is weak.Because of the incompressible nature of quantum Hall states, we expectthat the graphene bilayer octet is well described at integer filling factors byHartree-Fock (HF) mean-field theory. The importance of quantum fluctuationcorrections to the ground state can be assessed using a weak-coupling theoryof the octet’s elementary excitations.7.2 Octet Hunds RulesThe octet HF Hamiltonian [92] contains single-particle pseudospin splittingfields and direct and exchange interaction contributions:〈nτα|H HF |n ′ σβ〉 = E H (ρ τ − ρ β ) − ∑ ( )X+n 2 n ′ nn 1+ ξ τ ξ σ X − n 2 n ′ nn 1 ρn 1 n 2τσαβ(7.4)n 1 n 2+ (ξ τ ∆ LL δ n,1 δ n ′ ,1 − ∆ Z2 ξ αδ nn ′ − ∆ V2 ξ τδ nn ′)δ αβ δ τσ ,where n = 0, 1 are LL indices, τ, σ = t(b) are valley indices, α, β =↑ (↓) arespin-indices, and ξ τ(α) = 1(−1) for t(b) layer and ↑ (↓) spins respectively. InEq. (7.4) ρ τ = ∑ nα ρnn τταα is the total electron density in layer τ. The densitymatrixρ n 1n 2τσαβ = 〈c† n 2 σβ c n 1 τα〉 must be determined self consistently by occupying90

the lowest energy eigenvectors of H HF . The Hartree-field E H captures theelectrostatic contribution to the bilayer capacitance, E H = (e 2 /εl B )(d/2l B ),and the exchange fields capture fermion quantum-statistics:∫X ξ n 2 n ′ nn 1=d 2 p(2π) 2 v ξ(p)F n2 n ′(p)F nn 1(−p). (7.5)In Eq.( 7.5) v ± are the symmetric and antisymmetric combinations of same(s) and different (d) layer electron-electron interactions (v s = 2πe 2 /εq v d =v s e −qd ), and the form factors (F 00 (q) = e −(ql B) 2 /4 , F 10 (q) = (iq x +q y )l B) e −(ql B) 2 /4 / √ 2 =[F 01 (−q)] ∗ and F 11 (q) = (1 − (ql B ) 2 /2)e (−ql B) 2 /4 ) reflect the character of thetwo different quantum cyclotron orbits.The solution of the Hartree-Fock equations for balanced bilayers (∆ V =0) is summarized in Fig.[ 7.1] using a Zeeman field strength corresponding toB = 20T. The large gaps (∼ (π/8) 1/2 in e 2 /εl B units) separating occupiedand empty states at the odd integer filling factors of primary interest justifyour weak-coupling theory. The octet filling, proceeding in integer incrementsstarting from filling factor ν = −4, follows a Hunds rule behavior: first maximizespin-polarization, then maximize layer-polarization to the greatest extentpossible, then maximize Landau-level polarization to the extent allowed by thefirst two rules. For balanced bilayers the layer symmetric states (S) are filledbefore the layer antisymmetric states (AS). The first four states to be filled are(S,n = 0, ↑),(S,n = 1, ↑),(AS,n = 0, ↑) and (AS,n = 1, ↑) in this order. Thissequence is then repeated for the next four states with down (↓) spin. TheHunds rules imply that the Landau-level pseudospin is polarized at all odd91

ΕHF0.50.0 4 4 4 4 44 2 22 20.51.01.51111111122 112211 22 1111224 2 0 2 4Ν2222Polarization100.2Figure 7.1: (Color online) Filling factor dependence of the integer filling factorHF theory occupied state ( spectrum of the bilayer graphene octet at ∆ V = 0).Energies of occupied (red - solid lines) and unoccupied (blue - dashed lines) arein units of (π/2) 1/2 e 2 /εl B . The Zeeman field ∆ Z value in these units is 0.023at a magnetic field of 20T. Octet space fractional pseudospin polarizationsoffset for clarity: spin(red boxes), valley(green circles) and LL pseudospin(bluetriangles).integer filling factors between ν = −4 and ν = 4. The physics of this new typeof pseudospin polarization is the main focus of this chapter. An importantdistinction between layer and Landau-level polarization is that the former isassociated with spontaneous inter-layer phase coherence whenever a Landaulevel occupies both layers simultaneously, whereas the latter polarization isdriven by the Landau-level dependence of the microscopic Hamiltonian.92

Octet quantum Hall ferromagnets have an interesting and intricate dependenceon the external potential ∆ V . Because the two-layers are close together,a small value of ∆ V is sufficient to change the character of the layerpolarization from the XY spontaneous-coherence form, to an Ising polarizationform in which one layer is occupied before the other. We find that for∆ V larger than a critical value ∆ ∗ V , the layer filling proceeds by filling thetop layer first. (For ν = −3, ∆ ∗ V= 0.1023(0.40) meV at B = 20(50) Tesla.)As we explain later, this filling sequence has qualitative consequences for theodd-integer filling factor LL pseudospin polarized states.7.3 Landau-Level Pseudospin DipolesWe now focus on the LL pseudospin fluctuations of a state with oddintegerfilling factor, freezing spin and layer degrees of freedom. The collectiveexcitation spectrum of graphene bilayer octets as a function of ν and ∆ V willbedescribed in full detail elsewhere [93]. Fluctuating LL spinors are linear combinationsof n = 0 orbitals (even with respect to their cyclotron orbit center)and n = 1 orbitals (odd with respect to orbit center), and therefore carryan electric dipole proportional to the in-plane component of their pseudospin.Because dipole-dipole interactions are long-range, they play a dominant rolein the QHF long-wavelength effective action[84]. We find that∫S[⃗m] = dt [∫ d 2 q A ⃗ · ∂ t ⃗m − E[⃗m] ] , (7.6)93

where the first term is the Berry-phase contribution[84,94] and for small fluctuationsaway from m z = 1 (full n = 0 polarization)E[⃗m] = e2εl B∫d 2 q [ 12|q| (⃗q · ⃗m)2 + ˜∆ LL2 (m2 x + m2 y )] . (7.7)where ˜∆ LL = ∆ LL /(e 2 /ǫl B ). The mass terms in Eq.( 7.7) are due to thesingle-particle splitting between n = 0 and n = 1 levels and the interactionterm is due to electric-dipole interactions. The absence of interactioncontributions to the mass terms is a surprise, since the interaction is Landaulevelpseudospin dependent. We address this point below. Because of the inplaneelectric dipoles associated with LL pseudospinors, the long-wavelengthpseudo-spinwave collective mode dispersion is not analytic: ω → (∆ 2 LL +∆ LL e 2 q/ǫ) 1/2 , and for ∆ LL → 0 is proportional to q 3/2 when exchange interactionsare included in the energy functional. The in-plane dipoles are alsoresponsible for the intra-Landau-level cyclotron resonance discussed below.To explain the absence of interaction contributions to the mass termsand address shorter-wavelength fluctuations it is necessary to derive the actionmicroscopically. It is convenient to temporarily restrict fluctuations toone space direction by considering Landau-gauge states in which the LL pseudospinsat different guiding centers X fluctuate independently:|ψ[z]〉 = ∏ X(z 0X c † 0X + z 1Xc † 1X)|0〉, (7.8)where the spinor components z nX satisfy the normalization constraint |z 0X | 2 +94

|z 1X | 2 = 1. The corresponding imaginary-time action is∫ β0S[¯z, z] = S B + E = dτ ∑ ¯z nX ∂ τ z nX + ∑ ( 1 ∑ [H(X − X ′ ) (7.9)2XnXX ′ n i− F(X − X ′ ) ]¯z n1 Xz n3 X¯z n2 X ′z n 4 X ′ + ξ∆ LL¯z 1X z 1X ′),where S B is the Berry’s phase term and E = 〈ψ[z]|(H+H int )|ψ[z]〉 is the energyfunctional. In Eq. (7.9) the direct(H) and exchange(F) energy contributionsdepend on the LL pseudospin labels,H n 1,n 2n 3 ,n 4(X) =F n 1,n 2n 3 ,n 4(X) =1 ∫ dqL y 2π v qF n1 n 4(q)F n2 n 3(−q)e −iqxX ,1 ∑vL 2 q δ qy,XF n1 n 3(q)F n2 n 4(−q). (7.10)qThis action can be identified as the Schwinger boson[94] coherent state pathintegral representation of a model with pseudospins at each guiding center.We can introduce a bosonic creation operator a † nX corresponding to ¯z nX andlet E[¯z, z] → H[a † , a].To analyze fluctuations around the HF mean field state, we use thelinear spin wave approximationa 0X → 1 − 1 2 a† X a X a 1,X → a X . (7.11)Taking the continuum limit 1/L y∑X = ∫ dX/(2πl B ), the action describingharmonic fluctuations can be written in Fourier space as S = S 0 + δS whereδS = e2εl B∫ β0dτ ∑ q[( 1 √ π2 2 + ξ q)a † q a q + λ ]q2 (a qa −q + a † q a† −q) , (7.12)95

withξ q = |ql ∫B|2 e −(ql B )22 −λ q = |ql ∫B|2 e −(ql B )22 −dp ( 1 − p2 )J0 (ql B p)e −p22 + ξ2˜∆ LL ,dp p22 J 2(ql B p)e −p22 , (7.13)In Eq.( 7.12) we have restored [95] two-dimensional wavevectors to recognizethe system’s spatial anisotropy. The first and second terms in the above expressionscapture the direct(H) and exchange(F) contributions respectivelyand J 0 and J 2 are the zeroth and second order Bessel functions. It can beverified that Eq.( 7.12) reduces to Eq.( 7.9) for q → 0. The quadratic actionin Eq. (7.12) has the familiar Bogoliubov form and the energy dispersion ofthe collective mode is given by:√ω(q) = e2 ( 1 π (εl B 2 2 + ξ q) 2 − |λ q | 2) 1/2. (7.14)As shown in Fig.[ 7.2], this collective mode has a roton minimum atql B ∼ 2.3 and approaches the Hartree-Fock theory band splitting for q → ∞as expected.[82] The surprising absence of interaction contributions to the gapat q = 0 can be understood by examining the dependence of the uniform stateinteraction energy on global rotations in LL pseudospin space:2E[z]N φ√= − e2 π[|z 0 | 4 + 3 ]εl B 2 4 |z 1| 4 + 2|z 0 | 2 |z 1 | 2 , (7.15)The factor in square brackets above is 1−|z 1 | 4 /4, independent of z 1 to quadraticorder. Notice that because ∆ LL < 0 for ν = −1, 3 the absence of interaction96

0.6q l B ∞0.50.4Ω0.30.20.1 V ⩵ V ⩵ V ⩵20meV10meV0.103meV0.00 2 4 6 8 10q l BFigure 7.2: Collective mode ω q of the Landau-level pseudospin polarized statein units of interaction strength e 2 /ǫl B = 11.2 √ B[Tesla] meV as a function ofql B at different values of the external potential difference ∆ V at a magneticfield of 20 T. The black(solid) line indicates the ql B → ∞ asymptote for∆ B = 0.contributions to the gap implies that the fully spin-polarized state is unstable.The ground state at these filling factors is instead[93] an XY state withspontaneous phase order.7.4 Intra-Landau-Level Cyclotron ResonanceFinally we show that the octet QHF will exhibit unusual intra-LL cyclotronmodes at odd filling factors, focusing on the fully polarized ν = −3, 197

cases. The dynamical conductivity σ ± = σ xx ± iσ xy can be evaluated usinglinear response theory. The projection of the current operator, j i = dH/dπ i ,onto the octet space can be expressed in terms of LL pseudospins:j i = ξ∆ Bmγ 1( √2lBm i + e c Aext i (t) ) , (7.16)where the ac electric field E i = (1/c)dA exti /dt. The ac conductivity (ξ = 1) ismost simply evaluated by solving the LL pseudospin equation of motion withthe j · A ext coupling included in the energy functional. We find thatσ ± (ω) = N φe∆ Bmγ 11i(ω ± ω LL )(7.17)In the absence of interactions the conductivity has intra-octet peaks at the LLband-splitting frequency ω LL , in addition to inter-Landau-level peaks which donot appear in the projected theory. The low-frequency absorption peaks shouldbe visible in microwave absorption experiments. The appearance of tunablelow-frequency peaks in σ(ω) is a surprise that might be quite interesting fromthe point of view of the quantum Hall localization physics, even in systemsfor which disorder dominates interactions. In normal quantum Hall systems,peaks in σ ± appear near the characteristic inter-Landau-level energy ω c and thestrong localization physics which leads to flat broad quantum Hall pleateausoccurs only in systems with ω c τ > 1. We conjecture that one requirementfor odd-integer filling factor plateaus within the graphene bilayer octet is thatω LL τ > 1. Since ω LL is proportional to ∆ V , the strength of the quantumHall effect can be tuned by a gate voltage which doesn’t influence either thesystem’s disorder or its total carrier density.98

As noted in [13] trigonal warping can be neglected in broad range of magneticfields given by l −1B> v 3m. However it is reasonable to ask the effect of trigonalwarping on the intra-LL cyclotron gap. To address this issue we performednumerical calulations on the four-band model, we find that at a magnetic fieldstrength of 10T and ∆ V ≈ 10meV the gap is reduced by < 2%. Therefore weanticipate that the intra-LL cyclotron resonance signal to be experimentallymeasurable above 10T.99

Chapter 8ConclusionThe primary focus of this thesis has been the study of electron-electroninteractions in Chiral 2DEGs. There has been recent interest in these systemdue to the experimental observation of single layer and bilayer graphene [3,4].Due to the high mobility inherent of graphitic nanostructures there is tremendouspromise for potential device applications [4]. These systems have uniqueelectron-electron interactions due to the presence of chiral band eigenstates.Chiral Fermions also respond in a different way when compared to normal2DEG electron in a magnetic field, the presence of additional degeneracies associatedto the chiral band structure especially make this a unique system interms of strongly correlated Quantum Hall Physics [33].Quasiparticles in graphene sheets behave like massless Dirac Fermionsmathematically similar QED 3 , interacting via non-relativistic long-range Coulombinteractions. We have developed a theory of electron-electron interactions ingraphene sheets based on the random-phase-approximation. In particular wehave shown that the tendency of Coulomb interactions in graphene to favorstates with larger net chirality leads to suppressed spin and charge susceptibilities.This suppression is a consequence of the quasiparticle chirality switchwhich enhances quasiparticle velocities near the Dirac point. The renormal-100

ized velocity and quasiparticle spectral weight have a weak dependence on thedoping. This has important implications on density-functional applications tographene nanostrutures.Recent ARPES [62,63] experiments have reported a band gap in graphenewhich was interpreted as influence of electron-phonon [65] and electron-plasmoninteractions, or as the apparent band-gap opening at the Dirac point due tosubstrate effects [63]. Graphene ARPES spectra are influenced by disorder,coupling to the substrate, and by electron-phonon interactions, in additionto the electron-electron interaction effects considered in this thesis. Becauseinteractions effects scale with vk F energy scale, while phonon effects are fixedat optical phonon energy scales, these two contributions can be separated experimentallyby varying carrier density. Our RPA theory demonstrates thatbroad quasiparticle peaks and apparent energy gaps near the Dirac point areexpected even without substrate coupling. We expect that the present RPAtheory results, combined with progress in the preparation of samples suitablefor ARPES or for 2D to 2D tunneling spectroscopy [69], will enable furtherprogress.In graphene bilayer 2DES’s chiral bands lead to an additional degeneracydoubling [13] at the Fermi energy of a neutral system. This additionaldegeneracy leads to formation of LL dipoles, because dipole-dipole interactionsare long-range, they play a dominant role in the Qunatum Hall Ferromagnetlong-wavelength effective action. In particular leading to nearly gapless collectivemodes with an approximate q 3/2 dispersion. These systems exhibit a101

very rich phase diagram with new types of topologically charged excitationscurrently under investigation [93]. Even more interesting is the possibility ofstrongly correlated states in bilayer graphene, in particular there is evidenceto suggest that bilayer graphene might support novel nonabelian quantum hallstates [75]. It is safe to assume that Quantum Hall Physics in bilayer graphenehas more surprises in store for us to discover.102

Appendices103

Appendix AGraphene’s Lindhard FunctionThe Dirac Hamiltonian for graphene in momentum space can be writtenasH k = v⃗σ · ⃗k(A.1)here the fermi velocity v = 3 ta where t is the tight binding parameter and a is2the nearest neighbor spacing in graphene’s honeycomb lattice. We can definea set of gamma matrices γ 0 = iσ 3 , γ 1 = −iσ 3 σ 1 , γ 2 = −iσ 3 σ 2 which defines aClifford algebra {γ µ , γ ν } = 2g µν (here g 00 = −1, g 0i = g i0 = 0and g ij = δ ij ).In this basis the low energy effective interacting hamiltonian for a graphenesheet isH = v ∑ ¯ψk i⃗γ · ⃗kψ k + 1 ∑ ∑v(q)(2S¯ψ k1 +qiγ 0 ψ k1¯ψk2 −qiγ 0 ψ k2 − ˆN) (A.2)⃗ k ⃗q≠0 ⃗k 1 , k ⃗ 2where S is the sample area, ˆN is the total number operator, v(q) is the 2Dfourier transform of the interaction potential . The one-body noninteractingGreen’s function defined as G 0 ( ⃗ k, ω) = −i〈T(ψ ¯ψ)〉 is given byG 0 ( ⃗ k, ω, µ ≠ 0) = i −ωγ 0 + v⃗γ · ⃗k−ω 2 + v 2 k 2 − iǫ − π −ωγ 0 + v⃗γ · ⃗kδ(ω − v|v|k|⃗ k|)θ(µ − v| ⃗ k|),G 0 ( ⃗ k, ω, µ ≠ 0) = G 0 ( ⃗ k, ω, 0) + δG 0 ( ⃗ k, ω).(A.3)104

where µ is the Fermi energy of a doped graphene sheet. The dynamical polarizibility(Graphene’s Lindhard function) is:∫χ µν (|q|, Ω, µ ≠ 0) = −iχ µν (|q|, Ω, µ ≠ 0) = χ 0 µ,ν ((|q|, Ω, 0)) + δχ µν(|q|, Ω)d 2 k dω(2π) 2 2π Tr[ieγ µG 0 ( ⃗ k + ⃗q, ω + Ω, µ ≠ 0)ieγ ν G 0 ( ⃗ k, ω, µ ≠ 0)],(A.4)Here χ 0 µ,νis the polarizibility for a neutral graphene sheet with Fermi energy atcoincident with the Dirac point and δχ µ,ν incorporates Pauli blocking effectsdue to doping. In the following we calculate the dynamical polarizibility byevaluating the half-filled (µ = 0) case first corresponding to χ 0 µ,ν , and then theδχ µ,ν contribution.A.1 Half-FillingThe free fermion propagator at half-filling with momentum ⃗ k and energyω is given byG 0 ( ⃗ k, ω, 0) = i −ωγ 0 + v F ⃗γ · ⃗k−ω 2 + v 2 k 2 − iǫ .(A.5)The dynamical polarizibility χ 0 µν(|q|, Ω) for neutral graphene is:χ 0 µν (|q|, Ω) = −i ∫d 2 k(2π) 2 dω2π Tr[ieγ µG 0 ( ⃗ k + ⃗q, ω + Ω, 0)ieγ ν G 0 ( ⃗ k, ω, 0)]. (A.6)We can separate the trace over the gamma matrices and define the threedimensionalvector k µ = (−ω, v ⃗ k) this gives the generalized three-dimensionallength k 2 = −ω 2 + v 2 k 2∫χ 0 µν (|q|, Ω) = −ie2v 2 Tr[γµ γ ρ γ ν γ σ ]d 3 k(2π) 3 (k + q) ρ k σ(k + q) 2 k 2 (A.7)105

The trace over the gamma matrices can be easily calulated usingTr[γ µ γ ρ γ ν γ σ ] = 2(g µρ g νσ − g µν g ρσ + g µρ g νσ )(A.8)This integral can be calculated by the well known methods commonly employedin Quantum Electrodynamics. At first glance the above integral inplagued with ultra-violet divergences, however as this a a low-energy effectivetheory there is a natural ultraviolet cutoff scale. We will use dimensional regularizationto deal with the ultraviolet divergences thereby getting a cutoffindependent expression for the dynamic polarizibility. Using Feynman parametersthis integral can be written asχ µν (|q|, Ω) = − ie2v 2 ∫d 3 ∫k 1(2π) 30dx 2[(k + q) µk ν + (k + q) ν k µ − g µν (k + q) · k)[(k + q) 2 x + (1 − x)k 2 ] 2 .We can complete the square and write the denominator as(A.9)x(k + q) 2 + (1 − x)k 2 = (k + xq) 2 + q 2 x(1 − x),(A.10)defining a new momentum l ≡ k + xq, we can see that the denominator justdepends on l 2 . Integrating over d 3 k = d 3 l is becomes easier as the integrandis spherically symmetric with respect to l. Performing this shift k → l − xq,the integral becomes∫χ µν = − ie2 d 3 ∫l 1v 2 (2π) 30dx 2l µl ν − 2q µ q ν x(1 − x) − g µν (l 2 + q 2 x(1 − x))[l 2 E + ∆]2 (A.11)where ∆ ≡ q 2 x(1−x). Employing symmetry considerations gives that terms inthe numerator containing odd powers in l vanishes, the rest can be evaluated106

from the general formula. I will also perform this integral in Euclidean spaced 3 l → id 3 l E∫ 1χ µν (|⃗q|, Ω) = 2e2v 20∫ d 3 l E −1/3g µν lE 2 dx+ (q2 g µν − 2q µ q ν )x(1 − x)(2π) 3 [lE 2 + ∆]2 (A.12)Usingwhich gives∫ d d l E 1(2π) d [lE 2 + = (−1)n Γ(n − d ) 2 ∆]n (4π) d/2 Γ(n)∫ d d l E lE2(2π) d [lE 2 + = (−1)n d∆]n (4π) d/2 2Γ(n − d 2 − 1)Γ(n)( 1 ) n−d2(A.13)∆( 1 ) n−d2 −1 (A.14)∆χ µν (|⃗q|, Ω) = − |q| (gµν − q µq ν) ∫ 12πv 2 q 20dx √ x(1 − x)(A.15)Restricting ourselves to the special case of Coulomb interaction µ = ν = 0χ 00 (|⃗q|, Ω) =⃗q 216 √ q 2 − Ω 2 ⃗q 2) + iv 2 q 2 − Ω 2Θ(v2 16 √ − v 2 q 2 )Ω 2 − v 2 q 2Θ(Ω2 (A.16)and analytically continuing Ω → iΩ just givesχ 00 (|⃗q|, iΩ) =This gives the Lindhard function for neutral graphene.⃗q 216 √ v 2 q 2 + Ω 2 (A.17)A.2 Pauli-blocking effectsTo calculate the µ ≠ 0 contribution to the dynamical polarizibility wespecifically restrict to Coulomb interaction, the numerator in the second term107

of A.4 is:Tr[γ 0 (−ωγ 0 + v⃗γ · ⃗k)γ 0 (−(ω + Ωγ 0 + v⃗γ ·⃗ k + q)](A.18)This trace can be easily calculated with the help of the definition of the Cliffordalgebra which givesTr[γ 0 γ 0 γ 0 γ 0 ] = 2 Tr[γ 0 γ i γ 0 γ 0 ] = Tr[γ 0 γ 0 γ 0 γ j ] = 0Tr[γ 0 γ i γ 0 γ j ] = 2δ ij(A.19)giving usTr[γ 0 (−ωγ 0 +v⃗γ·⃗k)γ 0 (−(ω+Ωγ 0 +v⃗γ· k + ⃗ q)] = 2[ω(ω+Ω)+v 2 ⃗ k· k + ⃗ q] (A.20)The crossterm (the imaginary times the real part of the greens function) callthis term δχ oo∫ dδχ oo (|⃗q|, Ω) = ie 2 2 ∫ [k dω 2[ω(ω + Ω) + v 2⃗ k ·(2π) 2 2π − iπ ⃗ k + ⃗q]δ(ω − v| ⃗ k|)θ(µ − v| ⃗ k|)[−(ω + Ω) 2 + v 2 | ⃗ k + ⃗q| 2 − iǫ]v| ⃗ k|+ 2[ω(ω + Ω) + v2 ⃗ k · ⃗ k + ⃗q]δ(ω + Ω − v| ⃗ k + ⃗q|)θ(µ − v| k + ⃗]q|)[−ω 2 + v 2 | ⃗ k| 2 − iǫ]v| ⃗ (A.21)k + ⃗q|Using the delta function to perform the frequency integral and shifting themomentum ⃗ k → − ⃗ k+⃗q in the second term of the ?? and analytically continuingΩ → iΩδχ oo = e2v∫ µ/v0∫dk 2π2π 0[dθ2π2k 2 + qk cosθ + i˜Ω| ⃗ k|2qk cosθ + (˜Ω 2 + q 2 ) − 2i˜Ω| ⃗ k| + 2k 2 + qk cosθ − i˜Ω| ⃗ k|2qk cos θ + (˜Ω 2 + q 2 ) + 2i˜Ω| ⃗ k|(A.22)in the above we scale the frequency by defining ˜Ω = Ω/v, and align k x in thedirection of ⃗q. We perform the above angular integral by substituting z = e iθ]108

and cosθ = z+z−12and using the identites∫ 2π0∫ 2πdθ0 a cosθ + b = 2πsgn(|z −| − |z + |)√b2 − a 2cosθdθa cosθ + b = 2π a − 2πbsgn(|z −| − |z + |)a √ b 2 − a 2(A.23)(A.24)where |z ± | = 1 a (−b ± √ b 2 − a 2 ). δχ oo then becomesδχ oo = e2 µ e 22πv 2+ √v∫ µ/v˜Ω 2 + q 2 0[dk 2k 2 − (˜Ω 2 + q 2 )/2 + 2i˜Ωk√2π(˜Ω 2 + q 2 − 4k 2 ) − 4i˜Ωk+ 2k2 − (˜Ω 2 + q 2 )/2 − 2i˜Ωk√(˜Ω 2 + q 2 − 4k 2 ) + 4i˜Ωk(A.25)The two integrands above are complex conjugates which implies that δχ oopurely real. This integral can be written in a compact analytical form asδχ oo = e2 µ2πv 2 − e 2 ⃗q 216π √ Ω 2 + v 2 q 2Re [sin −1 ( 2µ + iΩv|q|So the full χ 00 = χ 0 00(µ = 0) + δχ oo (µ ≠ 0) is given by)+(2µ + iΩv|q|) √ 1 − ( ]2µ + iΩ) 2v|q|(A.26)]χ 00 (|⃗q|, iΩ, µ ≠ 0) =−e 2 q 216 √ Ω 2 + v 2 q 2 + e2 µ2πv 2[e 2 ⃗q 28π √ Ω 2 + v 2 q 2Resin −1 ( 2µ + iΩv|q|)+(2µ + iΩv|q|(A.27)) √ 1 − ( ]2µ + iΩ) 2v|q|In the static limit (Ω = 0, q → 0) the above expression correctly reduces tothe noninteracting density of states per valley per spin at the Fermi energyD(ǫ) = µ/2πv 2 . Two limiting cases for A.27 are of particular interestlim χ 00(|⃗q|, iΩ, µ ≠ 0) =Ω→∞lim χ 00(|⃗q|, iΩ, µ ≠ 0) =q→∞q216Ω + O( 1q16vΩ 2 )(A.28)(A.29)109

Figure A.1: The surface plot shows graphene’s Lindhard function for ˜Ω = Ω/µand ˜q = q/vThe above expression indicate the divergences that need to be regularized inthe calculation for the energy and other observables. The low energy theoryof graphene described by massless Dirac Fermions has a natural ultravioletdivergence that needs to be regularized by a momentum scale cutoff Λ at largeq. Graphene’s Lindhard fucnction for q and Ω is plotted in the figure110

Appendix BCorrelation Self-energy of a quasiparticle ingrapheneThe random phase approximation(RPA) self-energy of an electron in agraphene layer can be written asΣ RPAs (k, ik n ) = ∑ i ∑ 1 ∑ G 0 sv ′(⃗ ( )k + ⃗q, ik n + iΩ n ) 1 + ss ′ cos(θ)qA β ǫs ′ =± ⃗q iΩ RPA (q, iΩ n ) 2n(B.1)ǫ RPA (q, iΩ n ) = 1 − v q Π(q, iΩ n ) where Π(q, iΩ n ) is the polarization bubblecalculated earlier, s, s ′ denote the band indices and θ is the angle between ⃗ kand ⃗ k + ⃗q. Here G 0 s ′(⃗ k + ⃗q, ik n + iΩ n ) is the finite temperature greens functionfor the quasiparticle. Σ RPA can be separated into Σ RPAsΣ RPAs (k, ik n ) = ∑ s ′ =±where Σ corrsis given asi ∑ 1 ∑G 0 sA β′(⃗ k+⃗q, ik n +iΩ n )⃗q iΩ nΣ corrs (k, ik n ) = ∑ i ∑ 1 ∑G 0 sA β′(⃗ k+⃗q, ik n +iΩ n )s ′ =± ⃗q iΩ nThe greens function is given by= Σ HFs( 1 + ss ′ cos(θ)2+ Σ corrs)[as]v q1 − v q Π(q, iΩ n ) +v q−v q(B.2)( )[ 1 + ss ′ cos(θ) v2]q Π(q, iΩ n )2 1 − v q Π(q, iΩ n )(B.3)G 0 s ′(⃗ k + ⃗q, ik n + iΩ n ) =iik n + iΩ n − ξ ⃗k+⃗q,s ′(B.4)111

where ξ ⃗k,s ′ = s ′ | ⃗ k| − µ is the energy measured from the fermi energy µ.Σ corrs (k, ik n ) = − ∑ 1 ∑( ) 1 + ssvq2 ′ cos(θ) 1 ∑ 1 Π(q, iΩ n )A 2 β iks ′ n + iΩ n − ξ ⃗k+⃗q,s ′ 1 − v q Π(q, iΩ n )=± ⃗qiΩ n(B.5)Here we are interested in evaluating retarded self-energy of the quasiparticleobtained by setting ik n → ξ ⃗k,s + iη as step towards evaluating the self-energyat the fermi surface. Unfortunately the processes of summation over Matsubarafrequencies and analytic continuation do not commute. We can howeverexpress the final retarded function as two terms:Σ corrs(k, ξ ⃗k,s ) = Σ lines (k, ξ ⃗k,s ) + Σ ress (k, ξ ⃗k,s ) (B.6)Σ line is the term that is obtained if we were allowed to interchange the twosteps of analytical continuation and contour intergrationΣ lines (k, ξ ⃗k,s ) = − ∑ ∫ ( )d 2 q 1 + ss ′ ∫cos(θ) dΩ 1 Π(q, iΩ)(2π) 2v2 q2 2π iΩ + ξ ⃗k,s − ξ ⃗k+⃗q,ss ′ ′ 1 − v q Π(q, iΩ)=±(B.7)then Σ res is just the contribution one gets by interchanging the orders of thetwo steps of analytical continuation and frequancy summation, below I showthat Σ res evaluated at the fermi energy is zero. Let us examine the frequencyintegralwhereI ss ′ ,k(ik n ) =f(iΩ) =∫ +∞−∞()dΩ2π f(iΩ) 11−iΩ + ξ ⃗k,s − ξ ⃗k+⃗q,s ′ ik n + iΩ − ξ ⃗k+⃗q,s ′Π(q, iΩ)1 − v q Π(q, iΩ)Σ ress (k, ik n ) = − ∑ ∫s=±d 2 q(2π) 2v2 q( 1 + ss ′ cos(θ)2)I ss ′ ,k(ik n )(B.8)(B.9)112

Similar to the electron gas situation the poles of f(iΩ) and its branch cutsgive no contribution to I s,k (ik n ). For example that f(iΩ) has a simple pole atΩ = Ω j with residue A j thenI ss ′ ,k(ik n ) =∫ +∞−∞()dω A j 12πi Ω − Ω j Ω − i(ξ ⃗k,s − ξ ⃗k+⃗q,s ′) − 1Ω + k n + iξ ⃗k+⃗q,s ′(B.10)The dΩ integral can be done by taking a semi-circular contour in the upperhalf plane (note that this choice is arbitrary and closing the contour in thelower half plane does not change our conclusions)[]1I ss ′ ,k(ik n ) = A jΩ j − i(ξ ⃗k,s − ξ ⃗k+⃗q,s ′) + 1Ω j + k n + iξ ⃗k+⃗q,s ′(B.11)+ Θ(ξ ⃗ k,s− ξ ⃗k+⃗q,s ′)A ji(ξ ⃗k,s − ξ ⃗k+⃗q,s ′) − Ω j−Θ(−ξ ⃗ k+⃗q,s ′)A j−k n − iξ ⃗k+⃗q,s ′ − Ω jAnalytic continuation ik n → ξ ⃗k,s + iη then just gives[ ]Θ(ξ⃗k,s − ξ ⃗k+⃗q,s ′) Θ(−ξ ⃗k+⃗q,s ′)I ss ′ ,k(ξ ⃗k,s ) = A j −i(ξ ⃗k,s − ξ ⃗k+⃗q,s ′) − Ω j i(ξ ⃗k,s − ξ ⃗k+⃗q,s ′ − Ω j(B.12)thus Σ res (ξ ⃗k,s ) just becomesΣ ress (k, ξ ⃗k,s ) = − ∑ ∫s ′ =±(d 2 q 1 + ss ′ cos(θ)(2π) 2v2 q2[Θ(ξ ⃗k,s − ξ ⃗k+⃗q,s ′) − Θ(−ξ ⃗k+⃗q,s ′)]) Π(q, ξ⃗k,s − ξ ⃗k+⃗q,s ′)(B.13)1 − v q Π(q, ξ ⃗k,s − ξ ⃗k+⃗q,s ′)113

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IndexAbstract, viAcknowledgments, vAppendices, 80Bibliography, 91Correlation Self-energy of a quasiparticlein graphene, 88Dedication, ivGraphene’s Lindhard Function, 81Chiral Hamiltonian, 1124

VitaYafis Barlas was born in Karachi, Pakistan the son of Mirza ShahidBarlas and Anwer Barlas. He recieved a B.S. degree in Physics and Mathematicsfrom University of Houston in May 2002. In August 2002 he joinedGraduate School at Univerity of Texas at Austin to pursue a PhD in Physics.Permanent address: 701 W North Loop Apt 106Austin, Texas 78751This dissertation was typeset with L A TEX † by the author.† L A TEX is a document preparation system developed by Leslie Lamport as a specialversion of Donald Knuth’s TEX Program.125