Dirac Fermions in Graphene and Graphite—A View from Angle

Dirac Fermions in Graphene and Graphite — a view from angle-resolved photoemission spectroscopy

Shuyun Zhou

M.A. (UC Berkeley) 2005 B.S. (Tsinghua University) 2002

A dissertation submitted in partial satisfaction of the requirements for the degree of

Doctor of Philosophy in

Physics

in the

GRADUATE DIVISION

of the

UNIVERSITY OF CALIFORNIA, BERKELEY

Committee in charge:

Professor Alessandra Lanzara, Chair Professor Dung-Hai Lee Professor Oscar Dubon

Fall 2007 The dissertation of Shuyun Zhou is approved.

Chair Date

Date

University of California, Berkeley Fall 2007 Dirac Fermions in Graphene and Graphite — a view from angle-resolved photoemission spectroscopy

Shuyun Zhou Abstract

Dirac Fermions in Graphene and Graphite — a view from angle-resolved photoemission spectroscopy

Shuyun Zhou Doctor of Philosophy in Physics

University of California, Berkeley Professor Alessandra Lanzara, Chair

The research in graphene has exploded in the past few years, due to its intriguing physics as an emerging paradigm for relativistic condensed matter physics as well as its great promise for application in next generation electronics. Understanding the low energy electronic structure of graphene is fundamental as most of the intriguing properties of graphene arise from its peculiar electronic dispersion, which resembles that of relativistic Dirac Fermions. This thesis presents a detailed study of the low energy electronic structure of graphene and its related three dimensional material - graphite - by using angle-resolved photoemission spectroscopy (ARPES), a direct probe of the electronic structure. In particular, the evolution of the Dirac Fermions in graphene and graphite as well as the effect of impurities is the focus of this thesis. This thesis is organized as follows. The first chapter is an introduction of the electronic structure of graphene and graphite, and the specialty of Dirac fermions compared to quasiparticles in conventional condensed matter systems. Chapter 2 is an introduction of the techniques used throughout this thesis - angle resolved photoemission spectroscopy (ARPES), X-ray photoemission spectroscopy (XPS) and low energy electron microscopy (LEEM). Chapter 3 discusses the growth and characterization of epitaxial graphene on SiC wafers. Chapters 4 and 5 present the ARPES results on epitaxial graphene, the evolution of the low energy electronic dynamics as a function of sample thickness and how to make graphene a finite band gap semiconductor. More specifically, chapter 4 discusses how a gap is induced between the valence and conduction bands by graphene-substrate interaction and chapter 5 shows how a reversible metal-insulator transition can be possibly induced in epitaxial graphene by hole doping. Chapter 6 and 7 show the ARPES results on three dimensional graphite samples. Chapters 6 shows the coexistence of Dirac fermions with massive quasiparticles in different momentum space of single crystalline graphite and the effect of impurity. Chapter 7 shows the surprising results of obtaining the band dispersions even in a partially polycrystalline graphite sample, the comparison with those obtained in single crystalline graphite, as well as the implication for ARPES study of partially polycrystalline samples in general.

Professor Alessandra Lanzara Dissertation Committee Chair

1 Contents

Abstract 1

Contents i

Acknowledgements iii

vitae vi

1 Introduction 1 1.1 Graphene - an emerging paradigm for relativistic condensed matter physics ...... 1 1.2 Electronic structure of graphene in the tight binding model ...... 3 1.3 Massless Dirac fermions and massive quasiparticles ...... 7 1.4 Electronic structure of inﬁnite layer graphite ...... 8

2 Experimental techniques 10 2.1 Angle resolved photoemission spectroscopy (ARPES) ...... 10 2.1.1 ARPES and the photoelectric eﬀect ...... 10 2.1.2 Three step model and sudden approximation ...... 13 2.1.3 Single particle spectral function description of ARPES ...... 15 2.1.4 Energy distribution curves (EDCs) and momentum distribution curves (MDCs) . . . . 16 2.1.5 Electron mean free path and the probing depth of ARPES ...... 18 2.1.6 State of the art ARPES ...... 19 2.2 X-ray photoelectron spectroscopy (XPS) ...... 20 2.3 Low energy electron microscopy (LEEM) ...... 22

3 Growth and characterization of epitaxial graphene 24 3.1 Growth of epitaxial graphene ...... 24 3.2 Characterization of epitaxial graphene ...... 26 3.2.1 Scanning electron microscopy (SEM) ...... 26 3.2.2 X-ray photoemission spectroscopy (XPS) ...... 27

i 3.2.3 Angle-resolved photoemission spectroscopy - ARPES ...... 29 3.2.4 Low energy electron microscopy (LEEM) ...... 30

4 Gap opening in epitaxial graphene 33 4.1 Dirac fermions in epitaxial graphene ...... 34 4.2 Departure from conical dispersion in single layer epitaxial graphene ...... 35 4.3 Gap interpretation vs many body interactions ...... 39 4.4 Thickness dependence of the gap and Dirac point energy ...... 40 4.5 Possible mechanisms of the gap ...... 44 4.5.1 Inter-Dirac-point scattering ...... 44 4.5.2 Quantum conﬁnement ...... 44 4.5.3 Breaking of the sublattice symmetry due to graphene-substrate interaction ...... 46 4.6 Conclusions ...... 49

5 Metal-insulator transition in epitaxial graphene by molecular doping 50 5.1 Motivation ...... 50 5.2 Valence band of graphene with adsorption of nitrogen dioxide ...... 51 5.3 Gas adsorption and metal-insulator transition in bilayer graphene ...... 53 5.4 Metal-insulator transition in single layer graphene ...... 55 5.5 Conclusions ...... 57

6 Dirac fermions and eﬀect of impurities in single crystalline graphite 59 6.1 Introduction ...... 59 6.2 Electronic structure of single crystal graphite ...... 60 6.2.1 Dispersions at H - massless Dirac fermions ...... 61 6.2.2 Dispersions at K - massive quasiparticles ...... 66 6.3 Eﬀect of defect states ...... 67 6.4 Conclusions ...... 70

7 ARPES study of partially polycrystalline graphite: HOPG 71 7.1 Electronic structure of polycrystalline materials measured by ARPES ...... 71 7.2 ARPES study of the electronic structure of polycrystalline graphite ...... 73 7.3 Explanation of the paradoxical coexistence based on a density of states argument ...... 75 7.4 Dispersions along the surface normal direction ...... 77 7.5 Comparison of data taken from HOPG and single crystalline graphite ...... 79 7.5.1 Interlayer coupling ...... 79 7.5.2 Low energy dispersions and Dirac fermions ...... 80 7.5.3 Defect states ...... 81

ii 7.6 Conclusions and implication of this study ...... 86

Bibliography 87

List of Figures 96

iii Acknowledgements

I am very lucky to work with a group of talented and kind people in my Ph.D., from whom I have benefited greatly. All the work presented here would not have been possible without them. First of all, my deepest thanks go to my advisor, Alessandra Lanzara. I am very lucky to have such an inspiring, creative and considerate advisor, who has made my Ph.D. fruitful and enjoyable. Besides the training, encouragement and support that she gave me in science, I also appreciate her willingness to care about students’ life beyond work. The past few years of working with her has been a fortune for me. Her endless enthusiasm for science, her open-mindedness and her wisdom to balance work and life, have set her as a role model for me. I could not ask for better colleagues than the people in Lanzara group. Dr. Gey-Hong Gweon taught me numerous things in my first three years, ranging from experimental skills to different physics topics. I appreciate his mentoring and the efforts that he has put in to make me a good experimentalist. Dr. Kyle McElroy taught me the importance of understanding the physics problems in different perspectives. Jeff Graf, Chris Jozwiak and I joined the group as the first generation graduate students and we shared the longest memories of our Ph.D. lives. Besides intelligent, they are the best colleagues who are always willing to share their knowledge and skills and provide immediate help whenever needed. Daniel Garcia’s talks are the most wonderful sound of the group and I appreciate his help to improve my speech skills, though I have to admit that I am not a fast learner in this aspect. David Siegel reminded me of the early years of my Ph.D. It is a pleasure to see that he has become very good at communications and experiments. The interaction with visitors Dr. Marco Portalupi and Dr. Andreas Bill, as well as undergrads Elizabeth Rollings, Annie Endozo and Max Watson, has also been very pleasant. The collaboration with many theorists has been quite beneficial and pleasant. Dung-Hai Lee is the longest theoretical collaborator. His deep physics insights and his enthusiasm for finding the most beautiful solutions are very inspiring. The collaboration with A.N. Castro Neto, F. Guinea S.G. Louie and C.D. Spataru has been a great pleasure. Alexei Fedorov has collaborated in most of the research work presented here. His assistance in the experiments and his support for trying new experiments is highly appreciated. I am grateful to Andreas Schmid and Farid El Gably for providing extensive help with the LEEM measurements at the most critical time. John Pepper is very talented in machining and I thank him for making modifications to accommodate the needs of some challenging experiments. I thank Prof. P. N. First and Prof. W. A. de Heer for providing the graphene samples at the beginning of this project. I thank Prof. Oscar Dubon and Prof. Michael F. Crommie for serving as my committee members for the qualifying exam. J. T. Robinson and J. W. Beeman have been very nice to help me with substrate treatment. I thank the friendly and helpful people on the ALS floor: X. J. Zhou, W. L. Yang, N. Manella, K. Tanaka, S.-K. Mo, Y.-D. Chuang, J. H. Guo, B. K. Freelon, B. S. Mun. The LBL security people have been very kind to escort me to work and to home in many late nights. My personal interaction with other physicists also helps me greatly. I thank Z. Hussain, Z.-X. Shen, Y.-R. Shen and B.-F. Zhu for being patient to listen to me and give me advice. The encouragement from M. A. H. Vozmediano and N. Berrah is also appreciated. Finally, special thanks to my husband Pu for his encouragement and support throughout all these years. Thanks to my parents for letting me travel a long way from home and visiting me for the last three months of my Ph.D. Thanks to my brother and sister-in-law for having taken good care of my parents, my little nephew for adding a lot of fun to the family.

iv Curriculum Vitæ Shuyun Zhou

Education 2007 University of California, Berkeley, CA, USA Ph.D., Physics 2005 University of California, Berkeley, CA, USA M.A., Physics 2002 Tsinghua University, Beijing, China B.S., Physics

Honors and Awards 2007 ALS Doctoral fellowship, Lawrence Berkeley National Laboratory 2007 Chinese Government Award for Outstanding Self-ﬁnanced Students Abroad 2006 ALS Doctoral fellowship, Lawrence Berkeley National Laboratory 2006 UC Berkeley Graduate Division Conference Travel Grant 2005 Scholarship from International School of Physics ‘Enrico Fermi’, Italy

Publications Graphene and Graphite: 1 Metal to insulator transition in epitaxial graphene induced by molecular doping. S. Y. Zhou, D. A. Siegel, A. V. Fedorov and A. Lanzara. In preparation. 2 Departure from the conical dispersion in epitaxial graphene. S. Y. Zhou, D. A. Siegel, A. V. Fedorov and A. Lanzara. Physica E, in press (2007). 3 Substrate induced band gap opening in epitaxial graphene. S. Y. Zhou, G.-H. Gweon, A. V. Fedorov, P. N. First, W. A. de Heer, D.-H. Lee, F. Guinea, A. H. Castro Neto and A. Lanzara. Nature Mat. 6, 770 (2007). 4 First direct observation of Dirac fermions in graphite. S.Y. Zhou, G.-H. Gweon, J. Graf, A.V. Fedorov, C.D. Spataru, R.D. Diehl, Y. Kopelevich, D.-H. Lee, S.G. Louie, A. Lanzara. Nature Phys. 2, 595 (2006). 5 Low energy excitations in graphite: the role of dimensionality and lattice defects. S.Y. Zhou, G.-H. Gweon, and A. Lanzara. Ann. Phys. 321, 1730 (2006). 6 Synthesis and characterization of atomically-thin graphite ﬁlms on a silicon carbide substrate. E. Rollings, G.-H. Gweon, S.Y. Zhou, B.S. Mun, J.L. McChesney, B.S. Hussain, A.V. Fedorov, and A. Lanzara. J. Phys. Chem. Solids 67, 2172 (2006). 7 Coexistence of sharp quasiparticle dispersions and disorder features in graphite. S.Y. Zhou, G. -H. Gweon, C. D. Spataru, J. Graf, D.-H. Lee, Steven G. Louie, A. Lanzara. Phys. Rev. B 71, 161403(R) (2005).

High Tc superconductors:

v 1 A universal high energy anomaly in angle resolved photoemission spectra of high temperature superconductors - possible evidence of spinon and holon branches. J. Graf, G.-H. Gweon, K. McElroy, S.Y. Zhou, C. Jozwiak, E. Rotenberg, A. Bill, T. Sasagawa, H. Eisaki, S. Uchida, H. Takagi, D.-H. Lee and A. Lanzara. Phys. Rev. Lett. 98, 067004 (2007).

2 Strong and complex electron-lattice correlation in optimally doped Bi2Sr2CaCu2O8+δ. G.-H. Gweon, S.Y. Zhou, M.C. Watson, T. Sasagawa, H. Takagi and A. Lanzara. Phys. Rev. Lett. 97, 227001 (2006).

3 Elastic scattering susceptibility of the high temperature superconductor Bi2Sr2CaCu2O8+δ: a comparison between real and momentum space photoemission spectroscopies. K. McElroy, G.-H. Gweon, S.Y. Zhou, J. Graf, S. I. Uchida, H. Eisaki, H. Takagi, T. Sasagawa, D. -H. Lee, A. Lanzara. Phys. Rev. Lett. 96, 067005 (2006).

4 Oxygen isotope eﬀect on electron dynamics in Bi2Sr2CaCu2O8+δ: angle-resolved photoemission spectroscopy. T. Sasagawa, A. Lanzara, G.-H. Gweon, S. Zhou, J. Graf, Suryadijaya, H. Takagi. Physica C-superconductivity and its applications 426, 436 (2005).

5 Anomalous isotope eﬀect on the electron dynamics of Bi2Sr2CaCu2O8+δ high temperature superconductor. G.-H. Gweon, T. Sasagawa, S.Y. Zhou, J. Graf, H. Takagi, D. H. Lee and A. Lanzara. Nature 430, 187 (2004).

6 Strong inﬂuence of phonons on the electron dynamics of Bi2Sr2CaCu2O8+δ. G. -H. Gweon, S.Y. Zhou, A. Lanzara. J. Phys. Chem. Solids 65, 1397 (2004). 7 Lattice dynamics and paired electrons in high temperature superconductors. A. Lanzara, G.-H. Gweon, S.Y. Zhou. Kluwers Academic Publishers B.V., Vol. NATO Science Series II - Mathematics, Physics and Chemistry (2004). Charge density wave:

1 Probing the band structure of LaTe2 using Angle Resolved Photoemission Spectroscopy. D.R. Garcia, S.Y. Zhou, G.-H. Gweon, M.H. Jung, Y.S. Kwon, A. Lanzara J. Elec. Spec. Relat. Phenom. 156-158, 58 (2007).

2 Revealing charge density wave formation in the LaTe2 system by Angle Resolved Photoe- mission Spectroscopy. D.R. Garcia, G.-H. Gweon, S.Y. Zhou, J. Graf, C.M. Jozwiak, M.H. Jung, Y.S. Kwon, A. Lanzara. Phys. Rev. Lett. 98, 166403 (2007).

Science Highlights 1 S.Y. Zhou et al, Substrate induced band gap opening in epitaxial graphene, Nature Mat. 6, 770 (2007) • “Graphene: Mind the gap” by Kostya Novoselov, News and Views - Nature Mat. Vol. 6, 720 (2007) 2 S.Y. Zhou et al, First direct observation of Dirac fermions in graphite, Nature Phys. 2, 595 (2006) • “First direct evidence of Dirac fermions in graphite”, ALS News Vol. 277, June 27, 2007 • Selected as Science Highlight for the Advanced Light Source (ALS) annual activity report 2006

3 G.-H. Gweon et al, Anomalous isotope eﬀect on the electron dynamics of Bi2Sr2CaCu2O8+δ high temperature superconductor, Nature 430, 187 (2004) • “Giant Eﬀects of Lattice-Vibrations on Electron Dynamics in a High-Temperature Supercon- ductor”, Japan Science and Technology Agency, Nov. 2005

vi • Selected as Science Highlight for the Advanced Light Source (ALS) annual activity report 2005 • Selected as Science Highlight for the Science and Technology Panel DOE, 2004 • “ALS Sheds More Light on High Tc Superconducting Mystery” by Lynn Yarris, The Berkeley Lab View, Sept. 03, 2004 • “Vibrations In Crystal Lattice Play Big Role in High Temperature Superconductors”, Science Daily, Aug. 25, 2004 • “Vibrations in crystal lattice play big role in high temperature superconductors”, PhysOrg.com, Aug. 22, 2004 • “Lattice vibrations play important role in high temperature superconductors” by Robert Sanders, UC Berkeley News, Aug. 16, 2004

Invited Talks and seminars 1 “Substrate-induced band gap opening in epitaxial graphene”, ALS/CXRO seminar, Lawrence Berkeley National Lab, Sept. 2007. 2 “From graphene to graphite - the dynamics of Dirac Fermions and many body interactions”, In- ternational Seminar and Workshop ‘Dynamics and Relaxation in Complex Quantum and Classical Systems and Nanostructures’, Max Planck Institute for the Physics of Complex Systems, Dresden, Germany, Sept. 2006. 3 “Dirac Fermions and many body interactions in grapheme and graphite”, condensed matter physics seminar, IMPMC, Universite Pierre et Marie Curie, Paris, France, Sept. 2006. 4 “Electron-lattice interaction in high temperature superconductors: a view from photoemission”, International school of physics ’Enrico Fermi’ - ‘Polarons in bulk materials and systems with reduced dimensionality’, Varenna, Italy, July 2005.

Contributed talks 1 “Self energy eﬀect and gap opening in epitaxial graphene”, American Physical Society March Meet- ing, Denver, CO, USA, March 2007. 2 “Dirac quasiparticles in graphite - a direct experimental view from photoemission”, American Phys- ical Society March Meeting, Baltimore, MD, USA, March 2006. 3 “Coexistence of sharp quasiparticle dispersions and disorder features in graphite”, American Phys- ical Society March Meeting, Los Angeles, CA, USA, March 2005. 4 “Nodal vs. oﬀ-nodal kink in optimally-doped Bi2212: an oxygen isotope dependent study”, Amer- ican Physical Society March Meeting, Montreal, Canada, March 2004.

vii Chapter 1

Introduction

1.1 Graphene - an emerging paradigm for relativistic condensed

matter physics

Graphene is a single layer of carbon atoms arranged in two dimensional (2D) hexagonal honeycomb lattice. Graphene is the building block for other sp2 bonding carbon allotropes. For example, fullerene is formed by wrapping up graphene into zero dimensional (0D) bucky ball; carbon nanotube is formed by rolling up graphene into 1D cylinder and graphite is formed by stacking graphene into 3D structure (Fig. 1.1). While fullerene and carbon nanotubes have been widely explored experimentally in the past few decades 1,2, graphene was presumed not to exist in the free state and it had only been a theoretical toy model 3 until the recent discovery of atomically thin graphene samples 4. The research interest in 2D graphene soon exploded when graphene was demonstrated to have relativistic massless Dirac fermions 5. In the past few years, graphene has emerged as a new paradigm for studying relativistic condensed matter physics, where condensed matter physics merges with relativistic quantum dynamics 4.

Figure 1.1. (a) Sp2 bonding and the resulting carbon allotropes from 0D to 3D.

1 The unit cell of graphene contains two geometrically diﬀerent carbon sublattices A and B (Fig. 1.2). The lattice vectors can be written as:

√ 1 3 a1 = a( , ) 2 2√ 1 3 a = a(− , ) (1.1) 2 2 2 √ where a= 3a0 with the C-C lattice constant a0=1.42 A.˚

Figure 1.2. Unit cell of graphene in the real space (a) and reciprocal space (b).

The corresponding reciprocal lattice vectors are: √ 4π 3 1 b1 = √ ( , ) 3a 2 2 √ 4π 3 1 b2 = √ (− , ) (1.2) 3a 2 2

Fig. 1.2 (b) shows the reciprocal space of graphene. The Brillouin zone (BZ) shows a hexagonal shape 4π and the high symmetry points Γ, K and M are deﬁned as the center (0,0), corner ( 3a , 0), center of the edge ( π , √π ) in the BZ respectively. In graphene, most of the interesting physics occurs near the the two a 3a 0 inequivalent BZ corners K and K , where the bands cross the Fermi energy EF .

1.2 Electronic structure of graphene in the tight binding model

The unit cell of graphene contains two carbon atoms and each carbon atom has four valence electrons. Three out of the four electrons form strong σ covalent bonds at high binding energy, which gives graphene exceptional structural rigidity within its layers. The fourth electron forms delocalized π orbital, which crosses 0 the Fermi energy EF at the BZ corners (K and K ) (Fig. 1.3), giving graphene excellent conductivity and all the intriguing physics. The electronic structure of graphene can be calculated using the tight binding model 6. The tight binding model 7 is valid when the overlap of the wave function is large enough so that corrections to the isolated atoms is needed while at the same time not too large to invalidate the atomic description. In this case, the

2 Figure 1.3. The band structure of graphene.

wave function for electrons can be approximated by a linear combination of the atomic wave functions. Thus the tight binding model is also known as the linear combination of atomic orbitals (LCAO) approximation.

In the atomic description, the Hamiltonian for each carbon atom localized at r1 and r2 is

∇2 H = + V (r − r ) (1.3) 1,2 2m a 1,2

where Va is the atomic potential on each carbon atom.

The atomic wave functions centered on each carbon site φ(r − r1,2) satisfy

H1,2φ(r − r1,2) = E1,2φ(r − r1,2) (1.4)

In graphene, the total Hamiltonian is

∇2 H = + Σ (V (r − r − R) + V (r − r − R)) (1.5) 2m R 1 2

The total Hamiltonian can be rewritten as the Hamiltonian for atom 1 or 2 with some corrections ∆H1 and ∆H2 which take care of the potential created by all other atoms. H = H1 + ∆H1 = H2 + ∆H2. The Bloch wave function is ik·R Ψ(r) = ΣRe φ(r − R) (1.6)

where the wave function φ is a linear combination of the localized wave functions φ(r − r1,2)

φ(r) = c1φ(r − r1) + c2φ(r − r2) (1.7)

2 2 Here c1 and c2 are constants and c1 + c2 = 1. The eigenenergies can be obtained by solving the Shr¨odinger equation

HΨ(R) = E(k)Ψ(R) (1.8)

3 Projecting this equation into φ(r − r1) and φ(r − r2)

< φ(r − r1)|H|Ψ(r) >= E(k) < φ(r − r1)|Ψ(r) > (1.9)

< φ(r − r2)|H|Ψ(r) >= E(k) < φ(r − r2)|Ψ(r) > (1.10)

Plugging in Eqn. (1.6) to Eqn. (1.10), we obtain

φ(r − r1)|H|Ψ(r) >

=< φ(r − r1)|H1 + ∆H1|Ψ(r) >

= E1 < φ(r − r1)|Ψ(r) > + < φ(r − r1)|∆H1|Ψ(r) > (1.11)

0 Applying the orthogonality of the Bloch functions < φ(r − R)|φ(r − R ) >= δR,R0 ,

< φ(r − r1)|Ψ(r) > ik·R = ΣRe < φ(r − r1)|c1φ(r − r1 − R) + c2φ(r − r2 − R) >

= c1 (1.12)

Considering only the onsite energy and the overlap between nearest neighbors (e.g. atom A and the three B atoms surrounding it in Fig. 1.2, R = 0, a1, a2), the second term in Eqn. 1.11 can be simpliﬁed as

< φ(r − r1)|∆H1|Ψ(r) >

= c1 < φ(r − r1)|∆H1|φ(r − r1) > ik·a1 ik·a2 +c2(1 + e + e ) < φ(r − r1)|∆H1|φ(r − r2) >

= c1β + c2γf(k) (1.13) where β, γ and f(k) are deﬁned as follows:

β ≡< φ(r − r1)|∆H1|φ(r − r1) >

γ ≡< φ(r − r1)|∆H1|φ(r − r2) > f(k) ≡ 1 + eik·a1 + eik·a2 (1.14)

Here β is a correction to the onsite energy. γ is the next nearest neighbor hoping integral. Eqn. 1.11 can be simpliﬁed as

c1β + c2γf(k) = c1E(k) (1.15)

Similarly Eqn. 1.10 can be simpliﬁed as

∗ c2β + c1γf (k) = c2E(k) (1.16)

Solving the secular equation β γf(k) c1 c1 ∗ = E(k) (1.17) γf (k) β c2 c2 and E(k) is s √ k a 3k a k a E(k) = β ± γ|f(k)| = β ± γ 1 + 4cos( x )cos( y ) + 4cos2( x ) (1.18) 2 2 2 √ Here a = 3a0. β is a small correction to the overall energy. Fig. 1.4(a) shows the π band dispersion of graphene. The valence and conduction bands touch only at the six corners of the BZ. Since each carbon atom contributes one electron, the conduction band is completely ﬁlled up to the Fermi level, where the valence and conduction bands merge. Because of this, graphene is known as a semi-metal or zero-gap semiconductor.

4 Figure 1.4. The band structure of graphene and the schematic cartoon.

1.3 Massless Dirac fermions and massive quasiparticles

The low energy electronic structure of graphene near the corners of the BZ is extremely important, since it determines the transport properties as well as various exotic behaviors observed in graphene. To obtain the low energy electronic structure near the zone corners K(K0), we expand f(k) to the ﬁrst order around the Brillouin zone corners K, k = K + κ: √ √ iκ 3a κxa 2π 3a f(k) = 1 + 2e y 2 cos( + ) = − (κ − iκ ) (1.19) 2 3 2 x y

Eqn. (1.17) reduces to √ 3a 0 κ − iκ c c − γ x y 1 = E(k) 1 . (1.20) 2 κx + iκy 0 c2 c2

Thus the Hamiltonian is √ 3a H = − γ(κ σ + κ σ ) = − v κ · σ (1.21) 2 x x y y ~ F √ 3aγ where σx and σy are the Pauli matrices, vF = . The dispersion relation in Eqn. 1.18 becomes E(k) = √ 2~ 3a ± 2 γ|k| = ±~vF k. This is analogous to the dispersion relation formulated in Einstein’s relativistic theory E = ±p(m2c4 + c2p2), with zero eﬀective mass m=0 and the speed of light c replaced by the Fermi velocity vF , which is ≈ 300 times smaller. This suggests that electrons in graphene are governed by a two-dimensional version of the relativistic quantum theory introduced by Dirac. Because of this, the low energy quasiparticles in graphene are also described as ‘Dirac Fermions’ and the points where the valence and conduction bands merge are called ‘Dirac points’.

Fig. 1.4(b) shows a schematic drawing of the low energy dispersions near EF . The peculiar linear dispersion for the valence and conduction bands results in many intriguing properties. First of all, the ‘Fermi surface’ in graphene contains only six points, rather than a true Fermi surface. Because of the finite number of points at the Fermi surface, the density of states is vanishingly small at EF . Moreover, the linear dispersion also gives rise to many properties in graphene that are different from those in two dimensional semiconductors 8. In semiconductors, the typical band dispersions show a quadratic behavior and the behavior of electrons can be described by non-relativistic theory formulated in Schrödinger

5 Figure 1.5. Schematic drawing of the dispersions near EF for massless Dirac fermions (a) and massive quasiparticles in conventional condensed matter systems (b).

equation. An electron in these systems is modeled as a quasi-particle with a finite (‘massive’ compared to −1 ∗ 2 d2E zero) effective mass m = ~ ( dk2 ) . The mass is usually different from the non-interacting electron mass and this mass renormalization is used to take into account the effects of electron-electron interaction and electron-phonon interaction. The velocity of electrons v = 1 ∂E changes a a function of electron binding ~ ∂k energy. The electron in graphene, however, shows a linear dispersion relation E(k) = ±~vF k and travels with a constant velocity.

1.4 Electronic structure of inﬁnite layer graphite

Graphite is formed by infinite layers of graphene usually stacked in an ABAB sequence. Because of the periodicity along the c axis direction, the BZ of graphite has a third dimension kz (Fig. 1.4) compared to graphene. In graphite, the in-plane C-C bonds are strong due to the small in-plane lattice spacing of 1.42 A,˚ while the out-of-plane coupling is much weaker due to the large interlayer spacing of 3.37 A.˚ Thus the interlayer interaction in graphite is dominated by weak van der Waals force. These structural properties impart quasi-two dimensionality to the electronic structure of graphite, while the inter-layer interaction modifies the electronic structure at certain kz values. The electronic structure of graphite has been calculated by tight binding model or ab initio 9,10,11,12,13. Similar to graphene, the strong in-plane σ bonds formed by 2s, 2px and 2py orbitals result in 3 σ bands at high binding energy. The out-of-plane π bonding formed by 2pz orbitals is much weaker, and the π bands are at much lower binding energy. Among all these bands, only the π bands cross the Fermi energy EF at the corners of the hexagonal Brillouin zone (BZ). These low energy π bands play the most important role in determining the electronic and transport properties of graphite. Figure 1.4 shows the theoretical band structure of graphite. We note that different from a perfect two dimensional graphene, the weak interlayer interaction in graphite results in a splitting of the π bands in the kz=0 plane (K-Γ-M-K). On the other ∗ ∗ hand, the π bands are degenerate in the plane of kz=0.5 c (H-L-A-H), where c is the reciprocal lattice constant along the stacking direction, and the electronic structure in this kz plane strongly resembles that of graphene. Therefore despite the splitting of the π bands, the overall dispersion of the π band still bears a

6 strong similarity to that of graphene. It is fundamental to understand how the electronic structure evolves from graphene to graphite as the number of layers increases.

Figure 1.6. Brillouin zone and electronic structure of bulk graphite.

7 Chapter 2

Experimental techniques

2.1 Angle resolved photoemission spectroscopy (ARPES)

2.1.1 ARPES and the photoelectric eﬀect

Photoemission describes the ejection of electrons from a metal when a beam of light is shine on the clean surface. It was discovered by Hertz in 1887 and explained by Einstein in 1905. The photoelectric eﬀect was very puzzling when it was ﬁrst discovered, with a few mysterious key points. First, the electrons were emitted immediately. Second, increasing the intensity of the light increases the number of photoelectrons, but not their maximum kinetic energy. Third, red light will not cause ejection of electrons, no matter how strong it is. Fourth, a weak violet light will eject only a small amount of electrons, but their maximum kinetic energies are larger than those ejected by intense light of longer wavelength. These results cannot be explained without invoking the quantum nature of light, the wave-particle duality. Einstein postulated that light is quantized and that each quanta photon carried energy hν. The maximum kinetic energy of the photoelectrons is

Emax = hν − φ, (2.1) where φ (typically 4 to 5 eV for most materials) is the work function of the materials, i.e. the minimum energy needed to excite an electron into the vacuum. In Einstein’s theory, light is not just a particle and not just a wave: it can be one or the other, depending on how it is measured. For his explanation of photoelectric effect, Einstein was awarded the Nobel prize in 1921. It was later realized that photoelectric effect could provide useful information about the electronic states inside the materials. As a result of energy conservation, the energy distribution of photoelectrons can provide information about the density of states in the material studied 14. Started from 1960’s, it was realized that momentum dependent band structure could be mapped from the angle and energy dependence of the photoemission spectra and the first angle resolved photoemission was demonstrated in 1974 15,16. This technique, today known as angle resolved photoemission spectroscopy (ARPES) is among the most powerful spectroscopic techniques as it provides direct information on the electronic binding energy and the crystal momentum of solids. A schematic drawing of the experimental setup used in ARPES experiment and typical data set is shown in Fig. 2.1. Information on Fermi surface topology and band structure can be directly extracted from the peak position of the photoemitted electrons as a function of energy and angle of emission (momentum). As a result of the uncertainty principle, the width of the ARPES peak can also give

8 information about the lifetime of electrons and many body interaction 17. The rapid improvements in the past few decades have made ARPES a powerful technique to reveal important information on the electronic structure and many body interaction in various materials 18.

Figure 2.1. (a) Schematics of ARPRES experiments (courtesy of J. Fink) and (b) typical ARPES data from graphite.

The basic principle behind ARPES is the energy and momentum conservation. The kinetic energy and momentum of the photoelectrons are related to the binding energy EB and crystal momentum ~k inside the solid by the conservation laws.

Ekin = hν − φ − |EB| (2.2) p pk = ~kk = 2mEkin · sin θ (2.3)

Here the in-plane momentum kk is conserved because of the translational symmetry of the single crystal and the negligible wave vector of the photons. For a two dimensional solid where there is no dispersion along z-axis (sample normal direction), the electronic dispersion is completely determined by kk. For a three dimensional sample, the electronic structure also depends on kz and the full momentum information including the out-of-plane momentum kz is important. The extraction of kz, however, is more complicate because of the lack of translational symmetry at the interface between the sample and the vacuum. Thus kz cannot be extracted directly without a priori assumption. However, kz can be extracted by modeling the final state of photoelectrons, e.g. the ‘free-electron model’ or a calculated band structure 19,20. The simplest and most frequently used assumption is the free-electron approximation, where the final state is approximated as a free electron state. This is only an approximation as the photoemission process takes place in the presence of a crystal potential. The higher the photon energy is, the better this approximation is, because the effect of the crystal potential becomes weaker for higher kinetic energy. Though the free-electron approximation is a simple model, it has been a useful and accurate method to extract the kz values. In the free-electron model, the dispersion of the final state is assumed to be that of a free electron in a potential Vin: 2 E = ~ (k2 + k2) − V , (2.4) k 2m∗ k z in ∗ where m is the effective mass, which is usually taken as the free-electron mass m. Vin (so called ‘inner potential’) is a parameter that can be determined from the periodicity and the symmetry in the measured

9 dispersion E(kz). Thus kz values can be determined by r 2m 2 kz = (hν − φ − EB + Vin) − k (2.5) ~2 k

Therefore, diﬀerent kz values can be accessed by changing the photon energy. In particular, at normal emission (kk=0), kz is related to the kinetic energy Ek by

p 2 kz = 2m(Ek + Vin)/~ . (2.6)

2.1.2 Three step model and sudden approximation

N In the photoemission process, the initial state is the N electron ground state ψi and the ﬁnal state is N the N-1 electron state ψf . The intensity of the photoelectrons is proportional to the transition probability N N wfi for an excitation between the N-electron ground state ψi and one of the possible ﬁnal states ψf , which can be approximated by the Fermi’s golden rule

2π N N 2 N N I ∝ wfi = | < ψf |Hint|ψi > | δ(Ef − Ei − hν) (2.7) ~ N N−1 k N N−1 where Ei = Ei −EB and Ef = Ef +Ekin are the initial and ﬁnal state energies of the N-particle sys- k tem, and EB is the binding energy of electron that is photoemitted with kinetic energy Ekin and momentum k. The interaction with the photon is treated as a perturbation given by e e H = − (A · p + p · A) = − A · p (2.8) int 2mc mc where p is the electronic momentum operator and A is the electromagnetic vector potential. Here the commutator relation [p, A] = −i~∇ · A and the dipole approximation ∇ · A = 0 are used. A rigorous approach to the photoemission process is the one-step model in which photon adsorption, electron removal and electron detection are all included in the Hamiltonian and treated as a single coherent process. However, due to the complexity of the one-step model, photoemission process are usually discussed within the three step model. In this model, the photoemission process is divided into three discrete steps.

1. Photoexcitation of electrons in the solids. 2. Propagation of the photoelectrons to the surface. 3. Escape of photoelectrons from the solid into the vacuum.

The total photoemission intensity is given by the product of three independent terms: the total probability of the optical transition, the scattering probability for the photoelectrons and the transmission probability through the surface potential. The first step contains all information about the intrinsic electronic structure. In evaluating the photoelectron intensity in terms of the transition probability, the sudden approximation is usually applied. In this approximation, an electron removal is assumed to be sudden and the effective potential of the system changes discontinuously at that instant. Thus the initial and the final states of the N electron system can be separated and the wave functions can be factorized into the photoelectron and the remaining N-1 electrons. The final state is

N k N−1 ψf = Aφf ψf (2.9)

k where A is an antisymmetric operator so that the N-electron wave function satisﬁes the Pauli principle, φf is N−1 the wave function of the photoelectron with momentum k, ψf is the ﬁnal state wave function of the N-1

10 N−1 electron system left behind, which can be chosen as an excited state with eigenfunction ψm . The initial state is

N k N−1 ψi = Aφi ψi (2.10) k N−1 N where φi is the one electron orbital and ψi is ψi = ckψi , ck is the annihilation operator of an electron with momentum k is the initial N-1 particle. With the above approximation, the interacting matrix becomes

k 2 δ N−1 N I(k,Ekin) = Σf,iwf,i ∝ Σf,i|Mf,i| Σm|cm,i| (Ekin + Em − Ei − hν) (2.12)

2 N−1 N−1 2 where |cm,i| = | < ψm |ψi > | is the probability that the removal of an electron from state i will leave 2 the (N-1)-particle system in the excited state m. In the non-interacting case, |cm,i| will be unity for only particular m and zero for all the others, thus the ARPES spectra will be given by a delta function at the N−1 electron binding energy (see Fig. 2.2). In strongly correlated systems, ψi will overlap with many of the N−1 18 eigenstates ψm and the ARPES spectra will not show only single delta functions .

Figure 2.2. Spectral function for noninteracting (left panel) and interacting (right panel) systems, from Damascelli et al 18.

2.1.3 Single particle spectral function description of ARPES

In strongly correlated systems, the interpretation of the ARPES data is based on the Greens’s function formalism. ARPES intensity I(k,E) can be written as

I(k,E) = I0(k, ν, A)f(E)A(k,E) (2.13) where f(E) is the Fermi function and A(k,E) is the single particle spectral function. The spectral function is the imaginary part of the Green’s function G(k,E) 1 A(k,E) = − ImG(k,E) (2.14) π

0 For non-interaction systems with one electron energy Ek, the Green’s function is 1 G0(k,E) = 0 (2.15) E − Ek − i

11 where is inﬁnitely small. According to Equation(2.14), the spectral function A0(k,E)is a delta function 0 A0(k,E)=δ(E-Ek) for a noninteracting system. In an interaction system, the electron-electron correlation 0 00 is expressed in terms of the electron self energy Σ(k,E)=Σ (k,E)+iΣ (k,E). The real part contains information about the energy renormalization and the imaginary part gives information about the lifetime of an 0 electron with energy Ek and momentum k propagating in the many body systems. In this case the Green’s function and the spectral function are

1 G(k,E) = 0 (2.16) E − Ek − Σ(k,E)

00 1 Σ (k,E) A(k,E) = − 0 0 2 00 2 (2.17) π (E − Ek − Σ (k,E)) + (Σ (k,E))

2.1.4 Energy distribution curves (EDCs) and momentum distribution curves (MDCs)

Figure 2.3. Typical two dimensional ARPES data and the analysis of EDCs and MDCs.

In ARPES, a typical scan shows two dimensional intensity as a function of binding energy and crystal momentum, see Fig. 2.3. There are two ways to analyse the data. One way is to fix momentum at a particular value, and study intensity as a function of energy - energy distribution curves (EDCs). Another way is to fix energy at a particular value, and study intensity as a function of momentum - momentum distribution curves (MDCs). Each of these two methods has its own advantages and disadvantages, and the analysis of these two is complimentary to each other. EDC gives a global overview of the spectral function and the line shape. It provides information about whether a well-defined quasiparticle peak is present or not, whether the spectral function can be interpreted in the Fermi liquid theory or not 21. Moreover, subtle features such as small peaks at low binding energy,

12 presence of energy gaps 22 and more complicate lineshape (e.g. peak-dip-hump structure) due to self energy interaction can be observed clearly 18. However, fitting of the EDCs is generally difficult for various reasons 23. First, from Eqn. 2.13, EDC is affected by the Fermi function, which shows a step function at the Fermi energy. Second, the EDC contains a background which is not included in A(k, ω) and there is no well-established method to distinguish and subtract this component. This problem is less severe in the laser-based ARPES 24. Also, the line shape of the EDCs are usually complex and there is no unique theory to model them. The MDC analysis is relatively simple. MDC eliminates the effect of the Fermi function and the energy dependent background. More importantly, the lineshape of MDC can usually be fitted with a Lorentzian. As discussed above, the measured intensity in ARPES is proportional to the single particle spectra function 0 A(k,E). From Eqn.2.17, A(k,E) is a δ function with a pole at E − Ek = 0 when there is no interaction, i.e. 0 0 Σ(k,E) = 0. In the presence of interaction, A(k,E) is maximum at E − Ek − Σ (k,E) = 0. In the region 0 00 where Ek = vF k, if we assume that Σ (k,E) is independent of k, then A(k,E) is a Lorentzian function. The peak positions from the MDC fit gives direct information about the renormalized band dispersion and the MDC width is directly related to the imaginary part of the self energy. With some modeling of the bare 0 25 electron band dispersion Ek, both the real part and imaginary part of the self energy can be extracted . Because of the advantage of easy modeling, MDC analysis has been widely used in studying many body interactions, e.g. electron-phonon interaction in metallic surfaces 26,27 and strongly correlated materials 17,28. One caveat of the MDC analysis is that it does not distinguish whether there are real quasiparticles or not. For example, non-dispersive MDC peaks can still be observed inside the gap region 29. Thus seeing a peak in the MDC does not guarantee existence of real quasiparticles.

2.1.5 Electron mean free path and the probing depth of ARPES

Figure 2.4. The universal inelastic mean free path of electrons with kinetic energy from 2 to 2000eV, from H¨ufner 19.

The probability for electrons to escape from the solid into the vacuum is strongly related to the electron mean free path. Fig. 2.4 shows the universal electron mean free path λ as a function of the electron kinetic energy for a few metals. Between a few eV and 2000 eV, the mean free path is only 5 A˚ to 20 A,˚ which means only electrons on a few A˚ near the surface will be able to escape from the solids and be detected in ARPES. Therefore, to obtain information about the bulk properties of the solid, ARPES has to be done on atomically clean surfaces, which is usually achieved by cleaving the sample to expose a fresh surface or by annealing the sample to get rid of the gas adsorbed. In addition, UHV conditions are required to prevent contamination from the adsorbed contaminants. For example, with a sticking coeﬃcient of 1, which means that each molecule or atom striking on a surface also sticks to it, it only requires an exposure of 2.5

13 Langmuirs (1L=10−6 torr·s) to cover the surface with 1 monolayer of molecules. At a pressure of 10−9 torr, it only takes 40 minutes to adsorb one monolayer of molecules. Thus vacua of even better than 10−10 torr is usually needed to get spectra from clean surfaces.

2.1.6 State of the art ARPES

In the past two decades, ARPES has undergone rapid improvements in energy and momentum resolution as well as efficiency in data acquisition, with the availability of the high flux from the second and third generation synchrotron facilities and the development of the two dimensional electron spectrometer. Fig. 2.5 shows the improvement of the resolution in the past three decades, demonstrated by the spectra taken from Ag(111) surface 30. To date, typical ARPES experiment is done with a resolution of 10-30 meV energy resolution at the synchrotron, while better resolution can be possibly achieved at the price of lower photon flux.

Figure 2.5. The improvement of the ARPES resolution in the past three decades, from Reinert et al 30.

In recent years, the development in laser source has provided exciting opportunities to use laser as an alternative light source for ARPES. The recent demonstration of photoemission experiment using the fourth harmonic generation of a laser source also created a lot of excitement in photoemission 31,24, since it provides a possibility of using the convenient table top laser for ARPES with a few advantages. First, the energy resolution achievable is as good as 0.36 meV, order of magnitude better that the state of art synchrotron based experiment. Second, according to Fig. 2.4, the electron mean free path at a few eV photon energy is signiﬁcantly increased, which means that ARPES with photons of a few eV is more bulk sensitive. Third, the light from a laser source is highly coherent and the laser pulse provides the possibility of doing time-resolved experiments to study the electronic dynamics. There are also certain limitations, for example, the relatively low photon energy greatly limits the momentum regions that can be probed compared to the synchrotron light source.

14 Figure 2.6. The setup of the 6.994 eV laser and the demonstrated energy resolution of 0.36 eV, from Kiss et al 31.

2.2 X-ray photoelectron spectroscopy (XPS)

X-ray photoelectron spectroscopy (XPS), also known as electron spectroscopy for chemical analysis (ESCA), is a surface sensitive technique that measures photoelectrons emitted with X-ray (50-2000eV) to obtain quantitative analysis of the surface 32. Different from ARPES where ultraviolet light (less than 100 eV) is used, XPS radiation source is typically the Kalpha X-ray radiation from magnesium (1253.6 eV) or aluminum (1486.6 eV). Another difference between XPS and ARPES is that in XPS the photon is absorbed by an atom in a molecule or on a surface leading to ionization and emission of a core electron, while in UPS the photon interacts with valence electrons leading to ionization by removal of the valence electrons. The high energy resolution XPS was developed in 1960s by the several significant improvements in the equipment by the group of K. M. Siegbahn. In recognition of his extensive efforts to develop XPS into a useful analytical tool, Siegbahn was awarded the Nobel Prize for physics in 1981. A typical XPS spectrum records the number of photoelectrons detected as a function of the binding energy. The energy of the photoelectrons is characteristic of the element and the configuration of electrons inside the atom, e.g. 1s, 2s, 2p, 3s etc. The different electron densities in the vicinity of the atoms can also result in a shift of the energy of the characteristic peaks, which can provide information about the particular chemical environment of the atoms on the surface. The intensity of the characteristic peaks is directly related to the amount of element within the area or volume probed. Thus XPS can yield quantitative information about the elemental composition of the surface, the empirical formula of pure materials, chemical and electronic state of the elements in the surface and the thickness of thin film on a different substrate within the probing depth (≈10nm of the surface). The quantitative interpretation of XPS intensities involves a modeling for predicting the magnitudes from the properties of the excitation source, the sample, the electron analyzer and the detection system 32. In general the XPS peak intensity dNk associated with the atomic subshell k is determined by the products of the following: X-ray flux - I0, number of the atoms - ρdxdydz, differential cross section for subshell k dσk -( dΩ ), acceptance solid angle of electron analyzer at (x,y,z) - Ω(Ekin, x, y, z), probability of electrons to escape from the sample −exp[−l/Λe(Ekin)], instrumental detection efficiency - D0(Ekin): dσ dN = I · ρdxdydz · k · Ω(E , x, y, z) · exp[−l/Λ (E )] · D (E ) (2.18) k 0 dΩ kin e kin 0 kin

where l is the traveling distance for electrons to escape from the sample surface into the vacuum, Λe is the

15 mean free path, Ekin is the kinetic energy. The dependence of dNk on the path length l can be used to obtain a quantitative analysis of the thickness of the thin ﬁlm on a substrate.

For a semi-infinite substrate with uniform film of thickness t and effective sample area A0(Ek), the intensity for peak k from substrate with Ekin = Ek at angle θ is

0 Nk(θ) = I0Ω0(Ek)A0(Ek)D0(Ek)ρdσk/dΩΛe(Ek)exp(−t/Λe(Ek)sinθ) (2.19)

The intensity for peak l from the thin ﬁlm on the top with Ekin = El is

0 0 Nl(θ) = I0Ω0(El)A0(El)D0(El)ρ dσl/dΩΛe(El)(1 − exp(−t/Λe(El)sinθ)) (2.20)

0 where Λe(Ek) and Λe(Ek) are the attenuation lengths in the substrate and the thin film respectively, ρ and ρ0 are the atomic density in the substrate and in the thin film respectively. Therefore the intensity ratio of the peaks from the thin film and the substrate is

0 Nl(θ) Ω0(El)A0(El)D0(El)ρ dσl/dΩΛe(El) 0 0 = (1 − exp(−t/Λe(El)sinθ))exp(t/Λe(Ek)sinθ) (2.21) Nk(θ) Ω0(Ek)A0(Ek)D0(Ek)ρdσk/dΩΛe(Ek)

The intensity ratio of the signals from the thin film to the substrate strongly depends on the film thickness, which can be used to extract information about the thin film thickness. Also, the ratio of peaks from the film and the substrate increases strongly as θ decreases, which suggests that a general method for increasing surface sensitivity is to use grazing angles.

2.3 Low energy electron microscopy (LEEM)

Low energy electron microscopy (LEEM) is a surface sensitive imaging technique which collects the backscattered electrons from a low energy electron beam with energies of 1-100 eV 33,34. The low energy of the electrons used distinguishes LEEM from other electron microscopy, e.g. transmission electron microscopy (TEM) where the typical electron energy is 100,000eV. The advantage of using low energy electrons is that a large fraction of electrons will be backscattered elastically and that the backscattered electrons in this low energy range are very sensitive to the physical and chemical properties of the top few layers on the surface. Thus depending on the energy of the incident electron beam, subtle differences in local atomic structure or composition can result in dramatic contrast in LEEM images. Defects such as atomic height steps or dislocations can be imaged in LEEM with a good contrast. When spin-polarized electron beam is used, the exchange scattering between the electrons and a ferromagnetic material can result in spin dependent electron reflectivity, allowing the study of magnetism. Another advantage of LEEM over other microscopy is that the images can be taken instantly, thus allowing time resolved study. These characteristics make LEEM an excellent tool for imaging the surface topography and studying the surface dynamics as well as magnetism. LEEM can also be used to study the quantum-well states on ultrathin samples on a substrate 35. When spatially coherent low energy electrons are injected on the sample, part of the electrons are reflected elastically from the surface, and part of the electrons will penetrate through the thin film and be reflected at the film- substrate interface. Thus the total reflected electron beam is composed of two coherently super-imposed beams, which can form interference pattern. The number of quantum oscillations can be used to determine the thin film thickness, which will be discussed in Chapter 3.

16 Chapter 3

Growth and characterization of

epitaxial graphene

This chapter describes the growth and characterization of epitaxial graphene using various experimental techniques. The samples are grown by thermal decomposition of SiC. The surface topography is studied with scanning electron microscopy (SEM) and low energy electron microscopy (LEEM). The sample thickness is determined by XPS, ARPES and LEEM.

3.1 Growth of epitaxial graphene

Graphene films were produced on the Si-terminated (0001) face of an n-type 6H-SiC single crystalline wafer purchased from Cree, Inc. The wafer was cut to pieces of a few mm in size by using a diamond saw, and cleaned with acetone and isopropyl alcohol in an ultrasonic bath. The clean wafers were then mounted with Ta-foil clips to a Mo sample holder and entered into an ultrahigh-vacuum preparation chamber (base pressure 1×10−10 Torr). Before growing graphene layers, the wafer was cleaned in situ by annealing up to 850 ◦C under silicon flux for 20 to 30 minutes. The silicon flux was produced by a silicon ingot heated by electron bombardment to approximately 1200 ◦C and was calibrated with a quartz crystal monitor (deposition rate about 3 A/min).˚ This procedure removes native surface oxides by the formation of volatile SiO, which sublime at this temperature 36. The growth process was monitored by LEED patterns, which were obtained at room temperature. The data reported below are mainly from three samples, henceforth referred to as samples A, B, and C. These samples are labeled in order of increasing thickness, a parameter controlled primarily by variation in the annealing temperature 37. Fig. 3.1 shows selected LEED patterns obtained at different stages during the growth of sample A. After the initial cleaning procedure under Si flux, the LEED pattern (not shown) displayed a 3×3 reconstruction with respect to the SiC substrate, as has been well documented in the literature 36,38,39. A subsequent 5 ◦ minute annealing at ≈ 1000 C in the absence of Si flux gives rise to the sharp pattern shown√ in panel√ (a), corresponding to the 1×1 spots of SiC. Further annealing for 5 min at ≈ 1100◦C produces the ( 3× 3)R30 reconstruction shown in panel (b), attributed to a structural model comprised of 1/3 layer of Si adatoms in threefold symmetric sites on top of the outermost SiC bilayer 40,41. Finally, annealing for 10 min at ≈ 1250◦C

17 a b

c d

e f graphite

SiC

graphite

Figure 3.1. (a-d) LEED patterns with a primary energy of 180eV, obtained at four diﬀerent stages ◦ during the growth of sample A. (a) 1×1 spots of SiC,√ after√ a 5 min anneal around 1000 C followed by the initial cleaning√ procedure√ under Si ﬂux. (b) ( 3 × 3)R30 reconstruction, after 5 min around√ ◦ ◦ 1100√ C. (c) (6 3 × 6 3)R30 reconstruction, after 10 min around 1200 C. (d) Sharper(6 3 × 6 3)R30 pattern, after 4 min around 1250 ◦C. (e) and (f) LEED patterns with a primary energy of 130eV taken at the same stages as (c) and (d), respectively. In (f), the appearance of the 1×1 spots of graphite, located slightly farther out from the center relative to the spots observed at similar positions in (b, c, and e), indicates that a thin graphene overlayer has been formed in this last step.

18 100 nm 500 nm

a 50 nm b

Figure 3.2. SEM images recorded on the surface of a sample grown in the same conditions as sample A. (a) Typical region of the surface, showing a pattern with a length scale on the order of hundreds of nanometers, and a finer pattern with a length scale on the order of 10 nm. (b) Region marked by a cluster of unidentified structures characterized by six-sided geometry, reflecting the underlying hexagonal lattice.

√ √ produces the complex (6 3×6 3)R30 pattern shown√ in panel√ (c). Notably, a slight increase of the annealing temperature gives rise to the more well developed (6 3 × 6 3)R30 pattern shown in panel (d). Panels (e) and (f) show LEED patterns obtained at a lower electron energy, under the same conditions as panels (c) and (d), respectively. Graphite diﬀraction spots clearly observed only in panel (f) indicate that a thin graphene overlayer is formed in the last step.

3.2 Characterization of epitaxial graphene

3.2.1 Scanning electron microscopy (SEM)

Fig. 3.2 presents SEM images of a graphene sample. The image shown in panel (a) exhibits two patterns, one in a large length scale, corresponding to hundreds of nm, and another in a small length scale, on the order of 10 nm. The small length scale coincides with a typical terrace size observed by scanning tunneling spectroscopy 42, while the large length scale may correspond to the single-crystal grain size. The image shown in panel (b) exhibits a similar pattern and, in addition, unidentiﬁed structures - white specks each enclosed by a large black region. Spread out widely over the sample surface, the white specks and enclosing black regions show facets reﬂecting the underlying hexagonal lattice. An investigation into potential causes for the formation of such structures is needed.

3.2.2 X-ray photoemission spectroscopy (XPS)

XPS measurements were carried out using Mg Kα radiation and normal emission geometry at 300K. Fig. 3.3 shows XPS spectra of the C 1s and Si 2p core levels and their line shape ﬁts. For all samples, the Si 2p spectrum consists of a single peak located at ≈ 101.6 eV, attributed to the SiC bulk. The C 1s spectrum consists of three components, labeled as X, G, and S. In the analysis that follows we will make use of the latter two, which are assigned to the graphite overlayer (G) and the SiC bulk (S). These assignments are in agreement with previous work 43,44. In particular, the small chemical shift of peak S relative to pure SiC by about 0.4 eV has been noted before, and was attributed to a Fermi level pinning associated with surface

19 Figure 3.3. XPS results showing data (connected symbols) and their ﬁts (solid lines) for samples A, B, and C. Individual Voigt functions and Shirley background are shown as lines as well. In the C 1s spectra, peaks G and S are identiﬁed with the graphite overlayer and SiC bulk. In each row, the two panels are plotted using a common intensity scale.

metalization 44,43,45. Peak X, broad and weak, is of a less clear origin. Previously, a similar feature was interpreted as arising from C-C bonding in a Si-depleted interfacial region between graphite and SiC 43,44. Using the relative intensities of three peaks (Si 2p, G, and S), we obtained information on the thickness of the graphite overlayer for each of the three samples studied. Assuming the simple model of a semi-inﬁnite substrate with uniform overlayer of thickness t, one ﬁnds that the ratio of the intensity NG of the graphite 32 peak to the intensity NR of a reference peak (either Si 2p or S) is related to t by the equation ,

0 0 0 NG T (EG)ρ CGΛ (EG)[1 − exp(−t/Λ (EG))] = 0 × F NR T (ER)ρCRΛ(ER) exp(−t/Λ (ER)) where E is the kinetic energy of photoelectrons associated with a given peak, T is the transmission function of the analyzer, C is the differential cross section (dσ/dΩ), ρ is the atomic density, Λ is the inelastic mean free path, F is a geometrical correction factor due to photoelectron diffraction, and the superscript 0 indicates quantities referred to the graphene overlayer as opposed to the SiC bulk. Mean free paths were estimated using the so-called TPP-2M formula 46 1. Differential cross sections were calculated using tabulated values 47

1Tougaard, S., QUASES-IMFP-TPP2M program at http://www.quases.com

20 Reference Sample A Sample B Sample C Si 2p 1.2 ± 0.6 2.4 ± 1.2 2.9 ± 1.5 C 1s (S) 1.5 ± 0.5 2.8 ± 0.8 3.4 ± 1.0

Table 3.1. Sample thickness (ML) deduced from XPS analysis, using two different reference peaks from the substrate. for total cross sections and asymmetry parameters. The factor F was computed using x-ray photoelectron diffraction data in the literature collected on similar samples 43. Solving the above equation for thickness t and dividing by an interlayer spacing of 3.35 A˚ gives the results summarized in Table 1. The excellent agreement between values obtained using two different reference peaks shows that this method works well. Note that using the C 1s peak (S) as a reference is obviously the preferred method, since in this case several factors (C, T , and F ) simply drop out of the calculation to within a good approximation. Indeed, the systematic deviation up to ≈ 20 % when the Si 2p peak is used as a reference can be attributed to the uncertainty of these factors, most probably that of C. In any case, the greatest degree of uncertainty in these analyses is introduced by the Λ values, which are generally considered to be accurate to roughly 20 %. The uncertainty reported in Table 1 resulted from consideration of possible errors in all parameters, especially Λ.

3.2.3 Angle-resolved photoemission spectroscopy - ARPES

Figure 3.4. (a-c) Dispersions of the π bands from single layer, bilayer graphene to trilayer graphene. (d-f) Dispersions near the K point from single layer graphene to trilayer graphene. From Partoens et al 13.

The most direct way to determine the sample thickness is by doing ARPES. From band structure calculation 13, the number of π bands increases from 1 to 3 from single layer graphene to trilayer graphene, see Fig. 3.4. Thus by counting the number of the π bands, the graphene sample thickness can be determined.

21 Figure 3.5. Intensity map at -1 eV as a function of kk and kz.

Figure 3.6. Measured dispersions from single layer graphene to four layer graphene, from Ohta et al 48.

This can also be achieved by measuring the dispersions along the out-of-plane direction kz. For a single layer graphene, there is no dispersion along kz direction (see Fig. 3.5). For bilayer graphene or trilayer graphene, the intensity oscillates periodically between the diﬀerent π bands (see Fig. 3.6) 48. When the number of layers is larger than 10, the electronic structure is hardly distinguishable from that of graphite.

3.2.4 Low energy electron microscopy (LEEM)

LEEM is a powerful technique to study the surface topography and the dynamics of the growth process. Fig. 3.7 shows the LEEM images in SiC before and after annealing at 900C under Si flux. After annealing, the SiC surface shows less defects and a smoother surface. Further annealing without Si flux leads to growth of the buffer layer 49,50 - a carbon layer with the same σ bands as graphene without the low energy π band - and eventually the growth of graphene. Fig. 3.8 shows the surface topography of the as-grown graphene. Clear color contrast between white, gray to black (labeled by black, blue and red colors in Fig. 3.8) can be clearly observed. The temperature at which these regions form increases from white, gray to black regions. Scans of the reflected electron intensity as a function of electron energy in these three characteristic regions show that the gray and black regions have one and two quantum oscillations, which correspond to single layer and bilayer graphene respectively.

22 Figure 3.7. LEEM images taken at an electron energy of 4.2 eV before and after annealing the SiC substrate under Si ﬂux with a ﬁeld of view of 5µm.

Figure 3.8. LEEM images taken at an electron energy of 6.6 eV with a 3µm ﬁeld of view and the energy scans for the buﬀer layer, 1ML and 2ML graphene.

Since the white regions are observed at lower temperature before the single layer graphene (gray region) is formed, these are attributed to the buﬀer layer.

23 Chapter 4

Gap opening in epitaxial graphene

Graphene is considered one of the most exciting materials in solid-state physics. Since its discovery in 2004 a variety of novel phenomena have been discovered and predicted, from an anomalous quantum Hall effect 51,52,53, ballistic transport 5, easy control of charge carriers by externally applying voltage 5, half metallicity 54, Kondo physics 55,56,57, superconductivity 58,59,60 etc. The secret to all of these fascinating phenomena lies in the unique nature of the electronic properties of graphene, where the low energy excitations resemble massless Dirac fermions. Although several experiments have shown results in agreement with the existence of Dirac fermions, so far direct experimental evidence has been limited. Here we present a detailed study of the electronic structure of graphene by using ARPES. How this evolves from single layer graphene to infinite layers graphite is also discussed. In general graphene can be obtained by rubbing a piece of graphite 5,61 on a SiO2 surface . This method however gives rise to very small samples of the order of few microns, which are difficult to investigate with ARPES, where the average beam size is of the order of 100 micron. The graphene samples we have investigated are epitaxial graphene grown by thermal decomposition of a SiC single crystal 62. As we will discuss below, the resulting graphene layer is tightly bounded with the substrate. The graphene-substrate interaction will modify the Dirac cone by inducing a gap between the conduction band and the valence band, making it an ideal candidate for next generation electronic devices.

4.1 Dirac fermions in epitaxial graphene

Figs. 4.1(a-c) show ARPES intensity maps at constant energies of EF , -0.4 eV and -1.2 eV. At EF (panel a), the intensity map shows a finite and almost circular contour centered around the K point. As the binding energy increases, the contour first decreases in size and becomes a point at -0.4 eV (panel b). Beyond -0.4 eV, the size of the contour expands again. This behavior is consistent with the presence of Dirac fermions, where a conical dispersion centered at the K point is expected. The Dirac point energy is shifted to 0.4 eV 48,63 below EF , which shows that the as-grown graphene is electron-doped . At higher binding energy (panel c) the high intensity region in the intensity map deviates from the circular shape. Similar trigonal distortions have been reported for graphite 64. Note that the intensity along the circular contour is not isotropic and is strongly suppressed on the left (first BZ) for energy above the Dirac point energy (panel a) and on the right (second BZ) for energy below the Dirac point energy (panel c). This is a well-known effect in graphite and is due to the ARPES dipole matrix element which suppresses or enhances the intensity in different BZs 65. Fig. 4.1(e) shows the overall dispersion of the π bands measured for a symmetric cut through the K point. Following the maximum in the intensity plot, two cones dispersing in opposite directions (one upward and another downward) can be clearly distinguished, in overall agreement with the presence of Dirac fermions.

24 Figure 4.1. (a-c) ARPES intensity maps taken at EF , -0.4 eV and -1.2 eV respectively on single layer graphene. The dotted line shows the Brillouin zone of graphene. (d) schematic drawing of the dispersion in single layer graphene and the relative energies for data shown in panels a-c. (e) Dispersion of single layer graphene measured along a high symmetric direction through the K point (see black line in the inset).

The Dirac point energy, defined as the midpoint between the valence and conduction bands at the K point, occurs at -0.4 eV since the sample is electron doped, as already discussed above. More importantly, in addition to this shift of the chemical potential and hence ED, deviations from a conical dispersion are observed in the vicinity of the Dirac point. The first, hardly visible on the energy scale of this figure, is at 66,67,68 approximately -0.2 eV below EF and is due to coupling to phonons . The second occurs near ED. In particular, we observe that the valence band and the conduction band do not merge at a single point at ED, but instead the top of the conduction bands and the bottom of the valence bands stop before EF and at the K point there is an intensity over an extended energy region between these two bands. Understanding the second anomaly near ED is the main focus of this chapter.

4.2 Departure from conical dispersion in single layer epitaxial

graphene

There are two possible scenarios that can account for the deviation from the conical dispersion near ED discussed in the previous section. These include many body interaction and the opening of a gap at the K point. As we will show below the latter is the most likely scenario to account for the anomaly.

Fig. 4.2 shows the detailed analysis of the anomaly near ED. We show the energy distribution curves (EDCs) in panel c and the momentum distribution curves (MDCs) in panel d for a cut taken at the K point (panel b). The EDCs show always two peaks with the minimum energy separation, being realized at K. The presence of two EDC peaks in the all momentum range, even at the K point, is a strong evidence in favor of a gap between the conduction and valence band. From the separation between the two EDC peaks we deduce

25 Figure 4.2. Observation of the gap opening in single layer graphene at the K point. (a) Structure of graphene in the real and momentum space. (b) ARPES intensity map taken along the black line in the inset of panel (a). The dispersions (black lines) are extracted from the EDC peak positions shown in panel (c). (c) EDCs taken near the K point from k0 to k12 as indicated at the bottom of panel (b). (d) MDCs from EF to -0.8 eV. The blue lines are inside the gap region, where the peaks are non-dispersive. (e) Angle integrated intensity, which shows a suppression of intensity near ED.

a gap of approximately 0.26 eV. Another way of extracting the gap is from the MDC peaks. As already discussed in Chapter 2 (section 2.1.4), MDCs are non-dispersive inside the gap region. The MDC peaks are non-dispersive within the same energy window, 0.26 eV around ED (blue lines in panel d).Clearly away from this region, the MDC peaks start dispersing again, in agreement with a conical dispersion. Another signature of the gap opening is that the angle integrated intensity, which is proportional to the density of states (DOS) integrated over one dimension, shows a suppression near the same region around ED (see Fig. 4.2(e)). This suppression of intensity is typical of a gap opening.

A peculiar feature of this gap is that there is non-zero intensity around ED (panel b) between the valence and conduction bands. The direct comparison with a cut away from the K point, where a gap is certainly present due to the conical dispersion (see Fig. 4.3 and Fig. 4.4(c)) suggests that this non-zero intensity should not be taken as evidence against the gap scenario. The finite intensity in the gap is also responsible for the non zero density of states at ED. The integrated intensity for the data of Figs. 4.3(a,b) is shown in panel (e). Though this one dimensional angle integrated intensity does not correspond to the density of states, which is integrated over two dimensions, it gives some hint whether a gap is present. At high binding energy the dispersion shows an almost linear behavior while near ED (indicated by broken line) it decreases. This shows that near ED there are fewer states, a result that is in agreement with the gap scenario. When moving away from the K point, the angle integrated intensity curve (black curve) show a larger anomalous region, which is in agreement with a larger gap. The similarity between the cut through the K point and cuts away from the K point where a gap definitely opens up further suggests that a gap is a more natural explanation for the anomalous region, rather than the interaction between electrons and plasmons. It is very likely that the non zero intensity is the result of the broad EDC peaks (Fig. 4.2(c)) which cause an overlap of the intensity tails from the top of the valence band and the bottom of the conduction band. Another possible source of such spectral weight may be due to edge states associated with the terrace or defect states 69. Although at this stage it is not clear why the EDC peaks are so broad, possible causes may be a self energy effect or distribution of gaps. Finally regardless of the origin, what the large EDC width implies in terms of actual device application remains to be seen in the future. One should note that ARPES lifetime determined as the inverse line width tends to underestimate the transport lifetime by as much as two orders of magnitude 70, and that in general one would expect a sharpening up of peaks as they are brought to the Fermi level, as would happen in device applications. Finally, the direct comparison between the two cuts in Fig. 4.3 allows to define a way to confidently determine the presence of a gap in the spectra and to extract the size of such gap. This can be done from

26 Figure 4.3. (a,b) Dispersions taken along a symmetric direction for cuts away and through the K point. (c,d) MDCs in the energies as labeled in panels a,b. (e) Angle integrated intensity as a function of energy for data in panel a (black curve) and panel b (gray curve).

Figure 4.4. (a) Schematic drawing for the cuts shown in (b) and (c) in the BZ of graphene and the conical dispersion (not drawn to scale). (b,c) Dispersions measured through and oﬀ the K point. The dotted white and dark gray lines are dispersions extracted from the MDCs. (d, e) Extracted dispersions and MDC width as a function of energy for data shown in panels b and c.

27 the EDCs as well as the region of energy where the MDC peaks are non-dispersive (Fig. 4.3(c-d)), which coincides with the regions of vertical intensity in the upper panels.

4.3 Gap interpretation vs many body interactions

An alternative picture to the gap scenario has been reported by A. Bostwick et al. 71, where data similar to Fig. 4.2 have been discussed in terms of electron-plasmon interaction 71. This interpretation is based on the departure of the dispersion from the anticipated behavior near ED, which we attribute to a gap, and the observation of an anomalous upturn of the MDC width near ED. However, these are not unique features of the K point and they occur every time a gap is present in the spectra. In Fig. 4.4, we present data taken along two lines (panel a): one through the K point (panel b), and another one parallel to it but far from the K point (panel c). These particular cuts are convenient because the intensity is strongly suppressed on one side of the K point, thus allowing the MDC to be fitted with single peaks. This allows for a more reliable fit of the data. The cut far away from the K point is considered because it definitely has a gap, due to the conical nature of the dispersion. Performing an MDC analysis first on the cut through the K point, the anomalous region of Fig. 4.2 is manifested through a kink in the dispersion and a sudden increase of the scattering rate (pointed to by arrows in panels d and e). Such anomalies might be considered to be due to self-energy effects. An appealing explanation is that a decay through plasmon emission is responsible for the deviation from the conical dispersion, as recently proposed 71,72,73. However, performing a similar analysis on the cut far away from the K point reveals similar features. In both cases, we can identify a region between the conduction and valence bands where the intensity decreases (see horizontal arrows), the dispersion deviates from the linear behavior (see dashed line in panel b-c), and the scattering rate shows a sudden increase (see panel d). These striking similarities cast doubt on the validity of the many-body interaction scenario 71, as this should be able to account for very similar features far away from the K point where a gap is certainly present and is the natural explanation of the data. We therefore propose that these similarities are simply a manifestation of a gap opening at the Dirac point. It is therefore misleading to discuss the dispersion and scattering rate in the gap region and in general to perform an MDC fit near the bottom or top of a band, despite the possibility that MDC peaks might be present inside the gap region (panel c). Such peaks exist in our data due to the finite width of the EDCs at the bottom of the conduction band and the top of the valence band. This can be better illustrated in the simulated data shown in Fig. 4.5. Dirac-like dispersions with finite band gaps of 150 meV (Fig. 4.5(a)) and 400 meV (Fig. 4.5(b)) and hence finite effective masses are used. The MDC width is modeled to scale linearly with energy with finite constant broadening to account for the finite MDC width at EF . Matrix elements are included to suppress one side of the dispersion by multiplying the results by a step function with a finite width. Ignoring the fact that there is a gap between the valence and conduction bands and performing an MDC analysis in this region, the fit shows a kink in the dispersion (Fig. 4.5(a, b)) and an increase in MDC width (Fig. 4.5(c)). The anomalous region where the MDC width increases scales with the gap region. This simulation shows that disregarding the absence of electronic states and simply fitting MDCs can sometimes produce misleading results.

4.4 Thickness dependence of the gap and Dirac point energy

Fig. 4.6 shows how the gap and the distance between ED and EF change as the graphene sample thickness varies. Panels (b) and (c) show the ARPES data for bilayer and trilayer graphene samples. Again the dispersions extracted from the EDCs (panels (e) and (f)) are plotted. In these two panels, two distinct cones can be identiﬁed for E

28 Figure 4.5. (a,b) Simulation of the conical dispersions with a gap of 150 and 400 meV. (c) Extracted MDC width as a function of energy for data shown in panels a and b.

Figure 4.6. Decrease of the gap size as the sample becomes thicker. (a-d) ARPES intensity maps taken on single layer graphene on 6H-SiC, bilayer graphene on 4H-SiC, trilayer graphene on 6H-SiC and graphite respectively. Data were taken along the black line in the inset of Fig. 4.2(a) except panel (c), which was measured along ΓK direction and symmetrized with respect to the K point to remove the strong intensity asymmetry induced by dipole matrix element 65. (e, f) EDCs taken from the raw data (without symmetrization) for momentum regions labeled by the arrows at the bottom of panels (b) and (c).

29 Figure 4.7. Dispersions measured in bilayer graphene on 6H-SiC (panel a) and more insulating 4H-SiC (panel b) substrates.

consistent check for the sample thickness determined by other methods 37,62. Panel (d) shows the ARPES data taken along a line through the H point in graphite, where the dispersion resembles that of graphene through K 64. Data shown in panels (a-d) allow us to determine how the electronic structure near K point varies as the sample thickness increases. First of all, as the sample thickness increases, ED shifts toward EF . From single layer to trilayer graphene, ED (marked by arrows in panels (a-c)) shifts from -0.4 eV to -0.29 64 eV then to -0.2 eV. For graphite, ED has been estimated to be at ≈0.05 eV above EF . More importantly, as the sample becomes thicker, the gap (labeled by light blue shaded area in panels (a-c)) decreases rapidly. From single layer to trilayer graphene, the gap decreases from 0.26 eV to 0.14 eV then to 0.066 eV. For 64 graphite, since the Dirac point energy is above EF , whether there is a gap or not cannot be directly addressed by ARPES. However, from band structure calculation, it is expected that the gap at the H point is ≈ 0.008 eV 9,74, which is almost negligible. Fig. 4.7 shows comparison of data taken on one bilayer graphene sample on a more insulating 4H-SiC substrate with resistivity of 105 Ω/cm (panel b) compared to another graphene sample on a 6H-SiC with resistivity of 0.2 Ω/cm (panel a). In both cases, the Dirac point energy appears to be shifted by a similar amount below EF , suggesting that the doping is most likely associated with the surface charges at the interface, rather than the carrier concentration of the substrate.

Fig. 4.8 summarizes the evolution of the Dirac point energy ED and the gap ∆ for various sample thickness. The layer dependence of both quantities suggests that, beyond 5 layers, epitaxial graphene behaves as bulk graphite 64. The amount of doping should decrease as the sample becomes thicker, because the surface layer probed by ARPES is farther away from the interface as the thickness increases. Also, the strong dependence of ED on sample thickness is a direct manifestation of the short interlayer screening length (≈ 5 layers 75) of graphene. This result shows that the sample thickness is an effective way of controlling doping in epitaxial graphene. Panel(b) shows the dependence of the gap on the sample thickness. A gap in bilayer graphene has been reported and attributed to the different potentials in the two graphene layers induced by doping or electric field 76,77,78. While this could contribute to the gap in bilayer and even trilayer graphene, it certainly is not the reason for the gap in the single layer graphene.

30 Figure 4.8. Thickness dependence of ED and ∆. (a,b) ED and ∆ as a function of sample thickness, for epitaxial graphene on 6H-SiC (black) and on 4H-SiC (blue). The error bar for the sample thickness 62 was taken from the XPS measurements . For graphite, ED is extrapolated from the dispersions 64 9,74 at kz≈π/c , and the gap is estimated from band structure calculation . (c, d) Two possible mechanisms to open up a gap at the Dirac point. (e) Schematic drawing to show the inequivalent potentials on the A (blue) and B (red) sublattices induced by the interface.

4.5 Possible mechanisms of the gap

In the following, we discuss possible scenarios and we propose that the gap is most likely induced by symmetry breaking due to the graphene-substrate interaction.

4.5.1 Inter-Dirac-point scattering

One possible scenario that can open up a gap is the scattering between the K and K0 points, the inter- Dirac-point scattering. This requires√ translation√ symmetry breaking. The two known reconstructions on epitaxial graphene 79, 6×6 and (6 3 × 6 3)R30◦ are obvious candidates for the source of this symmetry breaking. However, in order to mix K and K0, a large scattering wave vector is required. This is much longer than the reciprocal lattice vectors of both reconstructions mentioned above. High ordering process involving consecutive small scattering wave vectors will be weak in general. Another source of inter-Dirac- point scattering is impurity scattering, which, as recently shown, can mix the wave functions at the two K points 80,81. This however would give rise to a gap that strongly depends on the impurity concentration, in contrast to our finding. The gap is in fact the same in all the samples that we have studied, prepared under different conditions (with and without hydrogen etching of the SiC substrate) and on differently doped substrates, insulating vs slightly electron doped substrate.

4.5.2 Quantum conﬁnement

It has been predicted that a gap can be opened in nanometer size graphene ribbons as a result of quantum confinement 83,69,82. This section investigates whether the gap observed in the epitaxial graphene here can be interpreted as a result of quantum confinement or not. To study the role of electron confinement for the gap opening in epitaxial graphene, in Fig. 4.9 we study the dependence of the gap on the graphene terrace or domain size and compare it with the results reported for nanoribbons 82. The terrace size of single layer

31 Figure 4.9. (a-c) LEEM images taken at electron energy of 6.6 eV to show the surface topology of single layer graphene (gray area) and corresponding ARPES data (d-f) taken through the K point (see vertical line in the inset of Fig.1(d)) for three single layer graphene samples prepared under different growth conditions. The white, gray and black colors in panels (a-c) represent the regions of buffer layer, single layer and bilayer graphene respectively. The red lines in panels (d-f) are dispersions extracted by fitting the EDCs. (g) Plot of the extracted gap size from ARPES as a function of the representative terrace size (red segments in panels a-c) of the single layer graphene. The dotted line is the gap size in graphene nanoribbons due to quantum confinement taken from Han et al 82.

32 graphene is obtained from low energy electron microscopy (LEEM) data for each sample. The corresponding angle resolved photoemission spectroscopy (ARPES) data at the K point for each sample are shown in panels d-f. The direct comparison between LEEM and ARPES data allows one to readily correlate the width of the photoemission features and the size of the excitation gap to the morphology of the films and, in particular, to the mean size of the single layer graphene terraces. The main result of this figure is summarized in panel g. Although we see a slight increase in the magnitude of the gap for the smallest terrace size, it is clear that as the terrace size exceeds 50 nm the gap plateaus at 180 meV. In contrast, the predicted and observed quantum confinement gap for exfoliated samples is shown as the dashed line. Clearly, for most of the terrace sizes we studied, the vanishingly small gap induced by quantum confinement can not account for what is observed in our data. These results show that while quantum confinement may have some contribution for small terrace sizes, the gap observed in samples with larger terraces is an intrinsic property of single layer epitaxial graphene and cannot be simply explained by quantum confinement.

4.5.3 Breaking of the sublattice symmetry due to graphene-substrate interaction

In our opinion, the more likely scenario is the breaking of the A, B sublattice symmetry. This leads to the rehybridization of the valence and conduction band states associated with the same Dirac point (Fig. 4.10(a)), resulting in a gap at all the K and K0 points. A necessary prediction of this scenario is the breaking of the six fold rotational symmetry of graphene near the Dirac point energy. For energy well above and/or below ED, the symmetry is restored. For bilayer and trilayer graphene, the breaking of the A, B sublattice equivalence can be a direct consequence of the the AB stacking between different layers. Indeed, topographic Scanning Tunneling Microscopy (STM) images for bilayer graphene have clearly shown inequivalent A and B sublattices 80,84, similar to what has been observed for graphite 85. This simply derives from the fact that one sublattice has carbon atoms directly below it while the other does not. Naively it seems that this explanation will not work for single layer graphene. However, it is known that for epitaxially grown graphene, a buffer layer exists 50,49 (see Fig. 4.10(b)). ARPES study of the buffer layer has shown practically the same σ bands as graphene while very different π bands 49. This is because the π orbitals have hybridized with the dangling bonds from the substrate. The fact that the σ bands are unchanged suggests that, like graphene, the carbon atoms in the buffer layer have also the honeycomb arrangement with similar bond length. Consequently, although the buffer layer is electronically inactive (absence of π orbitals) 49, structurally it can break the A, B sublattice symmetry when a single layer of graphene grows upon it. This is particularly so in view of the small layer separation of ≈ 2 A˚ 50 and the AB stacking usually expected for very thin graphene samples. For the single and bilayer graphene, we use a tight binding model with symmetry breaking on the A and B sublattices to fit the symmetry breaking parameters to the observed energy gap. By fitting the dispersion, the symmetry breaking parameter in single layer graphene, defined as half of the difference between the substrate potentials on the A and B sublattices, is determined to be m ≈ 0.13 eV. In bilayer graphene, the symmetry breaking parameters in the top and bottom layers are m1 ≈ 0.49 eV, m2 ≈ −0.21 eV respectively. The magnitude of the symmetry breaking parameter is much bigger in the bottom graphene layer than that in single layer graphene, because it is sandwiched between the buffer layer and the top graphene layer. The reason for m2 to have the opposite sign is because of the AB stacking. This cancels part of the effect in the bilayer graphene and decreases the gap. Therefore, for AB stacking graphene, the eigen-functions average out for many layers, and the gap decreases rapidly. Fig. 4.11 shows additional support for the A, B sublattice symmetry breaking. Panels (a-d) show intensity maps taken on single layer graphene as a function of kx and ky at EF , -0.4, -0.8 eV and -1.0 eV respectively. The dominant features in these panels are the small pockets centered at the six corners of the Brillouin zone. Interestingly, around each corner, there are six faint replicas forming a smaller hexagon. The intensity associated with them is ≈ 4% of the main intensity. Closer inspection shows that the vectors connecting the center of the small hexagon to its six corners are nearly the same as the second shortest

33 Figure 4.10. Proposed mechanism for the gap opening and the structure of epitaxial graphene on SiC.

Figure 4.11. Breaking of the six fold symmetry in the intensity map near ED. (a-d) ARPES intensity maps taken on single layer graphene at EF ,ED, -0.8 eV and -1.0 eV respectively. Near ED (panel b), the intensity of the six replicas near K shows breaking of six fold symmetry. Note that to enhance the additional feature around ED, the color scale is saturated for the dominant features near K and the replicas. (e) ARPES intensity map of the calculated spectral function at ED in the presence of symmetry breaking on the two carbon sublattices.

34 √ √ reciprocal lattice vectors of the (6 3×6 3)R30◦ 37,62 observed in low energy electron diﬀraction LEED 37,62. As ED is approached, three among the six faint replicas become more intense (pointed by red arrows in Fig. 4.11(b)). This suggests the breaking of the six fold rotational symmetry of graphene down to three fold, and is consistent with the notion of A and B sublattices being inequivalent. In Fig. 4.11(e), we use a tight binding√ model√ to compute the intensity of the replicas at ED. The potential modulation imposed by the (6 3 × 6 3)R30◦ reconstruction has been added as a perturbation to the Hamiltonian, and the sublattice symmetry breaking has also been taken into account (see supplementary information). The result favorably agrees with the observation. We note that STM measurements on epitaxially grown single layer graphene do not show this symmetry breaking. This is because the main graphene signal measured is near EF , where no symmetry breaking is observed (see Fig. 4.11(a)). In addition to these faint replicas, we observe additional intensity enhanced along the edge of certain medium sized hexagons around ED (see gray broken and dotted lines in panel b). The origin of this intensity is still unclear. However, two observations can be made. 1) The center mid-sized hexagon around Γ (gray dotted lines) almost overlaps the ﬁrst Brillouin zone of SiC. 2) All other hexagons (e.g. gray broken lines) are not regular, i.e. the six sides forming the hexagon do not have the same length. Interestingly, they all pass through K and K0. Whether this reveals the presence of perturbation that can hybridize the states at K and K0 remains unclear.

4.6 Conclusions

To summarize, this chapter reported the presence of an energy gap at the K point in epitaxial graphene. Various possible scenarios were examined and we propose that the gap is most likely to be induced by the interaction between graphene and the substrate. This provides a possible way of engineering the band gap in graphene through the substrate. If one can change the strength of the interaction by changing the substrate on which graphene is grown, a control of the gap size can be possibly achieved. Since the epitaxial graphene is usually electron doped and the gap in this case is below EF , the next important step to make graphene a real semiconductor is to dope graphene with holes or by applying a gate voltage to move EF inside the gap region.

35 Chapter 5

Metal-insulator transition in epitaxial

graphene by molecular doping

5.1 Motivation

The metal to insulator transition (MI) in solids is among the most fascinating topics of research in condensed matter physics. To date many systems have been studied and different methods to induce MI transition have been revealed, such as applying magnetic field, pressure and most importantly doping as in the case of semiconductor. The ability to control the gap and charge carriers has made semiconductor an important material for microelectronics applications. In the past few years a great deal of attention has been turned to the possibility of developing single layer graphite, nicknamed ‘graphene’, as the material of choice for future devices, due to the relative ease of controlling the charge carriers from electrons to holes by applying a gate voltage 4. Although it would certainly be a major breakthrough to induce a MI transition in graphene, so far the only report of such a transition in a carbon material is for bulk graphite in the presence of high magnetic fields 86. Even though a gap has been predicted and reported in nano-structured graphene quantum dots 87 and exfoliated graphene ribbons 83,88,69,82, the tiny size (≈ nm scale) of the samples makes it difficult to be studied. Recently it has been argued that a gap can be induced in mesoscopically large epitaxial graphene on a semiconducting substrate 78,63. In this case, the gap is a few hundred meV below the Fermi energy EF . Thus epitaxial graphene would be an ideal system to induce a MI transition if one could control the charge carriers and move EF to the gap region. However, the control of charge carriers in epitaxial graphene is not as simple as in exfoliated graphene, since hole doping of a few hundred meV by applying a gate voltage or chemical methods has not been successfully demonstrated.

Motivated by the report that molecular adsorption of NO2 on exfoliated graphene can induce hole doping 89, we have investigated the eﬀect of such doping on the electronic structure of single layer and bilayer epitaxial graphene. We have discovered a reversible metal to insulator transition associated with an almost rigid shift of the band structure as a function of doping which gives rise to a doping independent Fermi velocity. These results demonstrate a unique property of epitaxial graphene, i.e. the presence of a gap between the valence and conduction bands and the achievable MI transition, which makes graphene suitable for electronic devices.

36 Figure 5.1. Dispersions through the K point taken from the as-grown single layer graphene (a) and from the sample with the highest doping of NO2 (c). Panels (b) and (d) show the angle integrated spectra for (a) and (c) respectively. In panels (c) and (d) note the appearance of the NO2 states at ≈ 5 and 11 eV below EF .

5.2 Valence band of graphene with adsorption of nitrogen dioxide

Before discussing the effect of molecular doping on the electronic structure of bilayer and single layer graphene, we shall prove that NO2 is present on the surface of our samples. Figure 5.1 shows ARPES data of the valence band of a single layer graphene sample before (panel a) and after (panel b) exposure to NO2. The exposure to NO2 leads to the appearance of two broad peaks centered at 5 eV and 11 eV below EF . Similar structures were reported previously in the photoemission studies of NO2 adsorbed on the surfaces of 90 91 W(110) and Si . In the case of W(110), the spectra of NO2 adsorbed on the surface are similar to that 90 of the gaseous N2O4, suggesting dimerization of NO2 on the surface . Previous study of electron energy loss spectroscopy (EELS) of NO2 adsorbed on graphite at 90 K has found vibrational modes which were 92 consistent with the formation of NO2 dimers , suggesting that a similar dimerization might also take place in the present study. An exact estimation of the amount of NO2 adsorbed on the surface of our samples is difficult due to the significant photo-stimulated desorption. In fact, to minimize the impact of the photo desorption on our data we had to intentionally reduce the photon flux and accumulate the spectra for only 30 seconds. This explains a relatively poor statistics of the data taken from graphene with adsorbed NO2.

37 Figure 5.2. (a) Dispersions through the K point taken in as-grown bilayer graphene. Data were taken along the red line through the K point shown in the inset. (b) Dispersions taken after 0.6 L 6 (1 Langmuir=10 torr · s) NO2 adsorption. (c) data taken 6 minutes after panel b. (d) EDCs taken at k1, k2, k3 and k4 as labeled on the top of panels a and b. The black and red arrows point to the midpoint of the leading edge, which is shifted to higher binding energy after NO2 doping. (e) Zoom in of data shown in panel b. The white dotted line is a guide for the eye for the dispersions. (f) Momentum distribution curve (MDC) at the energy labeled by the dotted black line in panel d. The dots are the raw data and the solid line is the ﬁt using three Lorentzian peaks simulating the cross-section of the hat-like dispersion in panel e.

5.3 Gas adsorption and metal-insulator transition in bilayer

graphene

Fig. 5.2 shows ARPES data taken on bilayer graphene before and after NO2 adsorption at 20K. Panel a shows dispersions through the K point in the as-grown graphene. The Dirac point energy is readily identified at the binding energy of 0.3 eV. Strictly speaking, the Dirac point, which separates the upward dispersing conduction band from downward dispersing valence band, is located within the gap. Existence of this gap between the conduction and valence bands in bilayer graphene has already been shown in previous 78,63 −8 ARPES studies . Panel b shows data taken right after dosing NO2 gas at 1×10 torr for 60 seconds (0.6 Langmuir of NO2 gas). Clearly, adsorption of NO2 leads to the shift of the entire band structure toward EF , which indicates hole doping of the bilayer graphene. Although the electron doping of epitaxial graphene has been studied 78, this is the first demonstration of hole doping of epitaxial graphene. Close inspection of the region near EF shows that it lies within the gap separating the conduction and valence bands with the latter band located above EF . To illustrate this, in Fig. 5.2(d) we plot the energy distribution curves (EDCs) at the momentum values where the bands are closest to EF for data shown in panel a (black curves) and panel b (red curves). It is obvious that after NO2 adsorption, the EDCs move toward higher binding energy, with the leading edge shifted by ≈ 30 meV. The shift of the leading edge is characteristic of a gap, showing unambiguously that the sample in panel b is semiconducting. This process is reversible: the sample becomes a metal again if we remove most of the NO2 from the surface by exposure to high photon flux, or by annealing the sample. Fig. 5.2(c) displays the data taken after desorption of NO2 by photon flux. The bands of graphene are shifted back to their original positions with the Dirac point at 0.3 eV below EF . It is clear however that a small fraction of NO2 is still present, since there is still considerable broadening by up to 200% in the photoemission signal in Fig.5.2(c), which might be associated with the scattering of electrons by the residual NO2 molecules. By thermal annealing at ≈ 400C the residual molecules can be completely removed, as confirmed by the sharpening of the peaks and the complete recovery of the line width of pure graphene. In addition to demonstrating molecular hole doping of graphene, the data in Fig.5.2(b) also allow an unobscured view of the top of the valence band which exhibits a characteristic hat-like shape (see Fig.5.2(e)). Such shape was calculated 77,93 but to the best of our knowledge it was not observed in the previous ARPES studies with the present degree of clarity, mainly because of the contribution from the conduction band (see

38 in Fig. 5.2(a) for example). Finally, we wish to attempt to estimate the amount of NO2 adsorbed on the graphene. The amount of carrier concentration has been estimated by measuring the area enclosed by the Fermi contours and through the Luttinger theorem. From panel a to panel b we detected a change of carrier concentration by 0.0052 electrons per unit cell (4 carbon atoms) upon exposing the graphene sample to 0.6L of NO2. Transport measurements suggested that as much as 1 electron is transferred to each molecule of 89 NO2 adsorbed . If we take this estimate, change of 0.0052 electrons per unit cell corresponds to 0.033 Langmuir of NO2. This suggests that with the dosage 0.6L NO2 molecules, only 5.5% of the molecules stick to graphene and remove 1 electron from graphene. This is reasonable considering the fast photodesorption of the NO2 molecules.

5.4 Metal-insulator transition in single layer graphene

In Figure 5.3 we show the effect of molecular doping on single layer epitaxial graphene for various amounts of NO2 doping. Data in panel a are taken on the as-grown sample, which is electron doped with the Dirac point located at 0.4 eV below EF . The dispersion near the Dirac point shows deviations from the expected conical dispersion as previously reported 63,71. The origin of these deviations has been a matter of intense debate in the past year and has been attributed either to a kink structure caused by electron- plasmon interaction 71 or to the opening of a gap at the K point 63. By hole doping we have the unique ability to directly access this region and to investigate its origin. Panels b-f show data taken for the progressive adsorption of NO2 on a single layer graphene sample. Similar to the case of bilayer graphene, we can dope the single layer sample and move the chemical potential from the conduction band all the way down to the valence band. The maximum shift of 0.8 eV corresponds to a gate voltage as large as 300 V in the case of the 5,52 exfoliated graphene . When the chemical potential falls in the anomalous region near ED (panels c and d), extrapolation of the valence band shows that it lies below EF , while the conduction band is above EF . Therefore a semiconducting graphene is achieved. To further support this, in panel g we show the EDCs at the K point where peaks are closest to EF . When the chemical potential falls within the conduction or valence bands (curves a, b, e, f), a peak near EF (dotted line) can be observed, while when the chemical potential falls within the anomalous region near ED (curves c, d), a depletion of the spectral weight near EF and a shift of the leading edge toward higher binding energy is evident. The results shown here demonstrate that a metal to insulator to metal transition can be induced in hole doped epitaxial graphene. These results further supports that the origin of the anomalous region in the dispersion near the Dirac point energy is due to a gap 63 and not to many body interactions 71. As previously pointed out 63, a peculiar feature of the gap is the non-zero intensity inside the gap region. The source of such spectral weight may be due to edge states associated with the terrace or defect states 69. Figure 5.4 summarizes the effect of hole doping in single layer graphene. Rigid shift of the overall band structure, which appears to take place upon doping, offers an easy way of estimating the amount of doping. Figure 5.4(a) shows the shift of the Dirac point energy by increasing the charge carrier concentration through molecular doping. For the as-grown single layer graphene (Fig. 5.3(a)), the separation of the Fermi wave −1 vectors is 0.096 A˚ . Assuming that the electron pocket at EF is a circle, this converts to an electron concentration of 1.4×1013 cm−2. For the highest doping achieved the distance between the Fermi vectors is 0.125 A˚−1 which corresponds to a hole concentration of 2.4×1013 cm−2. Figure 5.4(b) plots the Fermi velocity vF , a quantity that governs the low-energy quasiparticle dynamics, as a function of carrier concentration. vF 1 ∂E is extracted from the slope of the dispersion near EF , v= . Surprisingly we find that the Fermi velocity ~ ∂k is nearly constant for all doping to within ±20% of the initial value, though small changes expected on a scale which is beyond the resolution of the present experiment 94 cannot be excluded. This independence of the Fermi velocity is different from the earlier report of a huge change of the Fermi velocity by electron doping through Ca 67 in epitaxial graphene. Whether this suggests the presence of a strong electron-hole asymmetry or the hybridization of the Ca band with the C π band 70 has to be confirmed. Finally, we note that a similar doping independent Fermi velocity has been reported for the nodal quasiparticles in high temperature superconductors, which are also Dirac fermions, through the insulator to superconducting

39 Figure 5.3. (a-f) Data taken through the K point for the as-grown (a) and various dopings with NO2 adsorption (b-f) in single layer graphene. The white lines are dispersions extracted from the MDC peaks when they can be clearly resolved. The dotted lines in panels c and d are linear extrapolation of the dispersion. (g) EDCs taken at the momentum regions (indicated by a small tick mark on top of each panel) where the bands are closest to EF .

Figure 5.4. (a, b) Plot of the shift in ED and Fermi velocity as a function of the carrier concentration for data shown in Fig. 5.3 In panel b, the data are extracted by ﬁtting the dispersion between EF and -0.3 eV (circles) and between EF and -0.1 eV (diamonds). The open symbols are extracted from the dispersions on the left and the ﬁlled symbols from the dispersions on the right.

40 transition 95. Whether this universality is a general property of Dirac Fermions is certainly an interesting topic that needs to be further investigated.

5.5 Conclusions

In summary, we have performed ARPES measurements of the electronic structure of epitaxial graphene with adsorbed NO2. Our data directly prove that NO2 induces hole doping of graphene and provides a possibility of tuning the charge carriers from electrons to holes in a wide doping range. We show how the peculiar electronic structure of epitaxial bilayer and single layer graphene containing an excitation gap results in a metal to insulator transition upon hole doping. Our results suggest a way of achieving semiconducting graphene which is an important step in making it a practical component of electronics devices. This study also opens up an interesting route to studying the physics of the hole doping regime of the Dirac cone.

41 Chapter 6

Dirac fermions and eﬀect of impurities

in single crystalline graphite

6.1 Introduction

In the previuos chapters we have seen that, as the number of graphene layers increases, the interlayer interaction is turned on and the electronic structure evolves from a pure two dimensional structure, characterized by massless quasiparticles, to a more three dimensional electronic structure. This results in a kz ∗ ∗ dependent band structure with coexistence of massless Dirac fermions at kz=0.5 c where c is the reciprocal 64 lattice constant and massive quasiparticles at kz=0 . As in graphene, a variety of novel properties have been recently reported in graphite such as novel quantum Hall eﬀect 86, room temperature ferromagnetism 96, metal-insulator-like transition 86, and superconductivity 74. The recent discovery of superconductivity with 97,98 high critical temperature Tc of 11.5K in C6Ca has also attracted a lot of research interest in understanding the superconducting mechanism and searching for superconducting graphite intercalation compounds (GICs) with even higher Tc. Although several transport experiments have shown results in agreement with the existence of Dirac fermions in graphite 99, so far direct experimental evidence has been limited. This chapter presents a study of the electronic structure of single crystal graphite. The samples that we use here are natural graphite which is extracted from graphite mines and kish graphite which is the byproduct of steel making process.

6.2 Electronic structure of single crystal graphite

Since graphite has three dimensional BZ, the dispersion also depends on kz and it is important to determine the kz values. Throughout this chapter the kz values are estimated using the standard free- electron approximation of the ARPES ﬁnal state 20,19,100 (see Chapter 2). The inner potential needed for extracting the kz value was determined from the periodicity of the detailed dispersion at the Brillouin zone 101 center ΓA using a wide range of photon energies from 34 to 155 eV . The consistency of the kz values is conﬁrmed by the degeneracy of the π bands near H and a maximum splitting near K (Fig. 6.1), both in

42 agreement with band structure 12,74,13. The splitting of the π bands in graphite is a manifestation of the interlayer interaction, as already discussed in Chapter 1.

Figure 6.1. Dispersions measured near H and K, showing the general consistency of the extracted ∗ 0 kz values. (a) Dispersion near H (hν=140 eV, kz ≈ 0.50 c ) along HH direction, showing that the ∗ π bands are degenerate. (b) Dispersion near K (hν=80 eV, kz ≈ 0.07 c ) along direction parallel to HH0, where the π bands split into bonding (BB) and antibonding (AB) bands.

Although the global band structure of graphite has been studied extensively by ARPES 102,100,103,104,105,101, ∗ ARPES studies of the low energy dispersion of the π and π bands near EF are very limited. The only information for the electron and hole pockets comes from other measurements 106,107,108,109, where the interpretation is not straightforward. Our study is the first ARPES study of the low energy electronic structure of graphite uncovering the coexistence of relativistic Dirac fermions with linear dispersion near the Brillouin zone (BZ) corner H and quasiparticles that have a parabolic dispersion near another BZ corner K. In addition, we also report a large electron pocket that we attribute to defect-induced localized states. Thus, graphite presents a system in which massless Dirac fermions, quasiparticles with finite effective mass and defect states all contribute to the low-energy electronic dynamics.

6.2.1 Dispersions at H - massless Dirac fermions

Fig. 6.2 shows an ARPES intensity map measured near the BZ corner H. The out-of-plane momentum kz is 0.5 c∗. Following the maximum intensity in this map, a linear Λ-shaped dispersion can be clearly observed. The dispersion can be better extracted by following the peak positions in the momentum distribution curves (MDCs), momentum scans at constant energies, shown in panel b. Here, two peaks in the MDCs disperse 6 −1 linearly and merge near EF . The Fermi velocity extracted from the dispersion is 0.91±0.15×10 m · s , similar to a value 1.1×106 m · s−1 reported by a magnetoresistance study of graphene 51. We note that at low energy near EF , this linear dispersion is also observed along other cuts through the H point, with similar Fermi velocity. This linear and isotropic dispersion is in agreement with the behavior of Dirac fermions. Another way of probing the linear and isotropic dispersion is to study the intensity maps at constant energy. At EF (Fig. 6.3), the intensity map shows a small object near H. The details of this small object will be discussed later. With increasing binding energy, this object expands and shows a circular shape (panels b-c). We note that only the circular shape in the first BZ is clearly observed. This is attributed to the dipole matrix element 65, which suppresses or enhances the intensity in different BZs. However, taking the three fold symmetry of the sample, this circular shape in the first BZ is expected to extend to other BZs and

43 Figure 6.2. Linear Λ-shaped dispersion near the BZ corner H. (a) ARPES intensity map taken near ∗ the H point (photon energy hν=140 eV, kz ≈ 0.50 c ), along a cut through H and perpendicular to kx (see red line in the BZ shown in panel c). The inset shows a schematic diagram of the Dirac cone dispersion near EF in the three dimensional E-kx-ky space. (b) MDCs from EF to -2.0 eV. The MDCs are normalized to have the same amplitude and displaced by the same amount so that the dispersion can be directly viewed by following the peak positions at each energy. The dotted lines are guides to the eyes for the linear-dispersing peaks in the MDCs. (c) Three dimensional BZ for graphite with high symmetry directions relevant for this paper highlighted with red, green and blue lines.

44 Figure 6.3. Constant energy maps taken near the H point, showing that the electronic structure is isotropic in the kx-ky plane from EF to -0.6 eV. (a-e) ARPES intensity maps near H (hν=140 eV, ∗ kz ≈ 0.50 c ) taken at energies from EF to -1.2 eV. The circles are guides for the circular intensity pattern near the H point. Arrows in panels d and e point to deviation from the circle. (f) Schematic diagram of the dispersion for graphene near six BZ corners in the three dimensional E-kx-ky space.

thus the electronic structure is isotropic near H. As the energy changes to -0.9 eV, the constant energy map slightly deviates from the circular shape (see arrow in panel d). This deviation increases with binding energy and a trigonal distortion is clearly observed at -1.2 eV (panel e). This trigonal distortion is determined by the relevant tight binding parameters for graphite and further studies to analyze this trigonal distortion are in progress. Overall, Fig. 6.3 shows that from EF to -0.6 eV, the electronic structure is isotropic in the kx-ky plane. Similarly, the Fermi velocity measured within the ﬁrst BZ is 1.0×106m · s−1 with a ≤ 10% variation along diﬀerent directions, consistent with the circular constant energy maps shown here. Combining the results of Figs. 6.2 and 6.3, we conclude that from EF to -0.6 eV, the dispersion shows a cone-like behavior near each BZ corner H, similar to what is expected for graphene (panel f).

To resolve the details of the low energy dispersion and the small object at EF (Fig.6.2(a)), we show in Fig. 6.4 an intensity map measured near H with lower photon energy to give better energy and momentum resolution. In the intensity map, one can distinguish two bands dispersing linearly toward EF , as also clear in the MDCs (panel b) where two peaks can be observed for all binding energies. The extracted dispersion (open circles in panel a) from MDCs (panel b) shows two bands dispersing linearly toward EF , −1 with a minimum separation of 0.020±0.004 A˚ at EF . This linear dispersion near the H point, as well as the isotropic electronic structure shown in Fig. 6.4 from EF to -0.6 eV, is a basic characteristic of Dirac quasiparticles, which points to the presence of Dirac quasiparticles in the low energy excitations near the H point in graphite. Furthermore, from the extracted dispersions, the Dirac point is extrapolated to be 50±20 meV above EF , and thus the small object observed at EF is a hole pocket, in agreement with previous studies of the three dimensional band structure of graphite 74,99. Assuming an ellipsoidal shape for the hole pocket 110, the hole concentration is estimated to be 3.1±1.3×1018 cm−3, from the 0.020 A˚−1 separation of 111,112,110 the peaks at EF . This hole concentration is in agreement with reported values . We note that, given the current resolution of ARPES technique, we are not able to resolve the two hole pockets at the H point reported by the other experimental probe 113. In fact, the diﬀerence in energy between these two hole pockets at the H point is ≤ 1 meV 113, which is beyond the current resolution of the ARPES technique. The presence of holes with Dirac fermion dispersion is further supported by the angle integrated intensity

45 Figure 6.4. Detailed low energy dispersion near the H point shows that low energy excitations are Dirac fermions with the Dirac point slightly above EF . (a) ARPES intensity map near the H point ∗ 0 (hν=65 eV, kz ≈ 0.45 c ) along AHL direction (green line in the BZ shown in Fig. 6.2(c)). The open blue circles in panel a show the MDC dispersion. The dotted straight lines are guides for the linear dispersion. (b) MDCs at energies from EF to -0.2 eV for data shown in panel a. Note that, similar to Fig. 6.3, the intensity of the π band is strongly enhanced in the ﬁrst BZ (H→A direction), 65 due to the dipole matrix element . (c) Intensity obtained by integrating over both kx and ky for ∗ data taken near H (hν=140 eV, kz ≈ 0.50 c ). The intensity has been symmetrized with respect to the Dirac point energy (ED ≈ 50 meV) to compare directly with the expected intensity for Dirac fermions (inset). An overall linear behavior is observed with some weak additional intensity around 100 meV from the Dirac point energy. The origin of this additional weak intensity is unclear and needs further investigation.

46 Figure 6.5. Detailed dispersion near K, which shows that quasiparticles with finite effective mass and defect-induced localized states also contribute to the low energy electronic dynamics. (a) ARPES ∗ 0 intensity map near K (hν=50 eV, kz ≈ 0.08 c ) along ΓKM direction (blue line in the BZ shown in Fig. 6.2(c)). The open circles are the dispersions extracted from MDCs. (b) MDCs from EF to -50 meV for data in panel a. The open circles mark the peaks clearly resolved in the data. The inset shows the MDC dispersion from -10 to -50 meV, with the parabolic fit used to extract the effective mass.

(panel c), which is proportional to the two dimensional density of states, barring the matrix element. In this energy range, a linear behavior, similar to what is expected for Dirac fermions, is observed. In addition, the energy intersect is at ≈ 50 meV above EF , in agreement with the Dirac point energy extrapolated from the dispersions.

6.2.2 Dispersions at K - massive quasiparticles

The above figures show that near the H point, the low energy excitations in graphite are Dirac fermions characterized by linear and isotropic cone-like dispersion, in agreement with transport measurements in graphite where Dirac fermions are suggested to coexist with quasiparticles which have finite effective mass 99. To gain direct insight on the different types of quasiparticles, ARPES can provide a unique advantage by directly measuring the effective mass as well as accessing its momentum dependence. Fig. 6.5 shows the intensity map near another high symmetry point in the BZ corner, the K point. The dispersion (open circles) shows a parabolic behavior, in sharp contrast to the linear behavior observed near the H point (Figs. 6.2 and 6.4). This parabolic dispersion points to the presence of quasiparticles with finite effective mass. To determine the effective mass, we first extract the low energy dispersion, then fit the MDC and EDC dispersions with a parabolic function. In both cases, the effective mass is determined to be 0.069±0.015 me, where me is the free electron mass. This effective mass measured by ARPES shows some difference with values 112,111,114 reported by transport measurements , 0.052 and 0.038 me for electrons and holes respectively. This difference may be due to the fact that transport measurements are not momentum selective, and therefore the mass measured is the average mass over all kz values. On the other hand, ARPES is momentum selective and the value for the effective mass is for this specific kz value only. Taking this into account, the agreement between these measurements is reasonable. In summary, the data presented so far show that the low energy

47 Figure 6.6. Detailed dispersion near K, which shows an additional electron pocket induced by defect states. (a, b) Intensity maps near the K point measured in diﬀerent parts of the sample, which shows an additional large electron pocket. The open circles are dispersions extracted from EDCs shown in panel f. (c) MDC at EF from data shown in panel b. The black arrows point to the peaks from the large electron pocket which are separated by ≈ 0.1 A˚−1, while the gray arrow points to the peak from the π band. (d) EDCs from k0 to k12, as indicated in panel b. Open circles are the peak positions for the large electron pocket.

excitations in graphite change from massless Dirac fermions with linear dispersion near H (Figs. 6.2, 6.3, 6.4) to quasiparticles with parabolic dispersion and ﬁnite eﬀective mass near K (Fig. 6.5(a)).

6.3 Eﬀect of defect states

We now discuss another interesting feature observed in graphite, i.e., a large electron pocket near EF . Fig. 6.6(b) shows the intensity map measured in the same experimental conditions as Fig.6.5(a) except at a different spot. In panel b, a strong and large electron pocket within 50 meV below EF is the dominant feature. Weak signatures of the parabolic π band (as in panel a) can still be observed, as the MDC at EF (panel c) demonstrates. Here, in addition to the two main peaks (black arrows) corresponding to the electron pocket, a central weak peak (gray arrow) corresponding to the top of the parabolic band can also be distinguished. −1 From the separation (≈ 0.1 A˚ ) between the two main peaks at EF , the electron concentration is determined to be 8.0±0.7×1019 cm−3, which is an order of magnitude higher than the values reported 112,111. Moreover, from the dispersion (panel d), the effective mass is extracted to be 0.42 ± 0.07 me, which is also much larger than any mass reported by transport measurements 112,111,114. This large electron pocket, is observed in most of the samples measured, and thus it represents an important feature associated with graphite. We note that a similar large electron pocket has been reported recently and proposed to be associated with either edge states or dangling bonds 110, here we provide detailed characterization of this large electron pocket, which is important in revealing its origin. We propose defect- induced localized states as a possible explanation for this large electron pocket, based on the following reasons. First, the electron concentration and effective mass measured for this electron pocket are much larger than reported values. Second, although the parabolic π band in panels a and c is observed in all the

48 Figure 6.7. STM image of zigzag and armchair edges (a) and typical dI/dV from STS data at a zigzag edge, from Kobayashi et al 115.

samples measured and in diﬀerent spots (averaged over ≈100 µm) within the same sample, this large electron pocket strongly depends on the spot position within the same sample. Third, STM shows that zigzag edges can induce a peak in the local density of states at an energy (≈ -0.03 eV) similar to the electron pocket observed here (see Fig. 6.7) 115. In fact, it has been shown that low concentration of defects (e.g. edge states, vacancies, etc.) can induce self-doping to the sample 116,117. If this interpretation is correct, then further studies on this large electron pocket may shed light on the magnetic properties of nano-graphite ribbons, since it has been proposed that some defect-induced localized states are magnetic 83,118.

6.4 Conclusions

This chapter shows that in bulk graphite, Dirac fermions near the H point coexist with massive quasiparticles bear the H point. We have also observed a large electron pocket, which cannot be explained except by defect states, e.g. zigzag edges.

49 Chapter 7

ARPES study of partially

polycrystalline graphite: HOPG

7.1 Electronic structure of polycrystalline materials measured by

ARPES

The capability to resolve crystal momentum values of single particle excitations in ARPES has been based entirely on the translational symmetry along the surface of a single crystal and the resulting conservation of the crystal momentum parallel to the surface (kk) during the photoemission process. This holds despite the short photoelectron lifetime 119,120,121 which can severely broaden the resolution of the momentum perpendicular to the surface (kz). Indeed, even in the limit of an extreme kz broadening that results in no resolution of kz, strong ARPES dispersions are expected as a function of kk, since the one dimensional density of states (1D-DOS) Dz(E) ∝ dkz/dE obtained by integrating over kz is dominated by contributions from van Hove singularities in high symmetry planes 121. For example, Fig. 7.1 shows the dispersions measured on LaSe, where the dispersions on the two high symmetry planes at kz = 0 and kz = π/c are clearly observed 122. On the contrary, for those systems characterized by orientationally disordered domains, i.e. polycrystalline materials, the translational symmetry is preserved only within each domain. As a consequence, the dispersion measured by ARPES is the average dispersion over diﬀerent domains, or equivalently azimuthal angle φ, which in general leads to no dispersion. However, extending the 1D-DOS Dz(E) scenario for kz discussed above further to the plane, there is an interesting possibility that a layered polycrystalline sample, with a strong azimuthal disorder, can nevertheless give distinct dispersions in the radial direction. This would happen if the average dispersion is dominated by those along the high symmetry directions due to van Hove singularities in the angular density of states Dφ(E) ∝ dφ/dE. This possibility has not been demonstrated experimentally and photoemission studies on disordered samples have focused on angle-integrated features without any momentum information. Here we demonstrate the possibility of measuring band dispersions in partially polycrystalline highly oriented pyrolytic graphite - HOPG. HOPG is a synthetic graphite, which is formed by cracking hydrocarbon at high temperature followed by subsequent annealing. It consists of many crystallites on the order of µm

50 Figure 7.1. BZ and measured dispersions on LaSe, from Nakayama et al 122.

with random azimuthal orientations while along the c-axis direction the sample is highly oriented. There on the order of mm scale, HOPG appears as a polycrystalline sample . However, HOPG is widely used since large sample size can be obtained and it is commercially available. In fact, HOPG has been the most common host material for graphite intercalation compounds (GICs). Our ARPES study on HOPG shows that band dispersions can be indeed obtained even from azimuthally disordered samples.

7.2 ARPES study of the electronic structure of polycrystalline

graphite

ARPES data were collected at beam line 10.0.1 of the Advanced Light Source (ALS) at the Lawrence Berkeley National Laboratory, using an SES-R4000 analyzer. The wide angular mode with acceptance angle of 30◦ and angular resolution of 0.9◦ was utilized for most scans, while high resolution angular mode with acceptance angle of 14◦ and angular resolution of 0.1◦ was utilized for one scan. The total instrumental energy resolution was 15 meV at 25 eV photon energy and 25 meV for other photon energies used (40, 55, 60 eV). The sample used was a grade ZYA highly oriented pyrolytic graphite (HOPG), obtained commercially from Structure Probe Inc. The sample was cleaved in situ in an ultra high vacuum better than 1.0×10−10 Torr and measured at temperature 50 K. Fig. 7.2(a) shows an ARPES intensity map measured at the Fermi energy taken at 40 eV photon energy. Here we use a color scale such that black represents high intensity in the raw data and blue represents low intensity. According to band structure calculation, a constant kz cross section of graphite Fermi surface can be a small hole pocket, a small electron pocket, or a point located at the six corners of the hexagonal 123,9 Brillouin zone (dashed line in Fig. 7.2(a)), depending on the value of kz . Experimentally, the predicted small electron or hole pockets are diﬃcult to be resolved, and measurements on single crystalline samples have shown only small dots of high intensity at these corners 124, schematically drawn as shaded circles in Fig. 7.2(a). For the graphite sample under study, the Fermi energy intensity map, symmetrized by three fold rotations to ﬁll the entire Brillouin zone, shows a perfectly circular pattern 1, in contrast to what is expected for single crystalline graphite. This is attributed to the angular spread of the dots to a circle due to the azimuthal disorder of the sample 124.

1The intensity variation along the circle is attributed to photoemission matrix element and is not a major concern here.

51 Figure 7.2. (a) Fermi energy intensity map. The hexagonal Brillouin zone (dashed lines) and Fermi surface (shaded circles) expected for single crystalline graphite are drawn schematically. (b) Intensity map versus binding energy and in-plane momentum along the solid line in (a) taken at 60 eV photon energy. Arrow marks the Fermi energy crossing point (kF ). The inset shows EDC at kF taken at 25 eV photon energy in the high angular resolution (0.1◦) mode. (c) Second derivative of raw data in (b) with respect to energy. LDA dispersions along both Γ-K-M’ direction (solid lines) and Γ-M-Γ’ direction (dashed lines) are plotted for comparison. The Brillouin zones are labeled on top of this panel for the two high symmetry directions.

Fig. 7.2(b) shows an ARPES intensity map as a function of binding energy and in-plane momentum kk, corresponding to the momentum cut shown as a solid line in Fig. 7.2(a). Despite the strong azimuthal disorder giving a circular Fermi energy intensity map in Fig. 7.2(a), we observe, surprisingly, very clear dispersions over the entire energy range. Furthermore, at the Fermi energy crossing point kF , a sharp coherent quasi-particle peak is observed. This is shown in the inset, where an energy distribution curve (EDC), energy cut at a constant momentum, is plotted. Here the half width of the EDC peak is 20 meV (50 meV FWHM due to the asymmetry of the line shape), the sharpest peak observed in HOPG so far 104. In Fig. 7.2(c) we report the second derivative of the raw data of Fig. 7.2(b) with respect to energy. The second derivative method has been used in the literature to enhance the direct view of the ARPES dispersion. Local density approximation (LDA) band dispersions along two high symmetry directions Γ-K- 0 0 M (solid lines) and Γ-M-Γ (dashed lines) are plotted in the same ﬁgure for a direct comparison. Despite a polycrystalline sample implied by the Fermi energy intensity map, an excellent agreement is observed between the experiment and the theory. We can identify the dispersions between 4 eV and 23 eV as originating from the sp2 orbitals with strong intra-layer σ bonding (black lines), and the dispersions between Fermi energy and 11 eV as originating from the pz orbitals with weaker π bonding (white lines). We note that the calculated dispersions were stretched by 20% in energy throughout this paper, as suggested in the literature 125,104,11. The stretching of the LDA band dispersions is attributed to missing self-energy corrections in LDA, since ab initio quasiparticle calculations based on the GW method show that for graphite the quasiparticle band dispersion near the Fermi level is 15% larger 11. The direct comparison between Fig. 7.2(a) and Fig. 7.2(b,c) shows an apparent paradox in our data, namely, the coexistence of azimuthal disorder feature (Fig. 7.2(a)) with single crystalline features (Fig. 7.2(b,c)). This can be readily understood if we consider an angular average of the calculated dispersions. Such an angular average would be necessary if the sample consisted of many small single crystallites with strong azimuthal disorder.

52 Figure 7.3. Dispersions for azimuthal angles φ=0◦(panel a); 10◦(panel b) and 20◦(panel c). The φ angle is deﬁned in the inset of panel a. LDA band dispersions along Γ-K-M’ (solid lines) and Γ-M-Γ’ (dotted lines) are plotted for comparison. (d) Calculated dispersions for single crystalline graphite along an arc (shown in the inset) with radius equal to ΓK distance. (e) Calculated density of states Dφ(E) for single crystalline graphite by integrating over an arc from A to B (see inset of panel d). Singular peaks in Dφ(E) occur at energies corresponding to band energy extrema, some of which are shown in panel(c,d) as shaded circles and open circles for A (ΓK direction) and B (ΓM direction) respectively.

7.3 Explanation of the paradoxical coexistence based on a density

of states argument

Figs. 7.3(a-c) show ARPES cuts for three azimuthal angles, φ=0◦, 10◦, 20◦. A direct comparison between panels a, b and c shows no appreciable angular dependence of the dispersions, establishing that the azimuthally invariant electronic structure of Fig. 7.2(a) at the Fermi energy extends to the entire band width. Thus, these data strongly support the azimuthal disorder model described in the previous paragraph, and indicate that the ARPES data measured are actually a 1D-DOS Dφ(E) along the azimuthal direction 121 φ, in analogy with the well-known 1D-DOS Dz(E) along the kz direction . As in the latter case, then, one would expect that van Hove singularities arising from states along the high symmetry directions to contribute dominantly, and this gives an explanation why the measured dispersions accurately reflect the dispersions along the two high symmetry directions. The 1D-like van Hove singularities arise from states along high symmetry lines because these states have zero group velocity along the arc with constant kk magnitude. In Fig. 7.3(d, e) we show LDA calculations supporting this reasoning. In panel d, we show the dispersion for single crystalline graphite along an arc from A to B with radius equal to ΓK distance. Within this arc, the ΓK direction corresponds to point A and the ΓM direction to point B respectively. As expected, extrema in the band dispersion occur at the two high symmetry directions, A (shaded circles) and B (open circles). The calculated 1D-DOS Dφ(E) over this arc, and thus over the entire azimuthal angle range by symmetry, is shown in panel e. One can see diverging 1D van Hove singularities as sharp peaks occurring at energies where bands cross points A and B, which completely dominate over other contributions. This nicely explains why well-defined sharp peaks with large dispersions can be observed in this azimuthally disordered sample, despite the fact that the observed data come from averaging over all azimuthal directions. The data presented so far can be summarized as showing well-defined dispersions along the radial direction with a complete lack of dispersion along the azimuthal direction. Therefore, our data suggest that the graphite sample under study consists of finite size single crystalline grains much smaller than the analysis area (≈ 100 µm) with a complete azimuthal disorder. However, each grain is large enough to allow for highly dispersive quasiparticles to exist. In addition, we have measured the dispersion perpendicular to the surface, kz dispersion, using photon energy range from 34 to 155 eV at beam line 12.0.1 of the ALS, with a perfect agreement with previous results 100. This indicates that the crystalline order remains coherent along this

53 direction, i.e. perpendicular to graphene layers, over a length scale larger than the probing depth of ARPES (order of 10 A).˚

7.4 Dispersions along the surface normal direction

Graphite is formed by many layers of graphene and the BZ shows a three dimensional structure. In order to investigate the details of the low energy electronic structure such as the effect of interlayer coupling, it is important to obtain the full momentum information including the out-of-plane momentum kz. To extract the kz value, we first determine the inner potential Vin from the symmetry of the measured dispersion along ΓA (kk=0). Fig. 7.4(a) shows a few examples of angle-integrated EDCs over a wide photon energy range from 34 to 155 eV, from which the dispersion is extracted. There are two features in panel a, a main peak at higher binding energy associated with the π bands, and a hump at lower binding energy associated with the σ bands. The main peak from π bands shows oscillation between -7.2 and -8.4 eV as a function of photon energy, or equivalently kz. The extracted dispersion (panel b) from the peak positions can be described as a periodic oscillation riding on a linear slope. The linear slope can be explained by angle integration over 8 degrees 2 due to the angle average mode used for this data set alone among the data shown in this paper. The periodic behavior and the symmetry of the dispersion enable us to determine the inner potential (17±1 eV) and thus extract the kz values. The periodic dispersion bears a strong similarity with previously reported data in the 126,100 126,100 literature , but covers more than twice the kz range. We note that, as previously reported , the dispersion shows a periodicity of 2 c∗ rather than c∗. This doubling periodicity has been observed 100 and can be understood by combining the kz dispersion of π bands shown in panel c and the symmetry of the final states detected using the dipole selection rules 65. More specifically, due to the nonsymmorphic group in graphite, the final states detected changes in a repeated sequence of α − β − γ − β − α. We note that α and γ states are resulted from the splitting of the π bands, and thus the measured dispersion at the zone center is a reflection of the interlayer coupling.

7.5 Comparison of data taken from HOPG and single crystalline

graphite

In order to check how good the dispersions extracted from HOPG compared to single crystalline graphite, we present here a detailed comparison of the interlayer coupling, the low energy electronic dynamics and the eﬀects of disorder for data taken on these two diﬀerent kinds of samples.

7.5.1 Interlayer coupling

We now focus on the eﬀect of the interlayer coupling near the BZ corners. Fig. 7.5 shows the π bands as a function of in-plane momentum (kk) near the BZ corner for diﬀerent kz values. Data measured on both HOPG (panels a-d) and single crystal are shown (panels e-h). Panels a and c show ARPES intensity maps ∗ of the π bands taken near 0.5 c . The data show a Λ-shaped dispersion crossing EF near the apex (H point), in agreement with the dispersion of the Dirac quasiparticles. The Fermi velocity vF estimated from the −1 52,51 slope of the dispersion dE/dk=6 eV·Ais˚ vF = 0.91±0.15m·s , similar to reported values . Panels b and ∗ d show ARPES intensity maps taken at near kz=0 c . In this case one can clearly distinguish two Λ-shaped dispersions, similar to the one in panel a, and the other located at a higher binding energy. The intensity

2The momentum window corresponding to this angular window is 0.57 Acentered˚ at Γ for 60 eV and 0.79 Afor˚ 120 eV. Thus the binding energy averaging over this momentum window is expected to decrease by ≈ 0.1 eV, comparable to the measured decrease of binding energy by ≈ 0.2 eV from 60 to 120 eV photon energies.

54 Figure 7.4. (a) Angle-integrated intensity curves taken near normal emission measured at diﬀerent photon energies from 40 to 140 eV. Filled circles mark the peak positions of the π bands. (b) Extracted peak positions from the angle-integrated intensity curves as a function of kz. The dotted line is the guide to the periodicity of the dispersion. From the symmetry of the ﬁnal states detected, ∗ the inner potential is determined. (c) LDA band structure of the π bands at kz=0 and kz=0.5 c . The energies are stretched by 20%.

variation in the bonding and antibonding bands is attributed to the photoemission dipole matrix element 65. The splitting of the π bands can be confirmed in the (MDCs) shown in panels d and h, where two peaks each from the bonding and antibonding π bands are clearly observed. From the MDC dispersions shown and the peak positions in the energy distribution curve (EDC) at the K point (not shown), this splitting is estimated to be ≈ 0.7 eV. The splitting of π bands near the K point but not near the H point is in agreement with band structure calculation, which confirms the validity of extracting the kz values. This is the first clear demonstration of the splitting of the π bands near EF , while we note that some data in the literature 100,127,104 may now be seen as suggestive of this splitting, as discussed above. Furthermore, the linear Λ-shaped dispersions shown in panels a and c, strongly resemble those of Dirac quasiparticles, are signatures of Dirac quasiparticles.

7.5.2 Low energy dispersions and Dirac fermions

To gain more information on the low energy excitations, in Fig. 7.6 we show high resolution ARPES data measured along AHL0 direction for HOPG (panels a-b) and single crystal graphite (panels c-d). By following the maximum intensity in both panels (a, c), it is clear that the dispersion shows a linear behavior. In panels b and d we show the raw MDCs at diﬀerent binding energies. In all the MDCs, one can clearly distinguish a main peak and a weaker one. The extracted dispersion extracted from MDC (open circles in

55 Figure 7.5. ARPES intensity map measured on HOPG near the BZ corners at photon energies of ∗ ∗ 43 (kz ≈ 0.35 c ) and 55 eV (kz ≈ 0.10 c ) respectively. AB and BB label the antibonding and bonding π bands. (c-d) MDCs at -1.2 eV for data shown in panels a and b respectively. (e-f) ARPES intensity map measured on single crystal graphite near the zone corner at photon energies of 140 ∗ ∗ eV (kz ≈ 0.50 c ) and 80 eV (kz ≈ 0.07 c ) respectively. (g-h) MDCs at -1.2 eV for data shown in panels e and f.

panels a,c) shows a linear behavior, strongly resembling the behavior of Dirac quasiparticles. The Fermi velocity, or the eﬀective speed of light, is determined to be 0.8±0.2×10−6 m·s−1. By extrapolating the dispersion, the crossing point of these two linear bands (known as the Dirac point) lies 50±20 meV above EF , suggesting that the low energy excitations near H point are holes. Finally, by measuring the volume of the hole pocket 3, we estimate a total hole concentration 64 of 3.1±1.3×1018 cm−3, in agreement with transport measurements 112,111.

7.5.3 Defect states

Within each plane of graphene or stack of plane in graphite a varying number of imperfections can be found such as vacancies, when lattice sites are unfilled indicating a missing atom within a basal plane; stacking faults when the ABAB sequence of the layers planes is no longer maintained; and zigzag edge occurring near unfilled lattice sites 116,128. Because of the unique electronic structure of Dirac quasiparticles and the negligible density of states near the Fermi energy, graphite is extremely sensitive to topological defects which can strongly modify the electronic structure and the scattering process of quasiparticles, thus resulting in important changes in transport properties. For example, it has been predicted 116 that extended defects as lattice dislocation can lead to self doping effects and presence of localized states at the Fermi energy. Self doping effect results in electron or hole pocket at EF rather than a single point, as predicted for an ideal graphene system. Zigzag edge on the other hand induces localized states near the Fermi energy resulting in a peak in the local density of states near the Fermi energy 83,115. Finally the presence of vacancies strongly modifies the scattering process resulting in a minimum of the scattering rate at finite energy 116 instead of EF as what is expected for Fermi liquid. In Fig. 7.7, we introduce disorder/inhomogeneity features that are pronounced only in the case of HOPG samples. This figure shows the first derivative in energy of an ARPES map, from EF to -11 eV. The first derivative is a method which allows to enhance the dispersive features as well as rising or falling edges in the data, and thus is particularly useful for detecting nondispersive peaks and edges. In panel a we can

3 We assume that the hole pocket is an ellipsoid that occupies half of the BZ along kz direction, and the cross section in the −1 kx-ky plane plane has a diameter of 0.02 A˚ (measured from the separation of the peaks at EF ).

56 Figure 7.6. ARPES intensity map measured on HOPG near the BZ corners at photon energies of ∗ ∗ 43 (kz ≈ 0.35 c ) and 55 eV (kz ≈ 0.10 c ) respectively. AB and BB label the antibonding and bonding π bands. (c-d) MDCs at -1.2 eV for data shown in panels a and b respectively. (e-f) ARPES intensity map measured on single crystal graphite near the zone corner at photon energies of 140 ∗ ∗ eV (kz ≈ 0.50 c ) and 80 eV (kz ≈ 0.07 c ) respectively. (g-h) MDCs at -1.2 eV for data shown in panels e and f.

clearly distinguish three nondispersive features at -2.9 eV, -4.3 eV and -7.8 eV, indicated by arrows on the same figure. The nondispersive nature of the features can be checked by the EDCs shown in panels b and c. These features, appearing as sharp extended horizontal lines in panel a suggests the presence of nondispersive localized states. The energies of these nondispersive features occur at the top and bottom of the dispersive band and these features are strongly connected with the dispersive features associated with the π and σ bands. Thus we have associated the presence of such nondispersive features to the elastic scattering of electrons in either the initial state or the final state by inhomogeneity or disorder 101. We now discuss the more important effect of disorder in the low energy electronic properties, as observed in all types of graphite samples. In order to investigate this effect, we have performed position dependent ARPES study with main focus on the near EF states. Each spectrum is averaged over ≈ 100 µm, the spot size of the synchrotron beam. We find that some of the low energy properties of the electronic structure are indeed strongly position dependent and we associate them with the presence of disordered states. We note that similar behavior has been observed in both single crystal graphite and HOPG, suggesting that these are important properties associated with these materials.

In Fig. 7.8 we report typical ARPES intensity map near EF for an HOPG sample (panels a, b) and single crystal graphite (panel c, d), taken in different positions on the same sample. While panels a and c show a parabolic π band dispersion, it is clear that, as we change position on the sample (panels b, d) within the same sample we observe an additional weakly-dispersive electron-like feature, within 50 meV below EF (panels b, d), coexisting with the parabolic π band. Note that while the presence of the π band is position independent, observed in all the samples studied for all positions measured, this additional electron-like feature is strongly position dependent, for each of the sample studied. We now discuss the origin of this feature. While it seems appealing to associate this electron-like feature with the electron pocket predicted by band structure at kz=0, we note that its position dependence as well as the detailed analysis of the effective mass and carrier concentration (discussed in details later), clearly suggests a different origin for this feature.

Fig. 7.9 shows the details of this additional band near EF , from data taken on HOPG. The details of this electron-like band and the π band are analyzed from the MDCs and EDCs shown in panels b-g. Panel b shows the MDC at EF . A main peak in the center, associated with the π band, and two side peaks associated with the additional band, can be distinguished. As the binding energy increases, the low energy band is no longer present and the peaks in the MDC at -100 meV (panel c) are from the main π band. However, the dispersions of the low energy bands are diﬃcult to follow, partly due to the highly parabolic bands in this

57 Figure 7.7. (a) ARPES intensity map taken on HOPG sample at 26 eV photon energy (kz ≈ 0.13 c∗). (b) ARPES intensity map measured at the same conditions as panel a except on a diﬀerent spot inside the sample. The arrow points to the additional intensity near EF . (c) ARPES intensity map measured on single crystal graphite with 50 eV photon energy. The dotted line is extracted dispersion from EDCs. (d) ARPES intensity maps measured at the same conditions as panel c except on a diﬀerent spot. The dotted line is the dispersion extracted from panel c. The open circles are the dispersions extracted for the additional feature at low energy.

region that makes an MDC analysis hard to interpret. On the other hand, in the EDCs, this low energy feature shows up clearly as a small well-deﬁned peak at k1 (panel e), k12 (panel g) and a small hump in k7 (panel f) and thus we use EDC analysis (panel d) to extract the dispersion. The extracted EDC dispersions are overplotted in panel a for both the low and high energy bands. By ﬁtting the electron pocket, we can directly measure the mass of the electrons. This gives a value 0.42 ± 0.07 me, which is much larger than that of electrons and holes as measured in transport 112,111,114. Also, by estimating the volume of the large electron pocket 4, we obtain an electron concentration of 8.0±0.7×1019 cm−3. This electron concentration is again an order of magnitude higher than the value reported by transport measurements 112,111. One possible explanation for this large electron pocket is due to defect-induced localized states 116,83,129, e.g. states along a zigzag edge. Additional support comes from STM, where a peak in the local density of states at an energy (≈ -0.03 eV) similar to the weakly dispersing electron pocket discussed here 115, is observed near zigzag edges.

4The concentration of electrons can be estimated by the volume of the electron pocket. This is done by assuming that the electron pocket is an ellipsoid that occupies half of the BZ along kz direction, and the cross section in the kx-ky plane has a diameter of ≈ 0.1A˚−1. This gives a volume of ≈ 1.2±0.1×10−4 of the BZ size, which corresponds to the electron concentration here estimated.

58 Figure 7.8. (a) ARPES intensity map taken on HOPG sample at 26 eV photon energy (kz ≈ 0.13 c∗). (b) ARPES intensity map measured at the same conditions as panel a except on a diﬀerent spot inside the sample. The arrow points to the additional intensity near EF . (c) ARPES intensity map measured on single crystal graphite with 50 eV photon energy. The dotted line is extracted dispersion from EDCs. (d) ARPES intensity maps measured at the same conditions as panel c except on a diﬀerent spot. The dotted line is the dispersion extracted from panel c. The open circles are the dispersions extracted for the additional feature at low energy.

7.6 Conclusions and implication of this study

We report clear evidence that sharp quasiparticle dispersions, in agreement with band structure calculation along the high symmetry directions, can coexist with a circular Fermi energy intensity map, a definitive signature of azimuthal disorder 124. The measured dispersions from HOPG are compared with those measured from single crystal graphite. The good agreement between data taken on these two different kinds of samples suggests that more ARPES opportunities can be made available for layered materials even when high quality single crystals of large size are difficult to obtain. A typical example of such samples is the 97 recently discovered superconducting C6Ca , where intercalation is more favorable in HOPG than in single crystalline graphite, thus the resulting superconducting C6Ca is azimuthally disordered. This study suggests that even with such an azimuthally disordered sample, it is still promising to study with ARPES to obtain useful information.

59 ∗ Figure 7.9. (a) ARPES intensity map taken at 26 eV photon energy (kz = 0.13 c ) from a HOPG sample. Red and blue circles are the dispersions extracted from EDCs for the electron-like band and π bands, while the dotted black line is the dispersion extracted from MDCs. (b) MDC at EF . Blue, red and gray dotted lines are the peaks used to ﬁt the MDC. (c) MDC at -0.1 eV. (d) EDCs taken at momenta from k0 to k15 indicated by vertical tick marks on the top of panel a. The EDCs at k0 to k5 are scaled by diﬀerent factors so that all the peaks can be seen. (e-g) EDCs at k1, k7, k12.

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67 List of Figures

1.1 (a) Sp2 bonding and the resulting carbon allotropes from 0D to 3D...... 2 1.2 Unit cell of graphene in the real space (a) and reciprocal space (b)...... 2 1.3 The band structure of graphene...... 3 1.4 The band structure of graphene and the schematic cartoon...... 6

1.5 Schematic drawing of the dispersions near EF for massless Dirac fermions (a) and massive quasiparticles in conventional condensed matter systems (b)...... 8 1.6 Brillouin zone and electronic structure of bulk graphite...... 9

2.1 (a) Schematics of ARPRES experiments (courtesy of J. Fink) and (b) typical ARPES data from graphite...... 12 2.2 Spectral function for noninteracting (left panel) and interacting (right panel) systems, from Damascelli et al 18...... 15 2.3 Typical two dimensional ARPES data and the analysis of EDCs and MDCs...... 16 2.4 The universal inelastic mean free path of electrons with kinetic energy from 2 to 2000eV, from H¨ufner 19...... 18 2.5 The improvement of the ARPES resolution in the past three decades, from Reinert et al 30.. 19 2.6 The setup of the 6.994 eV laser and the demonstrated energy resolution of 0.36 eV, from Kiss et al 31...... 20

3.1 (a-d) LEED patterns with a primary energy of 180eV, obtained at four different stages during ◦ the growth of sample A. (a) 1×1 spots of SiC, after a 5√ min anneal√ around 1000 C followed by the initial cleaning procedure√ under√ Si flux. (b) ( 3 × 3)R30 reconstruction, after 5 ◦ ◦ min around√ 1100√ C. (c) (6 3 × 6 3)R30 reconstruction, after 10 min around 1200 C. (d) Sharper(6 3×6 3)R30 pattern, after 4 min around 1250 ◦C. (e) and (f) LEED patterns with a primary energy of 130eV taken at the same stages as (c) and (d), respectively. In (f), the appearance of the 1×1 spots of graphite, located slightly farther out from the center relative to the spots observed at similar positions in (b, c, and e), indicates that a thin graphene overlayer has been formed in this last step...... 25 3.2 SEM images recorded on the surface of a sample grown in the same conditions as sample A. (a) Typical region of the surface, showing a pattern with a length scale on the order of hundreds of nanometers, and a finer pattern with a length scale on the order of 10 nm. (b) Region marked by a cluster of unidentified structures characterized by six-sided geometry, reflecting the underlying hexagonal lattice...... 27

68 3.3 XPS results showing data (connected symbols) and their ﬁts (solid lines) for samples A, B, and C. Individual Voigt functions and Shirley background are shown as lines as well. In the C 1s spectra, peaks G and S are identiﬁed with the graphite overlayer and SiC bulk. In each row, the two panels are plotted using a common intensity scale...... 28 3.4 (a-c) Dispersions of the π bands from single layer, bilayer graphene to trilayer graphene. (d-f) Dispersions near the K point from single layer graphene to trilayer graphene. From Partoens et al 13...... 30

3.5 Intensity map at -1 eV as a function of kk and kz...... 31 3.6 Measured dispersions from single layer graphene to four layer graphene, from Ohta et al 48.. 31 3.7 LEEM images taken at an electron energy of 4.2 eV before and after annealing the SiC substrate under Si flux with a field of view of 5µm...... 32 3.8 LEEM images taken at an electron energy of 6.6 eV with a 3µm field of view and the energy scans for the buffer layer, 1ML and 2ML graphene...... 32

4.1 (a-c) ARPES intensity maps taken at EF , -0.4 eV and -1.2 eV respectively on single layer graphene. The dotted line shows the Brillouin zone of graphene. (d) schematic drawing of the dispersion in single layer graphene and the relative energies for data shown in panels a-c. (e) Dispersion of single layer graphene measured along a high symmetric direction through the K point (see black line in the inset)...... 35 4.2 Observation of the gap opening in single layer graphene at the K point. (a) Structure of graphene in the real and momentum space. (b) ARPES intensity map taken along the black line in the inset of panel (a). The dispersions (black lines) are extracted from the EDC peak positions shown in panel (c). (c) EDCs taken near the K point from k0 to k12 as indicated at the bottom of panel (b). (d) MDCs from EF to -0.8 eV. The blue lines are inside the gap region, where the peaks are non-dispersive. (e) Angle integrated intensity, which shows a suppression of intensity near ED...... 36 4.3 (a,b) Dispersions taken along a symmetric direction for cuts away and through the K point. (c,d) MDCs in the energies as labeled in panels a,b. (e) Angle integrated intensity as a function of energy for data in panel a (black curve) and panel b (gray curve)...... 38 4.4 (a) Schematic drawing for the cuts shown in (b) and (c) in the BZ of graphene and the conical dispersion (not drawn to scale). (b,c) Dispersions measured through and oﬀ the K point. The dotted white and dark gray lines are dispersions extracted from the MDCs. (d, e) Extracted dispersions and MDC width as a function of energy for data shown in panels b and c. . . . . 38 4.5 (a,b) Simulation of the conical dispersions with a gap of 150 and 400 meV. (c) Extracted MDC width as a function of energy for data shown in panels a and b...... 40 4.6 Decrease of the gap size as the sample becomes thicker. (a-d) ARPES intensity maps taken on single layer graphene on 6H-SiC, bilayer graphene on 4H-SiC, trilayer graphene on 6H-SiC and graphite respectively. Data were taken along the black line in the inset of Fig. 4.2(a) except panel (c), which was measured along ΓK direction and symmetrized with respect to the K point to remove the strong intensity asymmetry induced by dipole matrix element 65. (e, f) EDCs taken from the raw data (without symmetrization) for momentum regions labeled by the arrows at the bottom of panels (b) and (c)...... 41 4.7 Dispersions measured in bilayer graphene on 6H-SiC (panel a) and more insulating 4H-SiC (panel b) substrates...... 42

69 4.8 Thickness dependence of ED and ∆. (a,b) ED and ∆ as a function of sample thickness, for epitaxial graphene on 6H-SiC (black) and on 4H-SiC (blue). The error bar for the sample 62 thickness was taken from the XPS measurements . For graphite, ED is extrapolated from 64 9,74 the dispersions at kz≈π/c , and the gap is estimated from band structure calculation . (c, d) Two possible mechanisms to open up a gap at the Dirac point. (e) Schematic drawing to show the inequivalent potentials on the A (blue) and B (red) sublattices induced by the interface...... 43 4.9 (a-c) LEEM images taken at electron energy of 6.6 eV to show the surface topology of single layer graphene (gray area) and corresponding ARPES data (d-f) taken through the K point (see vertical line in the inset of Fig.1(d)) for three single layer graphene samples prepared under different growth conditions. The white, gray and black colors in panels (a-c) represent the regions of buffer layer, single layer and bilayer graphene respectively. The red lines in panels (d-f) are dispersions extracted by fitting the EDCs. (g) Plot of the extracted gap size from ARPES as a function of the representative terrace size (red segments in panels a-c) of the single layer graphene. The dotted line is the gap size in graphene nanoribbons due to quantum confinement taken from Han et al 82...... 45 4.10 Proposed mechanism for the gap opening and the structure of epitaxial graphene on SiC. . . 47

4.11 Breaking of the six fold symmetry in the intensity map near ED. (a-d) ARPES intensity maps taken on single layer graphene at EF ,ED, -0.8 eV and -1.0 eV respectively. Near ED (panel b), the intensity of the six replicas near K shows breaking of six fold symmetry. Note that to enhance the additional feature around ED, the color scale is saturated for the dominant features near K and the replicas. (e) ARPES intensity map of the calculated spectral function at ED in the presence of symmetry breaking on the two carbon sublattices...... 48

5.1 Dispersions through the K point taken from the as-grown single layer graphene (a) and from the sample with the highest doping of NO2 (c). Panels (b) and (d) show the angle integrated spectra for (a) and (c) respectively. In panels (c) and (d) note the appearance of the NO2 states at ≈ 5 and 11 eV below EF ...... 52 5.2 (a) Dispersions through the K point taken in as-grown bilayer graphene. Data were taken along the red line through the K point shown in the inset. (b) Dispersions taken after 0.6 L 6 (1 Langmuir=10 torr · s) NO2 adsorption. (c) data taken 6 minutes after panel b. (d) EDCs taken at k1, k2, k3 and k4 as labeled on the top of panels a and b. The black and red arrows point to the midpoint of the leading edge, which is shifted to higher binding energy after NO2 doping. (e) Zoom in of data shown in panel b. The white dotted line is a guide for the eye for the dispersions. (f) Momentum distribution curve (MDC) at the energy labeled by the dotted black line in panel d. The dots are the raw data and the solid line is the ﬁt using three Lorentzian peaks simulating the cross-section of the hat-like dispersion in panel e...... 54

5.3 (a-f) Data taken through the K point for the as-grown (a) and various dopings with NO2 adsorption (b-f) in single layer graphene. The white lines are dispersions extracted from the MDC peaks when they can be clearly resolved. The dotted lines in panels c and d are linear extrapolation of the dispersion. (g) EDCs taken at the momentum regions (indicated by a small tick mark on top of each panel) where the bands are closest to EF ...... 56

5.4 (a, b) Plot of the shift in ED and Fermi velocity as a function of the carrier concentration for data shown in Fig. 5.3 In panel b, the data are extracted by ﬁtting the dispersion between EF and -0.3 eV (circles) and between EF and -0.1 eV (diamonds). The open symbols are extracted from the dispersions on the left and the ﬁlled symbols from the dispersions on the right...... 57

70 6.1 Dispersions measured near H and K, showing the general consistency of the extracted kz ∗ 0 values. (a) Dispersion near H (hν=140 eV, kz ≈ 0.50 c ) along HH direction, showing that ∗ the π bands are degenerate. (b) Dispersion near K (hν=80 eV, kz ≈ 0.07 c ) along direction parallel to HH0, where the π bands split into bonding (BB) and antibonding (AB) bands. . . 60 6.2 Linear Λ-shaped dispersion near the BZ corner H. (a) ARPES intensity map taken near the ∗ H point (photon energy hν=140 eV, kz ≈ 0.50 c ), along a cut through H and perpendicular to kx (see red line in the BZ shown in panel c). The inset shows a schematic diagram of the Dirac cone dispersion near EF in the three dimensional E-kx-ky space. (b) MDCs from EF to -2.0 eV. The MDCs are normalized to have the same amplitude and displaced by the same amount so that the dispersion can be directly viewed by following the peak positions at each energy. The dotted lines are guides to the eyes for the linear-dispersing peaks in the MDCs. (c) Three dimensional BZ for graphite with high symmetry directions relevant for this paper highlighted with red, green and blue lines...... 62 6.3 Constant energy maps taken near the H point, showing that the electronic structure is isotropic in the kx-ky plane from EF to -0.6 eV. (a-e) ARPES intensity maps near H (hν=140 eV, ∗ kz ≈ 0.50 c ) taken at energies from EF to -1.2 eV. The circles are guides for the circular intensity pattern near the H point. Arrows in panels d and e point to deviation from the circle. (f) Schematic diagram of the dispersion for graphene near six BZ corners in the three dimensional E-kx-ky space...... 63 6.4 Detailed low energy dispersion near the H point shows that low energy excitations are Dirac fermions with the Dirac point slightly above EF . (a) ARPES intensity map near the H point ∗ 0 (hν=65 eV, kz ≈ 0.45 c ) along AHL direction (green line in the BZ shown in Fig. 6.2(c)). The open blue circles in panel a show the MDC dispersion. The dotted straight lines are guides for the linear dispersion. (b) MDCs at energies from EF to -0.2 eV for data shown in panel a. Note that, similar to Fig. 6.3, the intensity of the π band is strongly enhanced in the first BZ (H→A direction), due to the dipole matrix element 65. (c) Intensity obtained by integrating ∗ over both kx and ky for data taken near H (hν=140 eV, kz ≈ 0.50 c ). The intensity has been symmetrized with respect to the Dirac point energy (ED ≈ 50 meV) to compare directly with the expected intensity for Dirac fermions (inset). An overall linear behavior is observed with some weak additional intensity around 100 meV from the Dirac point energy. The origin of this additional weak intensity is unclear and needs further investigation...... 65 6.5 Detailed dispersion near K, which shows that quasiparticles with finite effective mass and defect-induced localized states also contribute to the low energy electronic dynamics. (a) ∗ 0 ARPES intensity map near K (hν=50 eV, kz ≈ 0.08 c ) along ΓKM direction (blue line in the BZ shown in Fig. 6.2(c)). The open circles are the dispersions extracted from MDCs. (b) MDCs from EF to -50 meV for data in panel a. The open circles mark the peaks clearly resolved in the data. The inset shows the MDC dispersion from -10 to -50 meV, with the parabolic fit used to extract the effective mass...... 66 6.6 Detailed dispersion near K, which shows an additional electron pocket induced by defect states. (a, b) Intensity maps near the K point measured in different parts of the sample, which shows an additional large electron pocket. The open circles are dispersions extracted from EDCs shown in panel f. (c) MDC at EF from data shown in panel b. The black arrows point to the peaks from the large electron pocket which are separated by ≈ 0.1 A˚−1, while the gray arrow points to the peak from the π band. (d) EDCs from k0 to k12, as indicated in panel b. Open circles are the peak positions for the large electron pocket...... 68 6.7 STM image of zigzag and armchair edges (a) and typical dI/dV from STS data at a zigzag edge, from Kobayashi et al 115...... 69

7.1 BZ and measured dispersions on LaSe, from Nakayama et al 122...... 72

71 7.2 (a) Fermi energy intensity map. The hexagonal Brillouin zone (dashed lines) and Fermi surface (shaded circles) expected for single crystalline graphite are drawn schematically. (b) Intensity map versus binding energy and in-plane momentum along the solid line in (a) taken at 60 eV photon energy. Arrow marks the Fermi energy crossing point (kF ). The inset shows EDC ◦ at kF taken at 25 eV photon energy in the high angular resolution (0.1 ) mode. (c) Second derivative of raw data in (b) with respect to energy. LDA dispersions along both Γ-K-M’ direction (solid lines) and Γ-M-Γ’ direction (dashed lines) are plotted for comparison. The Brillouin zones are labeled on top of this panel for the two high symmetry directions...... 74 7.3 Dispersions for azimuthal angles φ=0◦(panel a); 10◦(panel b) and 20◦(panel c). The φ angle is defined in the inset of panel a. LDA band dispersions along Γ-K-M’ (solid lines) and Γ-M- Γ’ (dotted lines) are plotted for comparison. (d) Calculated dispersions for single crystalline graphite along an arc (shown in the inset) with radius equal to ΓK distance. (e) Calculated density of states Dφ(E) for single crystalline graphite by integrating over an arc from A to B (see inset of panel d). Singular peaks in Dφ(E) occur at energies corresponding to band energy extrema, some of which are shown in panel(c,d) as shaded circles and open circles for A (ΓK direction) and B (ΓM direction) respectively...... 76 7.4 (a) Angle-integrated intensity curves taken near normal emission measured at different photon energies from 40 to 140 eV. Filled circles mark the peak positions of the π bands. (b) Extracted peak positions from the angle-integrated intensity curves as a function of kz. The dotted line is the guide to the periodicity of the dispersion. From the symmetry of the final states detected, the inner potential is determined. (c) LDA band structure of the π bands at kz=0 and kz=0.5 c∗. The energies are stretched by 20%...... 78

7.5 ARPES intensity map measured on HOPG near the BZ corners at photon energies of 43 (kz ≈ ∗ ∗ 0.35 c ) and 55 eV (kz ≈ 0.10 c ) respectively. AB and BB label the antibonding and bonding π bands. (c-d) MDCs at -1.2 eV for data shown in panels a and b respectively. (e-f) ARPES intensity map measured on single crystal graphite near the zone corner at photon energies of ∗ ∗ 140 eV (kz ≈ 0.50 c ) and 80 eV (kz ≈ 0.07 c ) respectively. (g-h) MDCs at -1.2 eV for data shown in panels e and f...... 80

7.6 ARPES intensity map measured on HOPG near the BZ corners at photon energies of 43 (kz ≈ ∗ ∗ 0.35 c ) and 55 eV (kz ≈ 0.10 c ) respectively. AB and BB label the antibonding and bonding π bands. (c-d) MDCs at -1.2 eV for data shown in panels a and b respectively. (e-f) ARPES intensity map measured on single crystal graphite near the zone corner at photon energies of ∗ ∗ 140 eV (kz ≈ 0.50 c ) and 80 eV (kz ≈ 0.07 c ) respectively. (g-h) MDCs at -1.2 eV for data shown in panels e and f...... 81 ∗ 7.7 (a) ARPES intensity map taken on HOPG sample at 26 eV photon energy (kz ≈ 0.13 c ). (b) ARPES intensity map measured at the same conditions as panel a except on a different spot inside the sample. The arrow points to the additional intensity near EF . (c) ARPES intensity map measured on single crystal graphite with 50 eV photon energy. The dotted line is extracted dispersion from EDCs. (d) ARPES intensity maps measured at the same conditions as panel c except on a different spot. The dotted line is the dispersion extracted from panel c. The open circles are the dispersions extracted for the additional feature at low energy...... 82 ∗ 7.8 (a) ARPES intensity map taken on HOPG sample at 26 eV photon energy (kz ≈ 0.13 c ). (b) ARPES intensity map measured at the same conditions as panel a except on a different spot inside the sample. The arrow points to the additional intensity near EF . (c) ARPES intensity map measured on single crystal graphite with 50 eV photon energy. The dotted line is extracted dispersion from EDCs. (d) ARPES intensity maps measured at the same conditions as panel c except on a different spot. The dotted line is the dispersion extracted from panel c. The open circles are the dispersions extracted for the additional feature at low energy...... 83

72 ∗ 7.9 (a) ARPES intensity map taken at 26 eV photon energy (kz = 0.13 c ) from a HOPG sample. Red and blue circles are the dispersions extracted from EDCs for the electron-like band and π bands, while the dotted black line is the dispersion extracted from MDCs. (b) MDC at EF . Blue, red and gray dotted lines are the peaks used to ﬁt the MDC. (c) MDC at -0.1 eV. (d) EDCs taken at momenta from k0 to k15 indicated by vertical tick marks on the top of panel a. The EDCs at k0 to k5 are scaled by diﬀerent factors so that all the peaks can be seen. (e-g) EDCs at k1, k7, k12...... 85