Optical Properties of AA-Stacked Bilayer Graphene

Transcription

1 Optical Properties of AA-Stacked Bilayer Graphene by Calvin Jerome Tabert A Thesis presented to The University of Guelph In partial fulfilment of requirements for the degree of Master of Science in Physics Guelph, Ontario, Canada Calvin Jerome Tabert, August, 2012

2 ABSTRACT OPTICAL PROPERTIES OF AA-STACKED BILAYER GRAPHENE Calvin Jerome Tabert University of Guelph, 2012 Advisor: Professor Elisabeth J. Nicol Theoretical predictions of the AC conductivity of both monolayer and Bernal-stacked bilayer graphene have largely been in agreement with experimental observations. Due to the recent realization of AA-stacked samples, we provide theoretical predictions for this system. We begin this thesis with a review of the optical properties of graphene and provide a brief discussion of the previously studied Bernal-stacked bilayer. We then calculate the optical conductivity of AA-stacked bilayer graphene as a function of frequency for several interesting cases. We are primarily interested in the case of finite doping due to charging. Unlike the monolayer, we see a Drude absorption at charge neutrality as well as an interband absorption with strength twice that of the monolayer background conductivity which onsets at twice the interlayer hopping energy. We examine the behaviour as we vary the chemical potential relative to the interlayer hopping energy scale and compute the partial optical sum. We also consider the effect of adding a bias across the layers and find it serves merely to renormalize the interlayer hopping parameter. While interested in the in-plane conductivity we also provide the perpendicular conductivity of the AA-stacked bilayer. We then extend the ideas to the AAA-stacked trilayer. Based on proposed models for topological insulators discussed in the literature, we consider the effect of spin orbit coupling in both one and two layers on the optical properties of the AA-stacked bilayer which illustrates the effect of opening an energy gap in the band structure.

3 ACKNOWLEDGEMENTS I would like to thank my supervisor Dr. Elisabeth J. Nicol for her guidance and support in my learning and development without whom this thesis would not be possible. Additionally, I would like to thank Drs. Nickel and Dutcher for their ideas and support as members of my committee as well as Dr. Wickham for chairing my defence. I would like to thank James LeBlanc for his numerous discussions and ideas that helped keep me moving forward. I would also like to thank my beautiful wife Esther for her continued support and understanding of the hours required to complete this work. I would like to acknowledge those who brought me to the point of being able to attempt such an undertaking: my undergraduate professors for their guidance and assistance, my parents and family for their support in my education and the many friends that I made along the way. I would like to give a big thanks to my fellow grad students for making this such a fun time and my fellow teammates on The Net Force hockey team and DISCombobulated ultimate Frisbee and softball teams. iii

4 Contents 1 Introduction Atomic Examination of Graphene Formation of Graphene Tight-binding on the Honeycomb Lattice Energy Dispersion Wavefunctions Density of States Conductivity AB-Stacked Bilayer Graphene Energy Dispersion Conductivity Longitudinal Conductivity Perpendicular Conductivity AA-Stacked Bilayer Graphene Energy Dispersion Conductivity Longitudinal Conductivity Perpendicular Conductivity Biased AA-Stacked Bilayer Graphene AAA-Stacked Trilayer With Spin Orbit Coupling iv

5 3.4.1 Energy Dispersion Optical Conductivity Conclusion 76 A 78 A.1 Graphene Energy Dispersion A.2 Low Energy Expansion About the K Point A.3 Analytic Solution to the Conductivity B 86 B.1 Analytic Solution for AA-Stacked Bilayer Conductivity C 92 C.1 Sample C Code for Calculating the Conductivity of Graphene Bibliography 95 v

6 List of Figures 1.1 Graphene crystal lattice with emphasized triangular sublattices associated with A and B sites Two dominant stacking regimes of bilayer graphene. (a) Bernal or AB stacking. (b) AA-stacked bilayer Two graphene sheets twisted relative to each other giving rise to regions rich in AA and AB stacking Energy dispersion for graphene over and beyond the first Brillouin zone Comparison of the low energy linear approximation, Eqn. (1.24) (red) and the full low energy dispersion, Eqn. (1.23) (blue) for graphene about the K point ARPES image of the low energy dispersion of graphene taken from Ref. [1]. Electron density (per cm 2 ) is given by the yellow numbers Low energy dispersion for graphene over and beyond the first Brillouin zone Plane wave associated with the K point moving through the lattice Full density of states per unit area over the first Brillouin zone along with the density of states using the low energy expansion (a) Schematic of optical probing of a material. (b) Conductivity in terms of transitions between energy bands Feynman bubble for conductivity Conductivity for undoped graphene Conductivity for doped graphene Comparison between the analytic solution and numerical calculation of the longitudinal conductivity of graphene for µ/t = vi

7 2.1 Bernal (AB) stacking of bilayer graphene Top: Low energy dispersion for AB-stacked bilayer about the K point (a) without an asymmetry gap = 0, (b) with an asymmetry gap = 1.5γ. Bottom: Low energy density of states in units of 2γ/π( v F ) 2 (c) without an asymmetry gap, (d) with an asymmetry gap = 1.5γ (a) Band structure for unbiased AB-stacked bilayer graphene ( /γ = 0). (b) Conductivity of unbiased AB-stacked bilayer graphene for various values of chemical potential (a) Band structure for biased AB-stacked bilayer graphene ( /γ = 1.5). (b) Conductivity of biased AB-stacked bilayer graphene for various values of chemical potential Perpendicular conductivity for unbiased Bernal-stacked bilayer for various values of chemical potential AA-stacked bilayer graphene Low energy dispersion for AA-stacked bilayer graphene Double spin density of states for AA-stacked bilayer graphene in units of 2γ/π( v F ) Longitudinal conductivity for AA-stacked bilayer graphene. (a) µ < γ. (b) µ > γ. Inset: Band structure around one of the K points Comparison of the analytic solution for the conductivity of AA-stacked bilayer graphene to the full numerical work Imaginary part of the zero temperature longitudinal conductivity of AA-stacked bilayer graphene Partial optical sum in units of γ for various values of chemical potential. (a) µ < γ. (b) µ > γ. Inset: The evolution of the positive frequency spectral weight W found under half the delta function in the analytic solution for the conductivity, Eqn. (3.13), as a function of chemical potential Perpendicular conductivity of AA-stacked bilayer graphene Band structure for AAA-stacked trilayer graphene. The three colours represent the three decoupled monolayer dispersions vii

8 3.10 Conductivity for AAA-stacked trilayer graphene. Interband absorption edges appear at Ω = 2µ, 2 2γ µ and 2( 2γ + µ). These transitions are shown on the band structure given in the inset Band structure for SOC in both layers. (a) No SOC (b) < γ (c) > γ Density of states for SOC in both layers compared to that of no SOC in units of 2γ/π( v F ) Band structure for SOC in one layer. (a) No SOC (b) < γ (c) > γ Density of states for SOC in one layer compared to that of no SOC in units of 2γ/π( v F ) Conductivity for AA-stacked bilayer graphene with SOC in both layers (a) = 0.75γ and 1.2γ with no doping. (b) = 0.75γ with finite doping (µ = 0.5γ). (c) Various values of for µ = Conductivity for AA-stacked bilayer graphene with SOC in one layer (a) = 0.5γ for zero doping. (b) = 1.2γ for zero doping. (c) = 0.5γ with finite doping. (d) = 1.2γ with finite doping A.1 First Brillouin zone of the graphene lattice viii

9 Chapter 1 Introduction Carbon, the fourth most abundant element in the universe[2], is an integral component of all organic life. Capable of forming several allotropes, it is often found in the form of diamond and graphite. The variance in crystal structure between these allotropes leads to a vast difference in their physical properties. For example, graphite is a good conductor of electricity while diamond has a low conductivity. Graphite is extremely soft, hence its use in pencils, while diamond is one of the hardest known materials. Even their appearance is drastically different; diamond is transparent while graphite is opaque and black. The crystalline structure of graphite is such that carbon atoms bond strongly together to form a two-dimensional honeycomb lattice known in the free state as graphene; these planes then stack above each other to form the bulk sample. Although graphene was originally believed to be unstable in the free state[3], it was successfully isolated in 2004 by Andre Geim and Konstantin Novoselov at the University of Manchester using what has become known as the Scotch tape method. They used the adhesive tape to repeatedly split graphite into thinner and thinner pieces. The tape was then dissolved in acetone and, with several more steps, the flakes sedimented onto a silicon wafer, graphene was then identified using an optical microscope. Novelslov and Geim were awarded the 2010 Noble Prize for their work. This exciting discovery led to a plethora of research as graphene, which had been known to exhibit a linear low energy dispersion[4], promised to provide insight into the physics of massless Dirac fermions. This linear dispersion was shown to arise from 1

10 a simple nearest-neighbour-hopping-tight-binding Hamiltonian which maps on to a Dirac Hamiltonian for massless fermions with a Fermi velocity v F 10 6 m/s[5]. Some of the exotic features that have already been uncovered include an unusual quantum Hall effect[6, 7], giant Faraday rotation[8] and plasmarons[9]. These features and many more have been summarized in several reviews[10, 11, 9]. Bilayer graphene is also of particular interest as it too exhibits an unusual quantum Hall effect and its low energy tight-binding Hamiltonian maps to an equation for chiral fermions with an effective mass based on an interlayer hopping parameter γ. The crystal structure of graphene, shown in Fig. 1.1, is an example of a Bravais lattice with a two point basis. We can see that graphene is comprised of two triangular sublattices and thus while every carbon is chemically the same, they have topological differences. These inequivalent sites are labelled A and B and any primitive cell will contain two atoms and thus result in two energy bands. If we consider the reciprocal Figure 1.1: Graphene crystal lattice with emphasized triangular sublattices associated with A and B sites. lattice, we will have a hexagonal Brillouin zone which contains two K points labelled K and K, see Fig. A.1. Not only can graphene exist in the free state, but two or more layers can stack 2

11 above each other to form what is called few-layer graphene. When two sheets stack it is called a graphene bilayer. For bilayer graphene, there are two dominant ways in which the two layers can stack. While, like graphite, bilayer graphene typically adopts Bernal or AB stacking in which only half the atoms are aligned on top of each other and the other half sit over the center of the hexagon in the other layer, it is also capable of stacking in an AA formation in which the atoms in one layer are perfectly aligned with their counterparts in the neighbouring layer. These stackings are illustrated in Fig. 1.2(a) and (b) for Bernal stacking and AA stacking respectively. Due to δ 3 γ δ 1 δ 2 (a) γ (b) Figure 1.2: Two dominant stacking regimes of bilayer graphene. (a) Bernal or AB stacking. (b) AA-stacked bilayer. the preference of AB stacking, there has been little work done on the AA-stacked bilayer. Recently, however, Moiré patterns seen in scanning tunneling microscopy (STM) imaging point to alternative stackings where one layer is twisted relative to the other. In such systems, one may find regions rich in AB stacking as well as regions mainly comprised of AA stacking, see Fig There has also been evidence 3

12 Figure 1.3: Two graphene sheets twisted relative to each other giving rise to regions rich in AA and AB stacking of growing AA-stacked graphene[12] as well as naturally occurring samples[13]. These discoveries have brought forward the importance of further study into the AA-stacked system. For this thesis, we are specifically interested in the optical properties of graphene bilayers. When a metal is subjected to an applied electric field, one expects a current to be induced. When light is incident on a material, some is reflected while the remainder is absorbed or transmitted. In such a situation, the aforementioned electric field is that of the electromagnetic waves of light. Such experiments are known as optics experiments and thus the material property that we observe is called the optical conductivity. A famous example of AC conductivity is the Drude conductivity given by[14] σ(ω) = where Ω is the frequency of the incident photon and σ DC σ DC 1 iωτ, (1.1) is the DC conductivity (σ DC = ne2 τ ), where n is the electron density, e is the charge of an electron and m m is the electron mass. The Drude conductivity is dependent on the material specific scattering rate 1/τ. A more detailed discussion of conductivity follows in Section 1.5 as we examine the conductivity of monolayer graphene. 4

13 1.1 Atomic Examination of Graphene Before we explore the optical properties of graphene, we will begin by discussing the orbital hybridization that allows graphene to behave as it does. Carbon, the sixth element in the periodic table, has a nucleus comprised of six protons and six neutrons while the electron configuration is given by 1s 2 2s 2 2p 2. This means that the first orbital shell is filled while the second orbital shell contains four electrons, two in the 2s orbital and two in the 2p orbital which can be further broken into three orbitals labelled p x, p y and p z which are dumbbell-shaped and perpendicular to each other. As the complete 2 orbital contains eight electrons, these four electrons in the 2s and 2p orbital are valence electrons and thus able to engage in the formation of chemical bonds. An atom is most stable when the s and p orbitals of the outer shell are occupied. An s orbital, with quantum number l = 0, can have two electrons (one spin up and one spin down) while a p orbital, with l = 1 and thus m = 1, 0, 1, can have six electrons (one of each spin per m value). We can see that a carbon atom requires four more electrons to have a stable configuration. Carbon can attain these four electrons through the formation of a covalent chemical bond by which it is able to share another atom s valence electrons. The superposition principle of quantum mechanics states that when an orbital from one atom is in close proximity to an orbital of another atom, two new hybridization orbitals are formed[15]. There are two types of bonds that can be formed, π and σ bonds. In graphene, one s orbital and two p orbitals undergo a sp 2 hybridization which has a trigonal planar geometry such that the three resulting orbitals are in a plane at an angle of 120 to each other. When overlapped with the corresponding hybridized orbitals of another carbon, these three orbitals form the σ and σ bonds which cause each carbon site of graphene to have three nearest neighbours. These bonds are the strongest covalent chemical bond as they are formed by direct overlap of hybridized orbitals. The strong σ bond is what makes the carbon compound stable. The σ orbital is ignored as it is of such high energy that the electrons cannot be 5

14 excited to it[15]. Each σ bond is always completely occupied with two electrons of opposite spin, one from each atom. As a single carbon in graphene is connected to three carbon atoms, σ bonding accounts for three of the valence electrons. There remains one p orbital, p z, which has not hybridized and lies perpendicular to the hybridized plane. When it overlaps with the corresponding unhybridized orbital of the neighbour carbon it forms a π and π bond, or bonding and antibonding pair, which are weaker than the σ bonds due to significantly less overlap between p orbitals resulting from their parallel orientation. Unlike the σ bonds, there is a small energy difference between the π and π bonds and thus, while the π orbital is the ground state, electrons can access the π bond[15]. This results in fast electron transfer between π orbitals and therefore large currents. Each carbon atom contributes one electron to a π bond which acts as a charge carrier for the system. Bilayer graphene is made of two such sheets layered on top of each other weakly bound together through van der Waals forces[11]. 1.2 Formation of Graphene We will now discuss how samples of graphene are prepared. There are three primary ways by which monolayer, bilayer and few-layer graphene are fabricated[11]. The first method is mechanical exfoliation from bulk graphite[16, 17]. This so-called Scotch tape method involves peeling layers from a bulk sample of graphite using either adhesive tape or cleavage by mechanical rubbing. This method has proven highly effective as the layers are weakly bound together. After exfoliating, the number of layers in a sample is determined by optical and Raman spectroscopy[18]. This method is useful to create samples where graphene is not in contact with a substrate and has proven quite reliable in the preparation of films 10µm in size[16]. Another main method is that of de Heer et al.[19, 20, 21, 22] by which hexagonal silicon carbide crystals are heated in a vacuum to 1300 C allowing the silicon atoms to evaporate and leaving behind a pure film of graphene[11]. The final method involves placing a hydrocarbon gas near a metal foil surface[23]. 6

15 The hydrocarbon molecules can be decomposed and carbon atoms are then dissolved into the metal[11]. When the foil is allowed to cool, a carbon film can be formed on the surface. Chemical etching is then used to transfer the film to a polymer or semiconducting substrate[24]. This method is the most effective at producing samples with few defects and is becoming the dominant method[11]. A more detailed discussion of these methods can be found in the review by Abergel et al.[11]. 1.3 Tight-binding on the Honeycomb Lattice Now that we know what graphene is, it is time to delve into the mathematics which yields the exciting properties noted previously. The interesting low energy physics of graphene was shown to arise from a simple tight-binding model. The tight-binding model is an approach to calculate the electronic band structure of a system using an approximate set of wavefunctions based on the superposition of wavefunctions for the isolated atoms which comprise the lattice[14]. While this is a one-electron model, it provides the basis for more advanced many-body problems. As the name suggests, this model describes the properties of electrons which are tightly bound to the atom from which they came and have limited interactions with the potentials supplied by other atoms. With this simplification, the wavefunction of the electron will be very similar to the atomic orbital of the free atom. Thus the most important element of the model is the interatomic matrix elements which for us will be replaced by interand intralayer hopping parameters. A detailed description can be found in Ref.[14] and, here, we only provide a brief sketch. We begin our mathematical discussion by introducing the atomic orbitals φ m (r) which are eigenstates of the Hamiltonian for a single isolated atom H at. When this atom is placed in a crystal, the atomic wavefunction overlaps with the adjacent sites and are thus not true eigenstates of the full crystal Hamiltonian, H. This overlap will be less the tighter the electrons are bound to their respective atoms. The correction to the atomic potential needed to obtain the full Hamiltonian is given by U and 7

16 assumed very small. Our total Hamiltonian can then be written as H(r) = n H at (r n) + U(r), (1.2) where n indexes over all N atomic sites in the lattice. A solution to the single electron Schrödinger equation is approximated as some linear combination of atomic orbitals φ m (r n) given by ψ(r) = a m (n)φ m (r n), (1.3) m,n where m refers to the m th atomic energy level of an atom located at n. This hinges on the assumption that ψ(r) becomes very small when r exceeds a distance on the order of the lattice constant. Due to the symmetry of the crystal and thus the periodic nature of the potential, we utilize Bloch s Theorem which states[14] that when translated through a lattice vector, our wavefunctions can change only by a phase factor, id est, ψ(r + R) = e ik R ψ(r), (1.4) where k is the wavevector of the wavefunction. This puts restrictions on our coefficients which must now satisfy a m (n)φ m (r n + R) = e ik R a m (n)φ m (r n), (1.5) m,n m,n which then imposes the condition a m (R ) = e ik R a m (0). (1.6) If we normalize our wavefuntion to 1, we find a (0)a(0) = 1 N R 0 eik R α(r ), (1.7) 8

17 where α(r ) are the atomic overlap integrals which are usually neglected[25] giving a n (0) 1 N, (1.8) and therefore, ψ(r, k) 1 e ik n φ m (r n). (1.9) N m,n For tight-binding in graphene, it is sufficient to keep only the 2p z orbital at each site and due to the two-point basis, we will have the two Bloch basis wavefunctions ψ A (r, k) = 1 N n A e ik na φ A 1 (r n A ) (1.10) and ψ B (r, k) = 1 N n B e ik nb φ B 1 (r n B ), (1.11) where n A and n B correspond to the location of A and B sites respectively and φ A 1 and φ B 1 correspond to the wavefunction of the 2p z orbital at the A and B sites respectively. Our total wavefunction is made up of a linear combination of ψ A and ψ B. The tight-binding model, becomes intuitive when we work in second quantization notation. If we again use the atomic orbital as a basis state and only allow overlap between nearest neighbours, our Hamiltonian becomes H = t (c i,σ c j,σ + H.c.), (1.12) <i,j>,σ where c i,σ c j,σ are the creation and annihilation operators with corresponding Hermitian conjugate H.c., σ is the spin index, t is the hopping integral and < i, j > indexes the sum over nearest neighbours. Using this formalism, we will consider the Hamiltonian for tight-binding on the honeycomb lattice which leads to the aforementioned linear energy dispersion. 9

18 1.3.1 Energy Dispersion As previously noted, the graphene lattice contains two inequivalent carbon sites, labelled A and B. For both the A and B basis, we can specify two primitive lattice vectors: ( a a 1 = 2, a ) ( ) 3, a 2 = a 2 2, a 3, where a = a 1 = a 2 = 3a cc, (1.13) 2 with a cc the carbon-carbon distance approximately 1.42 Å[26] and thus a 2.46Å. The nearest-neighbour vectors are given by δ i : δ 1 = (a 1 + a 2 ), 3 δ 2 = δ 1 + a 1 = 2 3 a a 2, (1.14) δ 3 = δ 1 + a 2 = 1 3 a a 2, and the lattice vectors are n = n 1 a 1 + n 2 a 2 where n 1 and n 2 are integers. As the low energy dispersion that sparked so much interest in this material was shown to rise from a simple nearest-neighbour-hopping-tight-binding Hamiltonian, we will limit the Hamiltonian to such nearest neighbour hopping. In second quantization notation with a nearest-neighbour hopping amplitude of t 2.8 ev [27, 28], our Hamiltonian becomes H = t n,δ,σ ( ) a n, σ b n+δ, σ + b n+δ, σ a n, σ, (1.15) where a n, σ annihilates an electron on an A site with index n and spin σ and b n+δ, σ creates an electron on one of three nearest-neighbour sites with spin σ. To derive an energy dispersion, we need to transform to momentum space using the usual Fourier 10

19 transformations; we write a n, σ = 1 N e ik n a k, σ, (1.16) k b n+δ, σ = 1 e ik (n+δ) b k, σ. (1.17) N If we substitute Eqns. (1.16) and (1.17) into Eqn. (1.15), we obtain k H = t ( ) 1 e ik n a 1 1 k n,δ,σ N,σ e ik (n+δ) b 1 k,σ + H.c., k N k where we have used b 1 n+δ,σ a 1 n,σ = b 1 n,σ a 1 n δ,σ. Therefore, H = t N n,δ,σ k k ( ) e i(k k ) n e ik δ a 1 k,σ b 1 k,σ + H.c., but and thus 1 N H = t δ,σ e i(k k ) n = δ k k (1.18) n k ( ) e ik δ a 1 k,σ b 1 k,σ + H.c.. (1.19) We will suppress the sum over σ and include the spin degeneracy in later results. As a matrix, H = t k ( a 1 k b 1 k) δ eik δ 0 δ e ik δ 0 b 1 k a 1 k (1.20) or H = k ( ) a 1 k b 0 f(k) 1 k f (k) 0 a 1 k b 1 k, (1.21) 11

20 where f(k) = t δ eik δ. The energy dispersion is given by the eigenvalues of Ĥ Ĥ = 0 f(k) f (k) 0, (1.22) computed in the usual way det(ĥ Îλ) = 0 = λ f (k) f(k) λ = λ = ± f(k) 2 = ± f(k). Therefore, the energy dispersion of graphene is given by E G (k) = λ = ± f(k) ; so, we have [see Appendix A.1] ( E G (k) = ± a ) ( ) ( ) 3a a cos 2 k x cos 2 k kx y + 4 cos 2. (1.23) 2 As expected, due to the two-point basis of the lattice, we have two energy bands given by the ±. A plot of this band structure is given in Fig We can see that the energy dispersion is zero at each of the K and K points of the Brillouin zone. Figure 1.4: Energy dispersion for graphene over and beyond the first Brillouin zone. 12

21 Figure 1.5: Comparison of the low energy linear approximation, Eqn. (1.24) (red) and the full low energy dispersion, Eqn. (1.23) (blue) for graphene about the K point. As we are interested in the low energy physics of graphene, we will examine the low energy expansion about the K point which gives rise to many interesting properties. The K point of the Brillouin zone is K = (4π/3a, 0) and we can do a low energy Taylor expansion about this point to obtain the following energy dispersion [see Appendix A.2]: E G (k) = ± v F k, (1.24) where v F = 3ta/ m/s[5]. We can see that this is indeed linear. For charge neutrality, the Fermi energy corresponds to E = 0 at the K-point. A plot of the low energy expansion (red) compared to the full dispersion (blue) is given in Fig We can see that the agreement is excellent for low energy. This behaviour has been confirmed in angle-resolved photoemission spectroscopy (ARPES) experiments[1] and an example of the results can be seen for a doped system in Fig. 1.6 which has been taken from work done by Bostwick et al.[1]. From left to right, Fig. 1.6 illustrates the low energy band structure for increased doping from which we can see the Fermi level move up through the band structure. For low energies over the first Brillouin zone, our dispersion is now modelled by Fig. 1.7 where we have replaced the full dispersion by pairs of linear cones that intersect at zero energy and sit at each of the six K 13

22 Figure 1.6: ARPES image of the low energy dispersion of graphene taken from Ref. [1]. Electron density (per cm2 ) is given by the yellow numbers. Figure 1.7: Low energy dispersion for graphene over and beyond the first Brillouin zone. points Wavefunctions In the limit of small lattice constant, known as the continuum limit, we obtain the following Hamiltonian near the K point[29] HK (k) = ~vf 0 kx iky kx + iky 0 14 T = ~vf σ k = HK 0 (k), (1.25)

23 where σ = (σ x, σ y ) with σ x and σ y, the usual Pauli matrices, σ x = and σ y = 0 i i 0. (1.26) We can see that this will indeed yield the low energy form for the eigenvalues given by Eqn. (1.24). The form of the Hamiltonian implies that we are describing twodimensional massless fermions[29] with a linear energy dispersion. The momentum space wavefunctions for these Dirac particles are[11] Ψ K s,k = e ik r s, for H K (k) = v F σ k, (1.27) e iθ k and Ψ K s,k = e ik r eiθ k s, for H K (k) = v F σ k, (1.28) ( ) where θ k = arctan ky k x and s = ±1 for the eigenenergies E G = ± v F k, id est, for the upper π (electron) and lower π (hole) bands respectively. Another characterization of the wavefunctions is their chirality. This is identical to their helicity which is defined as the projection of their pseudospin on their momentum and can be written as[10] ĥ = 1 2 σ k k. (1.29) We can see that the helicity operator has the same σ k dependence as the Hamiltonian and the wavefunctions will therefore also be eigenstates of helicity[10], ĥψ K (r) = ± 1 2 ψk (r), (1.30) with an equivalent equation for the K point merely with inverted sign. As a result, at a given phase, the electrons about the Dirac points will have an opposite helicty to the holes. The helicity can also be understood by sending a plane wave through 15

24 the direct lattice with a reciprocal vector associated with the K point, that is, a wave of the form e ik r, which will yield the behaviour shown in Fig Due to the Figure 1.8: Plane wave associated with the K point moving through the lattice. periodicity of the lattice we know that e ik r = e ik (r+r) e iθ, (1.31) where K r = Kr due to the orientation of the wave, θ is the phase of the wave and R = na for integer n. This implies, e i(kr+θ) = 1 (1.32) and therefore KR + θ = 2mπ. We will take the case m = 0 to get θ = KR. 16

25 Substituting in for R and K, we get θ = 4π 3a na = n4π 3. (1.33) Therefore, if we start at one site on the lattice of phase 0 the next site a distance a away in the direction of the plane wave will have a phase of 4π/3 etcetera. If we sample the phase circling around the A site labelled in the figure, it will undergo a change in phase from 0 to 2π/3 to 4π/3 and finally back to 0 in the counterclockwise direction. Similarly, sampling the phase around the labelled B site we find the same changes in phase, however, in the clockwise direction. This feature is known as the pseudospin and we say the two sites have different chirality. The directions will be opposite for the K point. 1.4 Density of States Another property which garners a lot of interest is the electronic density of states. As the name suggests, this describes the number of states at a particular energy that an electron is free to occupy. A high density of states means there are a lot of available states while a zero density of states means an electron cannot have that energy; as a result, a saddle point in the energy dispersion, which gives rise to a flat region, will yield a high density of states as there are a lot of available states at that energy; we will see that this manifests itself as a van Hove singularity in the density of states. With our energy dispersion given by Eqn. (1.23), it is straight forward to calculate the density of sates. We know that the density of states per unit area can be expressed as[14] N(ε) = N f δ(ε E(k)), (1.34) k where N f is our degeneracy factor. We can replace the sum over k with a twodimensional integral in the usual way k 1 (2π) 2 d 2 k. 17

26 Eqn. (1.37) can be evaluated numerically by using the Lorentzian representation of the delta function 1 η δ(x) = lim η 0 π η 2 + x, (1.35) 2 with a broadening scattering rate of η. For the full density of states we use Eqn. (1.23) for E(k); we are also interested in the continuum limit in which we use the low energy approximation given by Eqn. (1.24). For the full density of states, 1 N(ε) = N f dk (2π) 2 x dk y δ(ε E(k)), (1.36) 1 st B.Z. where we are integrating over the first Brillouin zone with E(k) given by Eqn. (1.23). We have a two fold spin degeneracy and thus N f = g s = 2. Therefore N(ε) = 1 dk 2π 2 x dk y δ(ε E(k)) (1.37) 1 st B.Z. For the low energy approximation our degeneracy factor becomes N f = g s g v = 2 2 = 4 as we have a two-fold spin degeneracy and a two-fold valley degeneracy due to the two K points per Brillouin zone. In this approximation, our integral is no longer over the first Brillouin zone but over a region around the K point with a cutoff taken at k = 14/3 3 to yield the same number of states per unit cell. A plot of the density of states for both cases is given in Fig We can see in the full density of states that we have van Hove singularities at ω = ±t which correspond to the saddle points in the full energy dispersion (located at the M points of the Brillouin zone, see Fig. A.1). We can see that the linear approximation agrees well at low energy as we expect from the agreement in the low energy dispersion. 1.5 Conductivity As mentioned earlier, our primary interest in this thesis is an examination of optical conductivity. With the Hamiltonian given by Eqn. (1.15), we are now in a position to use the many body Green s function approach to calculate the AC conductivity. We 18

27 Figure 1.9: Full density of states per unit area over the first Brillouin zone along with the density of states using the low energy expansion. will begin our derivation with a brief discussion of conductivity from an energy band perspective. A schematic of optical probing of a material is given in Fig. 1.10(a). The Figure 1.10: (a) Schematic of optical probing of a material. (b) Conductivity in terms of transitions between energy bands incident photon has a frequency of Ω and therefore an associated electric field given 19

28 by E(Ω) = E o e iωt. (1.38) From Ohm s law, we know that an electric field will induce a current according to J(Ω) = σ(ω)e(ω), (1.39) where σ(ω), the conductivity, is a material dependent property that measures the response of the material to the electric field. In graphene this can be understood in terms of transitions between energy bands. Fig. 1.10(b) shows the low energy graphene dispersion near a single K point. All states below the Fermi surface are occupied and those above are unoccupied. For our work, we will use q = 0 which means we do not allow horizontal scattering in this diagram but only permit vertical transitions. This has been shown to be a good approximation[30] as the momentum of the photon is typically very small ( 10 7 m 1 for visible light which has a wavelength on the order of 100nm) compared to the dimensions of the Brillouin zone ( m 1 as the atom spacing is on the order of Å) and, thus, a typical optical interband transition excites an electron from a valence π band to a conduction π band with a negligible change in wavevector[31]. If the incident photon has sufficient energy it can excite an electron from an occupied state of energy ε to an unoccupied state of energy ε, leaving behind a hole; this is known as an interband transition. If we dope the system so that states up to energy µ measured from the Dirac point are occupied our lowest interband transition occurs at Ω = 2µ. We can also have intraband transitions if we allow infinitesimal scattering from a state just below the Fermi energy to a state just above (blue arrow in Fig. 1.10(b)); these transitions are only possible when a band crosses the Fermi surface such that part of the band is filled and the remainder is not. By varying the frequency of the incident photon, we can probe the important energy scales of the band structure. This can also be thought of in terms of a Feynman diagram shown in Fig We can see that a photon of energy Ω comes in and creates a hole-particle pair with energy (ω + Ω) and ω both of which have an associated Green s function which we will use to calculate 20

29 q=2π/λ Hole ω+ω, k+q Photon Ω, q~0 Photon Ω, q~0 Particle ω, k Figure 1.11: Feynman bubble for conductivity the conductivity in what is called the many body Green s function approach[32]. We begin by specifying a current density operator ĵ = e Ĥ k, (1.40) where Ĥ is given by Eqn. (1.15). As we are interested in the low energy conductivity, we will use f(k) = v F (k x + ik y ) = v F ke iθ where θ is the polar angle in momentum space about the K point. Therefore, Ĥ = v F 0 1 k x 1 0 and Ĥ = v F 0 i k y i 0 ; thus, Ĥ k i = v F σ i, (1.41) where σ i are the usual Pauli matrices. Therefore, ĵ = e Ĥ = ĵ x = e v F 0 1 k i

30 We know[14], therefore, v x = v F j = ev, (1.42) We now write the conductivity in terms of the spectral function of the Green s function in what is called the Kubo formula[32] σ αβ (Ω) = Im αβ (Ω + i0+ ), (1.43) Ω where σ αβ is the real part of the conductivity and Π αβ is the current-current correlation function. This turns into[33] σ αβ (Ω) = N f e 2 2Ω dω [f(ω µ) f(ω + Ω µ)] 2π d 2 k [ ] (2π) Tr ˆv 2 α Â(k, ω + Ω)ˆv β Â(k, ω). (1.44) Again, N f = g s g v where g s = 2 for the spin degeneracy and g v = 2 for the valley degeneracy. As we are in the continuum limit, the k integral is over a region surrounding one K-point (or valley). f(x) is the Fermi-Distribution function given by f(x) = 1 e βx + 1, β = 1 k B T. If we consider the zero temperature case, β 0 x > 0 f(x) = 1 x 0. This can then be modelled as the Heaviside step function. Therefore, dω µ dω [f(ω µ) f(ω + Ω µ)] = 2π µ Ω 2π. 22

31 To evaluate Eqn. (1.44), all we need is the spectral function A(k, ω) where Ĝ(z) = In our case, the Green s function is given by dω 2π Â(ω) z ω. (1.45) Ĝ 1 (z) = z Î Ĥ. (1.46) Using our Hamiltonian Ĝ 1 (z) = z f (k) f(k) z ; (1.47) thus, Ĝ(z) = 1 z f (k) z 2 f(k) 2 f(k) z. (1.48) From this, we can see that G 11 (z) = G 22 (z) = z z 2 f(k) 2. It is convenient to use partial fraction decomposition to write G 11 (z) = 1 2 [ ] 1 z f(k) + 1. (1.49) z + f(k) Given our relation between the Green s function and spectral function, Eqn. (1.45), we can see that A 11 (k, ω) = A 22 (k, ω) = π[δ(ω f(k) ) + δ(ω + f(k) )]. (1.50) 23

32 Identically, G 12 (z) = G f (k) 21(z) = z 2 f(k) 2 = f [ (k) 1 f(k) 2 which yields the spectral elements 1 z f(k) 1 z + f(k) ], (1.51) A 12 (k, ω) = A 21(k, ω) = f (k) [δ(ω f(k) ) δ(ω + f(k) )]. (1.52) f(k) It is important to note that we are treating z as real so that G (z) and A (z) serve merely to interchange f(k) and f (k). Since we will be working in the continuum limit, we will refer to ε k instead of f(k) with ε k = v F k. As we are interested in calculating the longitudinal conductivity, α = β = x, our velocity matrix becomes ˆv x = v F (1.53) Due to the symmetry in k, we can write d 2 k 2π (2π) = dθ kc 2 0 2π 0 kdk 2π, (1.54) where k C is some cutoff characteristic of the large band cutoff. All that remains is to evaluate Tr[ˆv x Â(k, ω + Ω)ˆv x Â(k, ω)] which we find to be v 2 F [A 12(k, ω + Ω)A 12(k, ω) + 2A 11 (k, ω + Ω)A 11 (k, ω) + A 12 (k, ω + Ω)A 12 (k, ω)]. We know that[see Appendix A] (1.55) A 12(k, ω + Ω)A 12(k, ω) (f (k)) 2 e 2iθ and A 12 (k, ω + Ω)A 12 (k, ω) (f(k)) 2 e 2iθ 24

33 and as there is no other angular dependence, these will average to zero in the angular integral 2π and so we will drop them now. Our conductivity equation then becomes 0 dθ 2π σ xx (Ω) = N fe 2 v 2 F 2Ω µ µ Ω dω 2π dθ kc 2π 0 2π 0 kdk 2π 2A 11(k, ω + Ω)A 11 (k, ω). (1.56) It is important to note that we have taken = 1 and there should really be an with all the ω and Ω terms. If we restore the with the dω and continue to use the implicit form of the delta functions, we obtain σ xx (Ω) σ 0 = 4 Ω µ µ Ω WC dω εdε[δ(ω + Ω ε)δ(ω ε) + δ(ω + Ωε)δ(ω + ε) 0 + δ(ω + Ω + ε)δ(ω ε) + δ(ω + Ω ε)δ(ω ε)], (1.57) where σ 0 = e 2 /4 and we have set ε k = v F k = ε. We can now get a numerical solution for the conductivity where we again use the Lorentzian form of the delta function given by 1 η δ(x) = lim η 0 π η 2 + x, 2 where the broadening η manifests itself as an effective transport scattering rate of 1/τ imp = 2η due to the convolution of the two delta functions in the conductivity formula[33]. A plot of the conductivity with a broadening of η = 0.007t for µ = 0 can be seen in Fig A remarkable feature is that the conductivity is perfectly flat at a value of σ 0 which is thus known as the background conductivity for graphene. A plot of the conductivity of doped graphene can be seen in Fig for µ/t = 0.2. We can see that we now have a Drude contribution to the conductivity at low energy which corresponds to intraband transitions. All interband transitions are Pauli blocked for energies below 2µ and therefore at 2µ we see a step in the conductivity. Above 2µ the conductivity levels out to the constant background value of the undoped 25

34 Figure 1.12: Conductivity for undoped graphene 4 µ/t=0.2 3 Drude Conductivity σ xx (Ω)/σ ο 2 2µ Ω/t Figure 1.13: Conductivity for doped graphene monolayer. The interband transitions correspond to transitions between bands of different chirality associated with the A and B sublattices which physically is expected as we are inducing a current which causes electrons to hop between A and B sites; mathematically, this can be seen from the off-diagonal form of the velocity matrix which only couples A and B sites. This behaviour has been extensively examined and confirmed experimentally[34, 35, 36, 37]. An example of such an experiment 26

35 is the one done by Mak et al.[37] in which the predicted behaviour was found for incident photons in the infrared to visible spectrum over the spectral range ev. This data was obtained by shining a beam of light onto a graphene sample sitting on an SiO 2 substrate and measuring the intensity of the reflected and transmitted light using a spectrometer. The absorbance and reflectance was then determined by normalizing the results with those obtained from the bare substrate. It is also possible to derive an analytic expression for the conductivity [see Appendix A.3] which has the form[38] σ xx (Ω) σ 0 = 4 µ δ(ω) + Θ(Ω 2 µ ). (1.58) A plot of our analytic solution compared to the full numerical work can be seen in Fig For the analytic solution, we have broadened the delta function with an Figure 1.14: Comparison between the analytic solution and numerical calculation of the longitudinal conductivity of graphene for µ/t = 0.2 effective scattering rate of 2η compared to that of the numerical work to account for the convolution of the two Lorentzians representing the two delta functions in our formula. We can see that the agreement is good with the deviations resulting from 27

36 not accounting for scattering beyond the delta peak at the origin. Following the formalism above, we can now begin an examination of the optical properties of bilayer graphene. In the following chapter, we will examine the ABstacked bilayer which has previously been studied by Nicol and Carbotte[33]. 28

37 Chapter 2 AB-Stacked Bilayer Graphene With the richness of interesting properties displayed in graphene, a lot of attention began to shift to the study of bilayer graphene. Two graphene sheets typically take an AB-stacked formation, more commonly know as Bernal stacking, in which the A atoms in one layer are stacked below the B atoms in the upper layer such that the A atoms in the upper layer sit above the center of the hexagons formed in the lower layer, see Fig Like the monolayer, Bernal-stacked bilayer graphene also γ Figure 2.1: Bernal (AB) stacking of bilayer graphene possesses remarkable properties. When the bilayer is placed in an appropriately configured field effect device, a tunable semiconducting gap can be generated. This causes the valence and conduction bands to no longer meet at the two Dirac points in the graphene Brillouin zone[39, 40, 41, 42, 43]. Many other remarkable attributes and 29

38 technological implications can be found in Refs. [44] and [10]. As we are interested in the optical conductivity of AA-stacked bilayer graphene, we will limit our discussion in this chapter to the optical conductivity of Bernal-stacked bilayer graphene which has been examined both with and without an asymmetry gap by Nicol and Carbotte[33]. We will only briefly discuss the results and for a more comprehensive discussion the reader is directed to Ref.[33]. 2.1 Energy Dispersion We begin our discussion of Bernal-stacked bilayer graphene by constructing the corresponding tight-binding Hamiltonian. The model we will use is similar to that of the monolayer where we allow intralayer hopping between nearest neighbours and interlayer hopping between sites of AB stacking in the two layers indexed 1 and 2. As we now have two layers, we will have four atoms per unit cell and thus expect four energy bands. In the tight-binding model, the lower graphene plane has an intralayer hopping Hamiltonian H 1 = t n,δ,σ ( ) a 1 n+δ,σ b 1 n,σ + H.c., (2.1) where δ i are the nearest neighbour vectors of the monolayer given by Eqn. (1.14). Similarly, the second layer has an intralayer hopping Hamiltonian H 2 = t n,δ,σ ( ) a 2 n,σb 2 n+δ,σ + H.c., (2.2) where δ i = δ i. The interlayer hopping Hamiltonian is H 3 = γ n,δ ( ) a 2 n,σb 1 n,σ + H.c., (2.3) where γ 0.4 ev is the hopping energy for transitions between sites of vertical stacking. We are also interested in the effect of adding a bias between the layers 30

39 which can tune the bilayer from a metallic to a semiconducting behaviour[33]. This is achieved by raising the energy on plane 2 by + /2 and lowering the energy on plane 1 by /2 to create an overall bias of. This yields the Hamiltonian H 4 = 2 ) (a 1 n+δ1,σ a 1 n+δ1,σ + b 1 n,σb 1 n,σ + 2 n,σ n,σ (a 2 n,σa 2 n,σ + b 2 n+δ 1,σb 2 n+δ 1,σ ). (2.4) Analogously to Chapter 1.3.1, we can transform our total Hamiltonian (H 1 + H 2 + H 3 + H 4 ) to k space and obtain the matrix H = k 0 0 f(k) 2 ( ) a 0 f 1 k b 2 k a 2 k b (k) k 0 f(k) γ 2 f (k) 0 γ 2 a 1 k b 2 k a 2 k b 1 k, (2.5) where again, f(k) = t δ eik δ = t δ e ik δ. The band structure is defined by the energy eigenvalues of the inner matrix and the four bands are given by ε 2 α(k) = γ f(k) 2 + ( ) α Γ, (2.6) Γ = γ f(k) 2 (γ ), (2.7) where α = 1 and 2. The main low energy physics occurs at the K and K points of the Brillouin zone; as in the case of the monolayer, f(k) can be expanded about the K point in the continuum approximation (limit of small lattice constant a) to f(k) = v F k where again v F = 3ta/2. Plots of the bands structure for = 0, γ 0 and 0, γ 0 can be seen in Fig. 2.2(a) and (b), respectively, along with the corresponding density of states, Fig. 2.2(c) and (d), in units of 2γ/π( v F ) 2 calculated as in Chapter 1.4. We can see for the unbiased case ( = 0) that the band structure is now quadratic in k at low energies although still linear at higher k. We have four bands as predicted; the two positive (negative) bands are separated by γ at the K 31

40 Figure 2.2: Top: Low energy dispersion for AB-stacked bilayer about the K point (a) without an asymmetry gap = 0, (b) with an asymmetry gap = 1.5γ. Bottom: Low energy density of states in units of 2γ/π( v F ) 2 (c) without an asymmetry gap, (d) with an asymmetry gap = 1.5γ. point and two of the bands are zero at the K point. We can see that a bias introduces an energy gap in the band structure, Fig. 2.2(b), and a mexican hat structure occurs in the low energy band ε 1 with a minimum value of E g1 = γ /(2 γ ) at f(k) = v F k 0 where k 0 = ( /2) ( 2 + 2γ 2 )/( 2 + γ 2 ). The relative maximum of the hat occurs at E 01 = /2 for k = 0[33]. The bias also introduces a square root singularity in the density of states which can be seen in Fig. 2.2(d). 32

41 2.2 Conductivity Longitudinal Conductivity Now that we have our Hamiltonian, Eqn. (2.5), we can proceed identically to Chapter 1.5 and evaluate the many body Green s function using the relation Ĝ 1 (z) = z Î Ĥ and thus obtain the spectral function from Eqn. (1.45) from which we can evaluate the conductivity, see Eqn. (1.44). Utilizing our relations, we obtain the Green s function G 11 G 12 G 13 G 14 G Ĝ(z) = 12 G 22 G 23 G 24 G 13 G (2.8) 23 G 33 G 34 G 14 G 24 G 34 G 44 where G 11 = ( 2z)( 2 + 4γ 2 4z 2 ) 4 f(k) 2 ( + 2z), (2.9) 8(z 2 ε 2 1)(z 2 ε 2 2) G 12 = G 13 = γf(k) 2 (z 2 ε 2 1)(z 2 ε 2 2), (2.10) (2z )γf(k) 2(z 2 ε 2 1)(z 2 ε 2 2), (2.11) G 14 = (4z2 4z f(k) 2 )f(k), (2.12) 4(z 2 ε 2 1)(z 2 ε 2 2) G 22 = ( 2z)( 2 + 4γ 2 4z 2 ) 4 f(k) 2 ( + 2z), (2.13) 8(z 2 ε 2 1)(z 2 ε 2 2) G 23 = (4z2 + 4z f(k) 2 )f (k), (2.14) 4(z 2 ε 2 1)(z 2 ε 2 2) G 24 = (2z + )γf (k) 2(z 2 ε 2 1)(z 2 ε 2 2), (2.15) G 33 = ( + 2z)[( + 2z)2 4 f(k) 2 ], (2.16) 8(z 2 ε 2 1)(z 2 ε 2 2) G 34 = 4γ(2z )(2z + ) 4(z 2 ε 2 1)(z 2 ε 2 2), (2.17) 33

42 and G 44 = ( + 2z)[( 2z)2 4 f(k) 2 ], (2.18) 8(z 2 ε 2 1)(z 2 ε 2 2) which is consistent with the results shown by Nicol and Carbotte[33]. As a consequence of Eqn. (1.45), we know that the spectral function will have the same relation in terms of the matrix elements. Before we calculate the matrix elements of the spectral function, we will determine which are needed. Recall Eqn. (1.44) from which we see that the spectral function comes in in the trace term ( ) Tr ˆv α Â(k, ω + Ω)ˆv β Â(k, ω). As we are interested in the longitudinal conductivity σ xx, α = β = x. From our velocity relation, Eqn. (1.42), we know that v x = v F From the Green s function we know that A 11 A 12 A 13 A 14 A Â = 12 A 22 A 23 A 24 A 13 A 23 A 33 A 34 A 14 A 24 A 34 A 44, (2.19) 34

43 from which we see that ( ) Tr ˆv x Â(k, ω + Ω)ˆv x Â(k, ω) =vf 2 [A 14(k, ω + Ω)A 14(k, ω) + A 24(k, ω + Ω)A 13(k, ω) + A 34 (k, ω + Ω)A 12(k, ω) + A 44 (k, ω + Ω)A 11 (k, ω) + A 13(k, ω + Ω)A 24(k, ω) + A 23(k, ω + Ω)A 23(k, ω) + A 33 (k, ω + Ω)A 22 (k, ω) + A 34 (k, ω + Ω)A 12 (k, ω) + A 12(k, ω + Ω)A 34 (k, ω) + A 22 (k, ω + Ω)A 33 (k, ω) + A 23 (k, ω + Ω)A 23 (k, ω) + A 24 (k, ω + Ω)A 13 (k, ω) + A 11 (k, ω + Ω)A 44 (k, ω) + A 12 (k, ω + Ω)A 34 (k, ω) + A 13 (k, ω + Ω)A 24 (k, ω) + A 14 (k, ω + Ω)A 14 (k, ω)]. We know that G 12 f(k) 2 ; therefore, as a consequence of Eqn. (1.45), A 12 f(k) 2. f(k) can be expressed in polar coordinates such that f(k) e iθ. Likewise, A 13 f(k), A 14 f(k), A 23 f (k) and A 24 f (k). Thus, A 14(k, ω + Ω)A 14(k, ω) e 2iθ, A 34 (k, ω + Ω)A 12(k, ω) e 2iθ, A 23(k, ω + Ω)A 23(k, ω) e 2iθ, A 34 (k, ω+ω)a 12 (k, ω) e 2iθ, A 12(k, ω+ω)a 34 (k, ω) e 2iθ, A 23 (k, ω+ω)a 23 (k, ω) e 2iθ, A 12 (k, ω + Ω)A 34 (k, ω) e 2iθ and A 14 (k, ω + Ω)A 14 (k, ω) e 2iθ. As there is no other angular dependence, these terms will go to zero when the angular average is conducted and we can thus ignore them in the trace. Keeping only the components of the trace which are non-vanishing under the angular average, we obtain the expression for the zero temperature longitudinal conductivity σ xx (Ω) = N f e 2 2Ω Ω µ Ω dω 2π d 2 k (2π) 2 v2 F [A 24(k, ω + Ω)A 13(k, ω) + A 44 (k, ω + Ω)A 11 (k, ω) + A 13(k, ω + Ω)A 24(k, ω) + A 33 (k, ω + Ω)A 22 (k, ω) + A 22 (k, ω + Ω)A 33 (k, ω) + A 24 (k, ω + Ω)A 13 (k, ω) + A 11 (k, ω + Ω)A 44 (k, ω) + A 13 (k, ω + Ω)A 24 (k, ω)]. We must now provide expressions for A 11, A 13, A 22, A 24, A 33 and A 44. This is done by 35

44 applying Eqn. (1.45) to the respective Green s function elements. We find where A 11 (ω, ) = [a 11 (α, )δ(ω ε α ) + a 11 (α, )δ(ω + ε α )], (2.20) a 11 (α, ) = ( ) α π 8 α=1,2 ( 2ε α )( 2 + 4γ 2 4ε 2 α) 4 f(k) 2 ( + 2ε α ), (2.21) ε α (ε 2 2 ε 2 1) A 13 (ω, ) = [a 13 (α, )δ(ω ε α ) + a 13 (α, )δ(ω + ε α )], (2.22) α=1,2 where a 13 (α, ) = ( ) α π 2 (2ε α )γf(k) ε α (ε 2 2 ε 2 1) (2.23) and where A 33 (ω, ) = [a 33 (α, )δ(ω ε α ) + a 33 (α, )δ(ω + ε α )], (2.24) α=1,2 a 33 (α, ) = ( ) α π 8 ( + 2ε α )[( + 2ε α ) 2 4 f(k) 2 ]. (2.25) ε α (ε 2 2 ε 2 1) The other spectral elements are related to the three specified above by A 22 ( ) = A 11 ( ), A 44 ( ) = A 33 ( ) and A 24 (, f(k)) = A 13 (, f (k)). Before we evaluate the conductivity, we need to restore the that should have gone with the ω so that dω dω. We have already assumed an in front of the ω in the δ-functions to match the units of energy. If we collect all the constant prefactors, we obtain N f e 2 v 2 F 16π 3 = N f 2 v 2 F σ o 4π 3, 36

45 where σ o = e 2 /4. Therefore, we obtain σ xx (Ω) σ o = N f 2 v 2 F 2π 3 Ω Ω µ Ω dω d 2 k[a 24(k, ω + Ω)A 13(k, ω) 1 st B.Z +A 44 (k,ω + Ω)A 11 (k, ω) + A 13(k, ω + Ω)A 24(k, ω) +A 33 (k,ω + Ω)A 22 (k, ω) + A 22 (k, ω + Ω)A 33 (k, ω) +A 24 (k,ω + Ω)A 13 (k, ω) + A 11 (k, ω + Ω)A 44 (k, ω) +A 13 (k,ω + Ω)A 24 (k, ω)]. (2.26) As we are interested in the low energy physics, we can again replace f(k) with ε = v F k such that the integral over d 2 k becomes d 2 k 2π 0 kc WC dθ kdk 2π 0 0 εdε ( v F ) 2, (2.27) where W C is some large cutoff typical of the large band cutoff of bilayer graphene. Thus σ xx (Ω) = N Ω WC f dω εdε[a σ o 2π 2 Ω 24(k, ω + Ω)A 13(k, ω) + A 44 (k, ω + Ω)A 11 (k, ω) µ Ω 0 + A 13(k, ω + Ω)A 24(k, ω) + A 33 (k, ω + Ω)A 22 (k, ω) + A 22 (k, ω + Ω)A 33 (k, ω) + A 24 (k, ω + Ω)A 13 (k, ω) + A 11 (k, ω + Ω)A 44 (k, ω) + A 13 (k, ω + Ω)A 24 (k, ω)], (2.28) where the integral over ε is now taken around a single K point instead of over the first Brillouin zone and we thus include a degeneracy factor of two to account for the two K points in the first Brillouin zone. Therefore, N f = 4 to account for the two-fold spin and valley degeneracies. To evaluate Eqn. (2.28) numerically, we again broaden out the delta functions contained in the spectral elements into a Lorentzian 1 η δ(x) = lim η 0 π η 2 + x 2 37

46 with a scattering rate of η which we take to be 0.02γ. We will begin by discussing the conductivity without an asymmetry gap. A plot of the zero temperature longitudinal conductivity for unbiased Bernal-stacked bilayer graphene can be seen in Fig. 2.3(b) next to a plot of the band structure for comparison. We can see that there is a large peak in the conductivity at γ which corresponds to the Figure 2.3: (a) Band structure for unbiased AB-stacked bilayer graphene ( /γ = 0). (b) Conductivity of unbiased AB-stacked bilayer graphene for various values of chemical potential. separation between the ε 1 and ε 2 bands. As we increase the doping, this peak becomes sharper and we see the onset of a Drude contribution due to intraband transitions. For all dopings, we see a step in the conductivity at 2 µ similar to monolayer graphene as well as features at 2µ γ and γ + 2µ above which we attain a flat background conductivity twice that of monolayer graphene. The conductivity of Bernal-stacked bilayer graphene with a finite asymmetry gap can be seen in Fig. 2.4(b) next to a plot of the band structure for comparison. The conductivity is shown for several values of chemical potential, namely, just below the maximum of the mexican hat structure and just above the mexican hat. We have a large peak in the undoped conductivity at approximately γ which corresponds to transitions between the two square root singularities in the density of states (see 38

47 Figure 2.4: (a) Band structure for biased AB-stacked bilayer graphene ( /γ = 1.5). (b) Conductivity of biased AB-stacked bilayer graphene for various values of chemical potential. Fig. 2.2(d)) or transitions between the two minima of the mexican hat structures of the ε 1 and ε 1 bands. As the system is doped into the mexican hat structure, this feature is diminished as some states in the structure are occupied and, therefore, Pauli blocked. The major peak now results from transitions between the ε 1 and ε 2 bands. We do, however, now have a Drude contribution as intraband transitions are permitted. As we increase the doping above the mexican hat, we no longer see a large jump in the conductivity. In all cases, at sufficiently high frequency, we retain a background conductivity twice that of monolayer graphene. Unlike the unbiased bilayer, we no longer have a peak at γ for all dopings. As we are primarily interested in the AB-stacked bilayer conductivity for a comparison with the AA-stacked case, we will reserve further discussion for Chapter 3 and the reader is encouraged to read Ref.[33] for a more detailed analysis Perpendicular Conductivity While the longitudinal conductivity of Bernal bilayer graphene is thoroughly examined in Ref.[33], in this thesis we are also interested in the perpendicular conductivity 39

48 that results from an electric field applied perpendicular to the two graphene sheets. While we retain the same Green s function as given by Eqn. (2.8), we are now interested in the c-axis conductivity (α = β = z) and thus need an expression for ˆv z, that is, 0 0 v v ˆv z =, (2.29) v v 0 0 which is found through a Peierls substitution as demonstrated in Ref.[33], where v = iγd/ with d the interlayer distance, about 3.6 Å and 3.3 Å for AA and AB stacking respectively[45, 46]. Utilizing Eqn. (1.44), we need to evaluate Tr(ˆv z Â(k, ω + Ω)ˆv z Â(k, ω)) which we find to be { } Tr ˆv z Â(k, ω + Ω)ˆv z Â(k, ω) = v 2 [A 23(k, ω + Ω)A 14(k, ω) A 13(k, ω + Ω)A 13(k, ω) + A 33 (k, ω + Ω)A 11 (k, ω) A 34 (k, ω + Ω)A 12(k, ω) + A 14(k, ω + Ω)A 23(k, ω) A 24(k, ω + Ω)A 24(k, ω) A 34 (k, ω + Ω)A 12 (k, ω) + A 44 (k, ω + Ω)A 22 (k, ω) + A 11 (k, ω + Ω)A 33 (k, ω) A 12 (k, ω + Ω)A 34 (k, ω) A 13 (k, ω + Ω)A 13 (k, ω) + A 14 (k, ω + Ω)A 23 (k, ω) A 12(k, ω + Ω)A 34 (k, ω) + A 22 (k, ω + Ω)A 44 (k, ω) + A 23 (k, ω + Ω)A 14 (k, ω) A 24 (k, ω + Ω)A 24 (k, ω)], where the spectral elements are identical to those given in the previous section. Again, several of these elements will vanish in the angular average. Recall, A 13 and A 14 f(k) e iθ, A 23 and A 24 f (k) e iθ and A 12 f(k) 2 e 2iθ. Dropping the 40

49 terms which vanish in the average, we obtain { } Tr ˆv z Â(k, ω + Ω)ˆv z Â(k, ω) = v 2 [A 23(k, ω + Ω)A 14(k, ω) + A 33 (k, ω + Ω)A 11 (k, ω) + A 14(k, ω + Ω)A 23(k, ω) + A 44 (k, ω + Ω)A 22 (k, ω) + A 11 (k, ω + Ω)A 33 (k, ω) + A 14 (k, ω + Ω)A 23 (k, ω) + A 22 (k, ω + Ω)A 44 (k, ω) + A 23 (k, ω + Ω)A 14 (k, ω)]. While A 11, A 22, A 33 and A 44 are given in the previous section, we now need expressions for A 14 and A 23 which we find to be A 14 (ω, ) = [a 14 (α, )δ(ω ε α ) + a 14 (α, )δ(ω + ε α )], (2.30) α=1,2 where a 14 (α, ) = ( ) α π 2 (4ε 2 α 4ε α f(k) 2 )f(k) ε α (ε 2 2 ε 2 1) (2.31) and where A 23 (ω, ) = [a 23 (α, )δ(ω ε α ) + a 23 (α, )δ(ω + ε α )], (2.32) α=1,2 a 23 (α, ) = ( ) α π 2 (4ε 2 α + 4ε α f(k) 2 )f(k). (2.33) ε α (ε 2 2 ε 2 1) Therefore, the real part of the zero temperature perpendicular conductivity can be written as σ zz (Ω) = N f e 2 2Ω µ µ Ω dω 2π d 2 k (2π) 2 v 2 [A 23(k, ω + Ω)A 14(k, ω) + A 33 (k, ω + Ω)A 11 (k, ω) + A 14(k, ω + Ω)A 23(k, ω) + A 44 (k, ω + Ω)A 22 (k, ω) + A 11 (k, ω + Ω)A 33 (k, ω) + A 14 (k, ω + Ω)A 23 (k, ω) + A 22 (k, ω + Ω)A 44 (k, ω) + A 23 (k, ω + Ω)A 14 (k, ω)]. (2.34) 41

50 For the unbiased bilayer, we derive an analytic formula where [ ] σ zz (Ω) Ω = σ 0 2(Ω + γ) + Ω Θ(Ω 2γ) Θ(Ω 2µ) + c(µ)δ(ω γ), (2.35) 2(Ω γ) c(µ) = µ(γ + µ) γ γ 2 ln2µ + γ γ [ µ(γ µ) + γ + γ 2 ln2µ γ ] Θ(µ γ) (2.36) γ and ( ) 2 σ 0 = e2 γ d σ 0, (2.37) 4 v F which again agrees quite well with our numerical work shown in Fig We can see Figure 2.5: Perpendicular conductivity for unbiased Bernal-stacked bilayer for various values of chemical potential. that for finite µ there is a strong response at γ much like in the in-plane conductivity. We also see an absorption edge occurring at 2max(γ, µ) which results from transitions between the lowest ( ε 2 ) and highest (ε 2 ) energy bands which are the bonding and antibonding bands of the A-B dimer strongly coupled by hopping γ[47]. It is not surprising that we see similar behaviour in the perpendicular conductivity as we do in the longitudinal conductivity. The lower bands (ε 1 and ε 1 ) represent hopping 42

51 between the non-dimer A and B sites in the two planes. Such transitions must occur by first hopping in the plane to the neighbour site which is part of the dimer (site of AB stacking) then hopping up the dimer bond and finally hopping to the non-dimered site in the second plane[47]. Having explored several cases for the conductivity of AB-stacked bilayer graphene, we will continue on to the purpose of this thesis and examine in detail the optical conductivity of AA-stacked bilayer graphene making appropriate reference and comparison with this chapter. 43

52 Chapter 3 AA-Stacked Bilayer Graphene Now that we have motivated and outlined the techniques we have used, we are in a position to present the optical conductivity of AA-stacked bilayer graphene. The AA-stacked bilayer is characterized by two monolayer sheets stacked directly on top of each other. Unlike the more familiar Bernal (AB) stacking, like sites sit right above or below each other, see Fig Naturally, graphite adopts Bernal stacking γ Figure 3.1: AA-stacked bilayer graphene as a small energy barrier exists between AB and AA stacking[48]. As a result of the lack of samples, little theoretical and experimental work has been done on the AAstacked bilayer. Recently, however, Lee et al.[12] succeeded in growing AA-stacked bilayer on (111) diamond. Liu et al. has since discovered that bilayer graphene of- 44