15 Triangular lattice Graphene Graphene is an atomic-scale honeycomb lattice carbon atoms monolayer. Conduction via p-orbitals of carbon. Tight binding. 16 Artificial Graphene Weak coupling Consider a 2D electron gas in a potential U(r) with hexagonal (triangular) symmetry and spacing L. p2 H U (r ) 2m The lattice translation vectors are L1 = (L, 0) L2 = (L/2,√3L/2) 17 18 There are two independent reciprocal lattice vectors in the Brillouin zone G1 2p (3, 3 ), 3L G2 2p (0,2 3 ), 3L G3 G1 - G2 The points K1, K2, K3 are connected by vectors Gi K′i are obtained from the Ki by reflection. G1 L1 2p G1 L2 2p G1 L1 0 G2 L2 2p 4p 1 , 0 K1 3L 4p K2 1/ 2 , 3 / 2 3L 4p K3 1/ 2 , - 3 / 2 3L 19 Below I measure energy in units of the bandwidth: K 2 (4p / 3L) 2 E0 2m 2m 20 We assume a periodic potential with a single Fourier component U (r ) 2W cosG1 r cosG2 r cosG3 r This potential has nonzero matrix elements only between states |k> and |k ± Gi> with matrix elements given by W. Perturbation Theory. Practically works up to W = W/E0 =2 A state close to the Dirac point, q ≪ 1, is described by degenerate perturbation theory as q c1 | 1 c2 | 2 c3 | 3 | j e i (K j q )r Kinetic energy: (K i q) 2 Ki q p2 ik E0 ik 2m 2m m 0 1 1 U ( r ) W 1 0 1 Potential energy: 1 1 0 Kinetic energy is a weak perturbation on the potential energy. 21 Diagonalize potential energy W U W W W W ( W ) 2 ( 2W ) W The eigen-energies of the U-matrix are (-W, -W, 2W). There is the double degenerate subspace!!!! In order to project in the double degenerate subspace of U we define: 0 | a 1 / 2 1 / 2 2/3 | b 1 / 6 1/ 6 22 23 Projecting the kinetic energy to this basis and shifting the zero energy level to E0 one finds: (K a | H | a K3) q vqx 2m ( 4 K1 K 2 K 3 ) q b | H | b vqx 2m (K 3 K 2 ) q b | H | a vqy 2 3m 0 1 2 0 -i v K 2p 2m 3Lm is the Fermi-Dirac velocity 1 0 , y , z x 1 0 i 0 0 - 1 Hence, in the Pauli matrix (pseudo-spin) representation the effective low energy Hamiltonian reads: H v( z qx x q y ) 24 One can perform the unitary transformation H → T †HT , where T represents two subsequent π/2 rotations around x- and y-axes in the pseudo-spin space: T xT y T yT z 1 1 i - 1 - i T 2 - 1 i - 1 i T zT x This transforms the Hamiltonian to the conventional form of a 2D Dirac Hamiltonian H vσ q However, in what follows we will use the “blue” form as it is slightly more convenient for the study of the edge states. H v( z qx x q y ) 25 Numerical diagonalization of the Hamiltonian is straightforward. The hole dispersion along a particular contour in the BZ, W=W/E0 =1 Two Dirac points of opposite parity. 26 27 Map of the total charge density (in units 1/L2) at the chemical potential tuned to the Dirac point. The average hole density is <n> = 8/3L2. At L = 50nm the average density is 1.1 × 1011cm−2. Even when the potential is strong, namely W = W/E0 = 1−2, the dispersion is rather close to the result obtained by perturbation theory. The charge density plot is fully connected with empty spots at positions of the potential maximums. So, in clear contrast to natural graphene at W < 2 the system is much closer to the nearly free electron regime than to the tight-binding one. 28 Chiral edge states in toplogical insulator Topological insulator Let us switch on the spin orbit interaction Hso. Whatever is the microscopic mechanism of the interaction the interaction must satisfy the following conditions 1)The interaction depends on spin s. This is true spin, not pseudospin. 2)The interaction is time reversal invariant. 3) The intercation is space inversion invariant Therefore matrix element of Hso between two plane wave must be of the following form p 2 | H so | p1 i([p1 p 2 ] s) An additional condition follows from the Bloch’s theorem. Since the spin-orbit interaction has period of the lattice the matrix element of Hso is nonzero only if p 2 p1 Gi 29 30 Near the Dirac cone 2 | H so | 1 i[K1 K 2 ] s i sz Hence the effective 3x3 spin-orbit Hamiltonian reads 0 i i H so i 0 i i i 0 Projecting this to the degenerate pseudospin states |a> and |b> we find H so 2sz y 31 Total Hamiltonian H v( z qx x q y ) 2sz y This is the K-Dirac cone, for the K’-Dirac cone we have to replace v→-v (opposite parity). One can perform the unitary transformation H → T †HT to transform to the conventional form H vσ q-2ηsz z The energy is 2 v 2 q 2 (2s z ) 2 v 2q 2 2 Due to the gap is opened at Dirac points If chemical potential is inside the gap this is insulator What is topological about this? Why this is different from usual band insulator? Topological chiral edge states Lateral confinement: let us limit the 2D crystal by the infinite wall potential 0 if y y 0 U conf if y y 0 The envelope wave function of the edge state at y > y0 iq x x y A e e Solution of Dirac Eq. H gives 2 v 2 q x2 v 2 2 1 v 2s z i vq x H v( z qx x q y ) 2sz y 32 33 In the explicit coordinate form this is eiq x e y | a | b x 1 iK 2 r iK 3 r | a e e 2 1 | b 2eiK 1r - eiK 2 r eiK 3 r 6 2p | a eiK 2 r eiK 3 r sin y0 0 3L Let us tune the position of the wall 2p y0 p n , n is integer 3L Hence, to satisfy (x,y0)=0 we need only to impose sz 1 / 2, /v, -vqx v 2s z i 0 vq x For the opposite parity Dirac cone the edge solution has different chirality 34 sz 1 / 2, /v, vq x Topological protection: The edge states are found at a special position of the confining wall. An explicit calculation at a different wall position/shape is more involved since the calculation must include admixture of high momentum states to the wave function. However, it is obvious that a variation of the wall position/shape does not influence the edge states since they are topologically protected. 35 The edge states support the spin polarized current at the edge of system. x TI

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