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

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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
Page 1: CopyrightbyYafis Barlas2008
Page 4 and 5: Dedicated to Ammi, Abbu and Amma.
Page 6 and 7: ditional love and support throught
Page 8 and 9: the Dirac point.In the second half
Page 10 and 11: Chapter 5. Plasmons and The Spectra
Page 12 and 13: List of Figures2.1 This figure show
Page 14 and 15: 4.3 “Lindhard” function χ (0)
Page 16 and 17: Chapter 1IntroductionThe discovery
Page 18 and 19: systems where the notion of ”quas
Page 20 and 21: tinuum model. We find that level re
Page 22 and 23: dispersion of 2.1 is ǫ J (⃗q)
Page 24 and 25: chanical exfoliation[9,10] . Graphi
Page 26 and 27: Fermi velocity much smaller than th
Page 28 and 29: G 2KG 1baK'Figure 2.3: Left: Lattic
Page 30 and 31: Figure 2.4: (Adapted from [10]) Lef
Page 32 and 33: eigenstates of the helicity operato
Page 34 and 35: Figure 2.6: Lattice structure of bi
Page 36 and 37: Identifying m α v 2 = |γ 1 cos(
Page 38 and 39: Figure 2.7: (Adapted from [26]) Thi
Page 40 and 41: The energy spectrum of a chiral sys
Page 42 and 43: Figure 2.8: (Adapted from [3] Hall
Page 44 and 45: Chapter 3Chirality and Correlations
Page 48 and 49: Figure 3.3: Dirac cone for n-doped
Page 50 and 51: 3.1 RPA Theory of GrapheneWe study
Page 52 and 53: δεx/εF2.11.751.41.050.7Λ = 10Λ
Page 54 and 55: 0-0.35δε RPAc /εF-0.7-1.05Λ = 1
Page 56 and 57: 10.9κ/κ00.80.70.60.50 1 2 3 4 5fF
Page 58 and 59: orbital response.Our results for th
Page 60 and 61: The physical origin of the velocity
Page 62 and 63: states of the interacting system. S
Page 64 and 65: Coulomb interaction is renormalized
Page 66 and 67: pole. Σ describes the interaction
Page 68 and 69: contribution Σ line . In the zero-
Page 70 and 71: have contributions that are analyti
Page 72 and 73: used in DFT. The accuracy of graphe
Page 74 and 75: Figure 4.2: In a weakly doped mater
Page 76 and 77: Z10.80.60.4(a)Λ = 10 1Λ = 10 2Λ
Page 78 and 79: Angle-resolved photoemission spectr
Page 80 and 81: most important features in Fig. 4.2
Page 82 and 83: F(x, y) → −παgrl(α 2 gr )|x|
Page 84 and 85: conduction band states and k and q
Page 86 and 87: 2D tunneling spectroscopy [69], wil
Page 88 and 89: 4433ΩεF2ΩεF210.000.07∞ 0.37 0
Page 90 and 91: 21.510.5ImΣ +ReΣ +A +ω−k+1k=0.
Page 92 and 93: Figure 6.1: This figure shows hall
Page 94 and 95: Laughlin wrote a variational wavefu
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Figure 6.2: The direction of the lo
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these topological excitation carry
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Figure 6.3: Phase diagram of a doub
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Chapter 7Octet Quantum Hall Ferroma
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the two low energy sites [13] (the
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the lowest energy eigenvectors of H
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Octet quantum Hall ferromagnets hav
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|z 1X | 2 = 1. The corresponding im
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0.6q l B ∞0.50.4Ω0.30.20.1 V ⩵
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As noted in [13] trigonal warping c
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ized velocity and quasiparticle spe
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Appendices103
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where µ is the Fermi energy of a d
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from the general formula. I will al
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and cosθ = z+z−12and using the i
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Appendix BCorrelation Self-energy o
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Similar to the electron gas situati
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[10] A. H. Castro Neto, F. Guinea,
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[36] D.C. Tsui, H.L. Stormer, and A
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Sarma, E.H. Hwang and W.K. Tse Phys
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[73] B.I. Halperin. Phys. Rev. Lett
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[93] Y. Barlas, Jules Lambert, R. C
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VitaYafis Barlas was born in Karach
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Ph. D. Thesis - The University of Texas at Austin

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