Graphene and Planar Dirac Equation

Graphene and Planar Dirac Equation Marina de la Torre Mayado 2016 Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 1 / 48

Outline 1 Introduction 2 The Dirac Model Tight-binding model Low energy effective theory near the Dirac points 3 Modifications of the Dirac model 4 References Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 2 / 48

Graphene Graphene is a one-atom thick layer of carbon atoms: It was isolated for the first time by A. K. Geim and K. S. Novoselov in 2004. They were awarded with the Nobel Prize in Physics in 2010: For groundbreaking experiments regarding the two-dimensional material graphene" Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 3 / 48

Graphene Graphene has many exceptional properties and it is one of the most interesting topics in condensed matter physics. The principal features of graphene are: 1 The quasi-particle excitations satisfy the Dirac equation (Weyl). 2 The speed of light c has to be replaced by the so-called Fermi velocity: v F c 300 106 m s Therefore, the quantum field theory methods are very useful in the physics of graphene. Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 4 / 48

The Dirac Model The Dirac model for quasi-particles in graphene was elaborated around 1984: D. P. DiVincenzo and E. J. Mele, Phys. Rev. B 29, 1685 (1984). G. W. Semenoff, Phys. Rev. Lett. 53, 2449 (1984). Its basic properties (linearity of the spectrum) were well known in 1947: P. R. Wallace, Phys. Rev. 71, 622 (1947). For more details see the complete review: A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov and A. K. Geim, Rev. Mod. Phys. 81, 109-162 (2009). Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 5 / 48

The Dirac Model A Bravais lattice is an infinite arrangement of points (or atoms) in space that has the following property: The lattice looks exactly the same when viewed from any lattice point A 1D Bravais lattice: A 2D Bravais lattice: Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 6 / 48

The Dirac Model A 3D Bravais lattice: A Bravais lattice has the following property, the position vector of all points (or atoms) in the lattice can be written as follows: 1D: R = n a 1, n Z. 2D: R = n a 1 + m a 2, n, m Z. 3D: R = n a 1 + m a 2 + p a 3, n, m, p Z. Where a i are called the primitive lattice vectors. Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 7 / 48

The Dirac Model The honeycomb lattice is not a Bravais lattice: Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 8 / 48

The Dirac Model But the honeycomb lattice can be considered a Bravais lattice with a two-atom basis: A primitive cell of a Bravais lattice is the smallest region which when translated by all different lattice vectors can cover the entire lattice without overlapping. The primitive cell is not unique. The Wigner-Seitz (WS) primitive cell of a Bravais lattice is a special kind of a primitive cell and consists of region in space around a lattice point that consists of all points in space that are closer to this lattice point than to any other lattice point. The Wigner-Seitz primitive cell is unique Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 9 / 48

The Dirac Model I can take the blue atoms to be the points of the underlying Bravais lattice that has a two-atom basis (blue and red) with basis vectors: d 1 = 0, d 2 = d i However red and blue color coding is only for illustrative purposes: All atoms are the same. Also, I can take the small black points to be the underlying Bravais lattice that has a two atom basis (blue and red) with basis vectors: d 1 = d 2 i, d2 = d 2 i Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 10 / 48

Tight-binding model Then, in graphene the carbon atoms form a honeycomb lattice with two triangular sublattices A and B: Where the lattice spacing is d = 1.42 A. Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 12 / 48

Tight-binding model In this structure, each carbon atom has six electrons: Two electrons filling the inner shell 1s, three electrons engaged in the 3 in-plane covalent bons in the sp 2 configuration, and a single electron occupying the p z orbital perpendicular to the plane. Hybridization sp 2 in graphene Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 13 / 48

Tight-binding model The three quantum-mechanical states are given by sp 2 1 = 1 2 2s 3 3 2p y sp 2 2 = 1 ( ) 2 3 2s + 3 3 2 2p x + 1 2 2p y sp 2 3 = 1 ( ) 2 3 2s + 3 3 2 2p x + 1 2 2p y Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 14 / 48

Tight-binding model The nearest neighbors of an atom from the sublattice A belong to the sublattice B. The three vectors δ 1, δ 2 and δ 3 relate A with the three nearest neighbors B: δ 1 = d( 1, 0) ( ) 1 3 δ 2 = d 2, 2 ( ) 1 3, δ 3 = d 2, 2 In the tight binding model only the interaction between electrons belonging to the nearest neighbors is taken into account. Thus, the Hamiltonian reads: H = t α A 3 ( a (r α )b(r α + δ j ) + b (r α + δ j )a(r α ) ) j=1 where t is the hopping parameter t 2.8 ev. Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 15 / 48

Tight-binding model The operators a (r α ) and a(r α ), b (r β ) and b(r β ) are creation and annihilation operators of the electrons in the sublattices A and B, respectively. We adopt natural units = c = 1 and consider the usual anti-commutation relations: {a(r α ), a (r β )} = δ α,β = {b(r α ), b (r β )} {a(r α ), a(r β )} = {a (r α ), a (r β )} = 0 {b(r α ), b(r β )} = {b (r α ), b (r β )} = 0 {a(r α ), b(r β )} = {a (r α ), b (r β )} = 0 Example: we consider an atom of type A in the position r 0 = (0, 0) and then 3 ( H 0 = t a (r 0 )b(r 0 + δ j ) + b (r 0 + δ j )a(r 0 ) ) j=1 Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 16 / 48

Tight-binding model In this model the possible states for the electron are: { 1 r0, 1 r0+δ 1, 1 r0+δ 2, 1 r0+δ 3 } So that, and then H 1 r0 = t 1 r0+δ 1 t 1 r0+δ 2 t 1 r0+δ 3 H 1 r0+δ j = t 1 r0 1 r0+δ 1 H 0 1 r0 = t 1 r0+δ 2 H 0 1 r0 = t 1 r0+δ 3 H 0 1 r0 = t Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 17 / 48

Tight-binding model Let us represent the wave function through a Fourier transform: ψ = ψ A (k) e ikrα a (r α ) + ψ B (k) e ikr β b (r β ) 0 α A β B The sublattice A is generated by shifts along the vectors δ 2 δ 1 and δ 3 δ 1 : where a 1 and a 2 are the basis vectors of the Bravais lattice ( ) ( ) 3 3 a 1 = δ 2 δ 1 = d 2, 3 3, a 2 = δ 3 δ 1 = d 2 2, 2 Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 18 / 48

Tight-binding model Therefore two momenta k 1 and k 2 are equivalent if for all integers n 1 and n 2. [n 1 (δ 2 δ 1 ) + n 2 (δ 3 δ 1 )] (k 1 k 2 ) 2πZ Representatives of the equivalence classes can be taken in a compact region in the momentum space, the Brillouin zone, which is an hexagon with the corners: v 1 = 2π ( 3d v 4 = 2π 3d ) 1, v 2 = 2π ( 3 3d 1, ( 1, 1 ), v 5 = 2π 3 3d 1, 1 ) 3 ( 1,, v 3 = 2π ( 3d ) 1, v 6 = 2π 3 3d The reciprocal lattice vectors are given by: b i a j = 2πδ ij, i, j = 1, 2; b 1 = 2π ( 1, ) 3 3d, b 2 = 2π ( 1, ) 3 3d 0, 2 ) 3 ( 0, ) 2 3 Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 19 / 48

Tight-binding model The reciprocal lattice: The opposite sides of this hexagon has to be identified and the corners v 1, v 3 and v 5 are equivalent, as well as v 2, v 4 and v 6. We calculate H ψ : 3 H ψ = t ψ B (k) e ikδj e ikrα a (r α ) + ψ A (k) j=1 α A 3 e ikδj e ikr β b (r β ) 0 j=1 β B Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 20 / 48

Tight-binding model The stationary Schödinger equation H ψ = E ψ becomes the matrix equation: ( ) ( ) ( ) 0 tx ψa ψa tx = E 0 ψ B ψ B where X = 3 j=1 e ikδj Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 21 / 48

Tight-binding model If E 0 the eigenvalues are: E = ±t X with the eigenfunctions: { txψb = Eψ A (1) tx ψ A = Eψ B (2) 1 Solution of Type 1 (ψ B = 1): { ( X ψ 1+ = X 1 ), ψ 2 = ( X X 1 ) } 2 Solution of Type 2 (ψ A = 1): { ( 1 ψ 3+ = X X ), ψ 4 = ( 1 X X ) } Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 22 / 48

Tight-binding model Let us write X = X e iφ, the Hamiltonian is then ( 0 t X e H = iφ t X e iφ 0 ) The eigenfunctions can be taken, Positive energies E = t X > 0: ( ψ + = 1 e i φ 2 ψ1+ + 1 e i φ 2 ψ3+ 1 2 2 2 e i φ 2 e i φ 2 ) Negative energies E = t X < 0: ( ψ = 1 e i φ 2 ψ2 + 1 e i φ 2 ψ4 1 2 2 2 e i φ 2 e i φ 2 ) Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 23 / 48

Tight-binding model These eigenfunctions satisfy: where in the case of two dimensions S ˆn ψ + = 2 ψ +, S ˆn ψ = 2 ψ S ˆn = 2 (σ 1 cos φ + σ 2 sin φ) = ( 0 cos φ i sin φ 2 cos φ + i sin φ 0 ) This is the pseudo spin index refers to the sublattice degree of freedom. Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 24 / 48

Tight-binding model The spectrum is symmetric: so that where 3 3 X 2 = ( e ikδj )( e ikδ j ) j=1 j =1 = 3 + 2 cos k(δ 1 δ 2 ) + 2 cos k(δ 1 δ 3 ) + 2 cos k(δ 2 δ 3 ) E(k) = ±t 3 + 2 cos k a 1 + 2 cos k a 2 + 2 cos k (a 1 a 2 ) k a 1 = 3d 2 k x + 3d 2 k y, k a 2 = 3d 3d 2 k x 2 k y, k (a 1 a 2 ) = 3dk y Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 25 / 48

Tight-binding model The dispersion law is ( ) ( ) E(k) = ±t 3d 3d ( ) 3 + 4 cos 2 k x cos 2 k y + 2 cos 3dky Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 26 / 48

Tight-binding model Scanning probe microscopy image of graphene Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 27 / 48

Tight-binding model The positive and negative parts of the symmetric spectrum coincide at the points where X = 0 and E = 0. Solutions of these conditions are the corners of the Brillouin zone:, Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 28 / 48

Tight-binding model Only two of these points are independent, let us take K + = v 6 = v 3 = 2π ( ) 2 0, 3d 3 K = v 6 = v 3 = 2π ( 0, 2 ) 3d 3 K + and K are two independent solutions called the Dirac points. Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 29 / 48

Tight-binding model Section of the electronic energy dispersion of graphene for k x = 0, k y = 0 and k x = 2π 3d : Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 30 / 48

Low energy effective theory near the Dirac points The next step is to expand the wave function around the Dirac points: ψ ± A,B (q) ψ± A,B (K ± + q) where we suppose that q is small compared to 1 d 1 KeV. We calculate X and X for k = K ± + q. Expanding to first order in momenta X ± = (X ± ) = 3 e i(k±+q)δj j=1 j=1 3 e i(k±+q)δj 3 i e ik±δj (qδ j ) = 3d 2 ( iq 1 q 2 ) j=1 j=1 3 i e ik±δj (qδ j ) = 3d 2 (iq 1 q 2 ) Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 32 / 48

Low energy effective theory near the Dirac points where K ± δ 1 = 0, K ± δ 2 = ± 2π 3, K ± δ 3 = 2π 3 and qδ 1 = dq 1, qδ 2 = d 2 q 1 + 3d 2 q 2, qδ 3 = d 3d 2 q 1 2 q 2 The Hamiltonian describing the low energy excitations near the Dirac points is found to be: H ± = 3td 0 iq 1 ± q 2 2 iq 1 ± q 2 0 Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 33 / 48

Low energy effective theory near the Dirac points The single-electron Hamiltonian can be written in the compact form: H ± = v F ( σ 2 q 1 ± σ 1 q 2 ) v F α ± q where v F = 3td 2 is the Fermi velocity, σ i, i = 1, 2 are the standard Pauli matrices. α ± = (α 1, α ± 2 ) = ( σ 2, ±σ 1 ). With d = 1.42 A and t = 2.8eV one obtains the value: v F 1 300. Therefore near each of the Dirac points, one obtains a 2D Weyl Hamiltonian describing massless relativistic particles and the dispersion relation is: E(q) = ±v F q = ±v F q 2 1 + q2 2 Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 34 / 48

Low energy effective theory near the Dirac points The electronic lineal energy dispersion of graphene: Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 35 / 48

Low energy effective theory near the Dirac points The eigenfunctions are: H + : H + = v F ( σ 2 q 1 + σ 1 q 2 ), E + (q) = ±v F q 2 1 + q2 2 1 Solution of Type 1 (ψ B = 1): { ψ + 1+ = ( iq1 +q 2 q 2 1 +q 2 2 1 ), ψ + 2 = ( iq1 q 2 q 2 1 +q 2 2 1 ) } 2 Solution of Type 2 (ψ A = 1): { ψ + 3+ = ( 1 iq 1 +q 2 q 2 1 +q 2 2 ), ψ + 4 = ( 1 iq 1 q 2 q 2 1 +q 2 2 ) } Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 36 / 48

Low energy effective theory near the Dirac points Let us write q 1 + iq 2 = q 2 1 + q2 2 eiθ, with θ = arctan(q 2 /q 1 ). Then, 0 q 2 H + 1 + q 2 2 ei( π θ) 2 = v F q 2 1 + q 2 2 e i( π θ) 2 0 The eigenfunctions are: Positive energies E = q 2 1 + q2 2 > 0: ( ψ + + = 1 2 Negative energies E = q 2 1 + q2 2 < 0: ( ψ + = 1 2 e i( π 4 θ 2 ) e i( π 4 θ 2 ) e i( π 4 θ 2 ) e i( π 4 θ 2 ) ) ) Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 37 / 48

Low energy effective theory near the Dirac points The eigenfunctions are: H : H = v F ( σ 2 q 1 σ 1 q 2 ), E (q) = ±v F q 2 1 + q2 2 1 Solution of Type 1 (ψ B = 1): { ψ 1+ = ( iq1 q 2 q 2 1 +q 2 2 1 ), ψ 2 = ( iq1 +q 2 q 2 1 +q 2 2 1 ) } 2 Solution of Type 2 (ψ A = 1): { ψ 3+ = ( 1 iq 1 q 2 q 2 1 +q 2 2 ), ψ 4 = ( 1 iq 1 +q 2 q 2 1 +q 2 2 ) } Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 38 / 48

Low energy effective theory near the Dirac points Again q 1 + iq 2 = q 2 1 + q2 2 eiθ, the Hamiltonian is then 0 q 2 H 1 + q 2 2 ei( π +θ) 2 = v F q 2 1 + q 2 2 e i( π +θ) 2 0 The eigenfunctions are Positive energies E = q 2 1 + q2 2 > 0: ( ψ+ = 1 2 e i( π 4 + θ 2 ) e i( π 4 + θ 2 ) ) Negative energies E = q 2 1 + q2 2 < 0: ( ψ = 1 2 e i( π 4 + θ 2 ) e i( π 4 + θ 2 ) ) Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 39 / 48

Low energy effective theory near the Dirac points The pseudo-spin structure in this low energy limit represents that: For H + : φ π 2 θ. For H : φ π 2 + θ. where S ˆn = 2 (σ 1 cos φ + σ 2 sin φ) = ( 0 cos φ i sin φ 2 cos φ + i sin φ 0 ) Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 40 / 48

Low energy effective theory near the Dirac points By introducing a four-component spinor: ψ T = ( ψ + ) A ψ+ B ψ A ψ B we can unify H ± in a single Dirac Hamiltonian, ( ) H+ 0 H = = v 0 H F α q iv F γ 0 γ a a where ( γ 0 σ3 0 = 0 σ 3 ) (, γ 1 iσ1 0 = 0 iσ 1 ) (, γ 2 iσ2 0 = 0 iσ 2 ) and q a = i a, a = 1, 2 and α a = γ 0 γ a. Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 41 / 48

Low energy effective theory near the Dirac points The eigenfunctions are now Positive energies E = q 2 1 + q2 2 > 0: and where θ = arctan(q 2 /q 1 ). ψ 1 = 1 2 ψ 2 = 1 2 e i( π 4 θ 2 ) e i( π 4 θ 2 ) 0 0 0 0 e i( π 4 + θ 2 ) e i( π 4 + θ 2 ) Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 42 / 48

Low energy effective theory near the Dirac points Negative energies E = q 2 1 + q2 2 < 0: ψ 3 = 1 2 e i( π 4 θ 2 ) e i( π 4 θ 2 ) 0 0 and ψ 4 = 1 2 0 0 e i( π 4 + θ 2 ) e i( π 4 + θ 2 ) Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 43 / 48

Low energy effective theory near the Dirac points By introducing a four-component spinor: ψ T = ( ψ + ) A ψ+ B ψ A ψ B S ˆn = so that = 2 ( 2 (σ1 cos φ + σ2 sin φ) 0 0 (σ 1 cos φ σ 2 sin φ) 0 cos φ i sin φ 0 0 cos φ + i sin φ 0 0 0 0 0 0 cos φ + i sin φ 0 0 cos φ i sin φ 0 S ˆn ψ 1 = 2 ψ 1, S ˆn ψ 2 = 2 ψ 2 ) S ˆn ψ 3 = 2 ψ 3, S ˆn ψ 4 = 2 ψ 4 Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 44 / 48

Low energy effective theory near the Dirac points Here γ µ, µ = 0, 1, 2 are 4 4 gamma matrices in a reducible representation which is a direct sum of two inequivalent 2 2 representations: {σ 3, iσ 1, iσ 2 }, {σ 3, iσ 1, iσ 2 } Each of these two-component representations for graphene quasiparticles wave-function is somewhat similar to the spinor description of electron in QED. However in the case of graphene this pseudo spin index refers to the sublattice degree of freedom rather than the real spin of the electrons. The whole effect of the real spin is just in doubling the number of spinor components, so that we have 8-component spinors in graphene (N = 4 species of two-component fermions). Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 45 / 48

Modifications of the Dirac model 1 Interaction with the electromagnetic field: + iea. The electromagnetic field is not confined to the graphene surface. External magnetic field (Anomalous Hall Effect), electromagnetic radiation, quantized fluctuations. 2 Quasi-particles maybe have a mass. This mass is usually very small and could be convenient to perform the Pauli-Villars regularization. 3 A more important mass-like parameter is the chemical potential µ, which describes the quasi-particle density. 4 Impurities in graphene are describing by adding a phenomenological parameter Γ in the propagator of quasi-particles. 5 Most experiments with graphene are done at rather high temperatures: Thermal field theory. Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 46 / 48

References Quantum Field Theory in Graphene I. V. Fialkovsky, D.V. Vassilevich, November 21, 2011 1:18 WSPC/INSTRUCTION FILE qfext11 3. arxiv:1111.3017v2 (hep-th) 18 Nov 2011. P. R. Wallace, Phys. Rev. 71, 622 (1947). D. P. DiVincenzo and E. J. Mele, Phys. Rev. B 29, 1685 (1984). G. W. Semenoff, Phys. Rev. Lett. 53, 2449 (1984). Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 47 / 48

References A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov and A. K. Geim, Rev. Mod. Phys. 81, 109-162 (2009). Introduction to Dirac materials and Topological Insulators Jérôme Cayssol. arxiv:1310.0792v1 [cond-mat.mtrl-sci] 2 Oct 2013. Introduction to the Physical Properties of Graphene Jean-Noël FUCHS and Mark Oliver GOERBIG. Lecture Notes 2008. http://web.physics.ucsb.edu/ phys123b/w2015/pdf_coursgraphene2008.pdf. Lecture notes. Farhan Rana 2009. https://courses.cit.cornell.edu/ece407/lectures/lectures.htm. Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 48 / 48