Electronic properties of Graphene and 2-D materials 2D materials – background Carbon allotropes Graphene Structure and Band structure Electronic properties Electrons in a magnetic field Onsager relation Landau levels Quantum Hall effect Engineering electronic properties Kondo effect Atomic collapse and artificial atom Twisted graphene Eva Y. Andrei Rutgers University Aug 27-29, 2018 E.Y. Andrei Graphene: a theorists invention 2016 E.Y. Andrei No long range order in 2-Dimensions David Mermin Herbert Wagner .. long-range fluctuations can be created with little energy cost and since they increase the entropy they are favored. …continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions D ≤ 2. … No Magnets ….No superfluids No long range order in 2D …No superconductors ... No 2D crystals E.Y. Andrei First 2D materials Andre Geim Konstantin Novoselov Graphene, hBN, MoS2, NbSe2, Bi2Sr2CaCu2Ox, E.Y. Andrei Can We Cheat Nature? ANY LAYERED MATERIAL SLICE DOWN TO ONE ATOMIC PLANE? A. Geim Making graphene P. Blake et.al, 2007 D. S. L. Abergel 2007 Novoselev et al (2005) S. Roddaro et. al Nano Letters 2007 Micromechanical cleavage by 300 nm SiO2 “drawing” Si K. Novoselov Andre Geim 300nm oxide 300nm oxide 200nm oxide White light 560nm green light White light E.Y. Andrei Properties: Mechanical Unsupported graphene with Cu particles • Young’s modulus ~ 2 TPa 0.5 µm graphene Young’s modulus E ~ 2TPa Stiffest materal ~5m ~1nm 1011 nano-particles S. Bunch et al,Nano Letters‘08 T. Booth et al,Nano Letters‘08 Properties: Optical , Transmittance at Dirac point: T=1-ap =97.2% P. Blake et al, Nano Letters, ‘08 R.R. Nair et al, Science (2008). Properties: Chemical Single molecule detection NO2, NH3, CO F Schedin et al,Nature Materials ‘07 2D Building Blocks Graphene family Transition Metal Dichalcogenides Phosphorene family (TMDCs) E.g.: Phosphorene, SnS, GeS, SnSe, GeSe E.g.: MoS2 ,WSe2 , NbS2, TaS2 E.g.: Graphene, hBN, Silicene, Germanene, Stanene. Group 13 Monochalcogenides Group 2 Monochalcogenides • About 40 2D materials are currently known. • About half of them E.g.: GaS, InSe have been isolated. E.g.: BaS, CaS, MgS • Others shown to be 10 stable using simulations. E.Y. Andrei Van der Waals heterostructures Nature 499, 419 (2013) E.Y. Andrei Stacking 2D Layers E.Y. Andrei The most important element in nature E.Y. Andrei Carbon chemical bonds Carbon: Z=6 4 valence electrons 2s22p2 sp3 2S + 2px + 2py + 2pz hybridize 4 sp3 orbitals: tetrahedron Diamond E.Y. Andrei Carbon chemical bonds Carbon: Z=6 4 valence electrons 2s22p2 sp2 2S + 2px + 2py hybridize 3 planar s orbitals: tetragon 1 Out of plane 2pz [“p” orbital] p electrons allow conduction s bonds: 2D and exceptional rigidity E.Y. Andrei Carbon chemical bonds Carbon: Z=6 4 valence electrons 2s22p2 sp2 2S + 2px + 2py hybridize 3 planar s orbitals: tetragon 1 Out of plane 2pz [“p” orbital] GraphiteE.Y. Andrei Carbon allotropes sp3 sp2 p s Diamond Graphite Diamond is Metastable in ambient conditions!! E.Y. Andrei break Carbon Allotropes sp2 3D 2010 2D “1D” 1996 “0D” Graphite ~16 century Graphene Single -layer 2005 Carbon nanotube Multi-wall1991 Single wall 1993 Buckyball 1985 E.Y. Andrei Graphene 2D materials – background Carbon allotropes Graphene Structure and Band structure Electronic properties Electrons in a magnetic field Onsager relation Landau levels Quantum Hall effect Engineering electronic properties Kondo effect Atomic collapse and artificial atom Twisted graphene E.Y. Andrei Tight binding model Periodic potential Atomic orbitals Eigenstates of the isolated atom: H = E at a a a • Overlaps between orbitals a corrections to system Hamiltonian. H (r) = H (r - R) DU (r at H k = Ek k R • Tight binding a small overlaps a small corrections DU E.Y. Andrei Tight binding model Build Bloch waves out of atomic orbitals (Bravais lattice!) ikR j ka (r) = e a (r R j ) Lattice vector R j Solutions have to satisfy: 1. Normalization k k =1 2. Eigenstate of the Hamiltonian k H k = Ek 3. Algebra aFind Solutions in terms of: • Nearest neighbor transfer (hopping) integral i H j = tij • Nearest neighbour overlap integral i j = sij E.Y. Andrei Band Structure Simple metal E Bravais lattice Parabolic dispersion p2 px E = 2m* How is graphene different? honeycomb lattice Not Bravais strong diffraction by lattice z unconventional dispersion Wallace, 1947 E.Y. Andrei Graphene honeycomb lattice 1. 2D 2. Honeycomb structure (non-Bravais) 3. 2 identical atoms/cell A B 2 interpenetrating Bravais (triangular) lattices E.Y. Andrei Honeycomb lattice – two sets of Bloch functions 1. 2D 2. Honeycomb structure (non-Bravais) 3. 2 identical atoms/cell A B 1 N k,r = eik .RA r R A A A N RA 1 N k,r = eik .RB r R B B B N RB 4p sum over all type K = ,0 atomic 3a B atomic sites wavefunction in N unit cells E.Y. Andrei Honeycomb lattice – two sets of Bloch functions 1. 2D 2. Honeycomb structure (non-Bravais) 3. 2 identical atoms/cell A B 4p K = (1,0) 3a 2 interpenetrating Bravais (triangular) lattices (r) = c c = k A A B B New degree of freedom –like spin up and spin down - but instead sublattice A or B a pseudo-spin E.Y. Andrei Tight binding model of monolayer graphene : Energy solutions tf k 1 sf k H = B ; S = * * tf k B sf k 1 H ; = H 3 A A A B B B ik . ik a / 3 ik a / 2 3 f k = e j = e y 2e y cos kxa 2 t A H B j =1 identical atoms : 0 = A = B Edward McCann arXiv:1205.4849 Secular equation gives the eigenvalues: detH ES = 0 Note: E t Es f k If atoms on the two sublattices are different: det 0 = 0 * t Es f k 0 E 2 A tf k E 2 t Es2 f k = 0 H = 0 tf * k B 0 t f k E = Solutions more complicated 1 s f k Graphene tight binding band structure ik ya/ 3 ik ya/ 2 3 kxa 0 t f k f k = e 2e cos E = 2 1 s f k Typical parameter values : 0 = 0, t = 3.033eV , s = 0.129 Anti- bonding Bonding E.Y. Andrei Band Structure Low energy excitations New degree of Freedom Dirac cones Pseudospin E G K’ K ky kx Dirac point (r) = c c = k A A 6B DiracB points, but only 2 independent points K and K’, others can be derived with translation by reciprocal lattice vectors. The Dirac points are protected by 3 discrete symmetries: T, I, C3 E.Y. Andrei Low energy band structure Conduction band a electrons Dirac cones K’ Fermi energy EF K Fermi “surface” =2 points Valence band a holes E.Y. Andrei Low energy band structure 1 Linear expansion near the K point ik ya/ 3 ik ya/ 2 3 kxa f k Consider= e two 2 none -equivalentcos K2 points: 4p K, K'= ,0; = 1 3a p and small momentum near them: 4p p k = ,0 px i ; py i 3a x y 3a Linear expansion in small momentum: 2 f k = px ipy Opa / 2 0 tf k 0 px ipy H = v 3at 6 * F vF = 10 m / s tf k 0 px ipy 0 2 E.Y. Andrei Low energy band structure 2. Dirac-like equation with Pseudospin For one K point (e.g. =+1) we have a 2 component wave function, A = B with the following effective Hamiltonian (Dirac Weyl Hamiltonian): 0 px ipy H = v = v s p s p = v s.p F F x x y y F px ipy 0 0 1 0 i 1 0 Bloch function amplitudes on s x =; ;s y = ;s z = 1 0 i 0 0 1 the AB sites (‘pseudospin’) mimic spin components of a relativistic Dirac fermion. Pauli Matrices Operate on sublattice degree of freedom E.Y. Andrei Low energy band structure 2. Dirac-like equation with Pseudospin To take into account both K points (=+1 and =-1) we can use a 4 component wave function, AK BK Note: real spin not included. = Including spin gives 8x8 Hamiltonian AK ' BK' with the following effective Hamiltonian: 0 p ip 0 0 x y px ipH(y K)0 0 0 H = v F 0 0 0 p ip x y 0 0 px H(K’)ipy 0 * H K = vFs.p H K ' = vFs .p HK and HK’ are related by time reversal symmetry E.Y. Andrei Massless Dirac fermions 3. Massless Dirac Fermions Dirac Cone Hamiltonian near K point: E 0 i H = v x y = v is is P F F x x y y y i x y 0 Px 1 E = sv p = 1 eikr F 2 i se Pseudospin vector = arctan(py / px ) polar angle • Is parallel to the wave vector k in upper band s = band index • Antiparallel to k in lower band (s=-1) Massless particle : m=0 ..Photons, neutrinos.. E = cp 6 But 300 times slower vF 10 m / s E.Y. Andrei No backscattering within Dirac cone 4. Absence of backscattering within a Dirac cone 1 = 1 eikr 2 i se angular scattering probability: 2 = 0 = cos2 / 2 = 0 under pseudospin conservation, backscattering within one valley is suppressed E.Y. Andrei Helicity (Chirality) 5. Helicity Hamiltonian at K point: Helical electrons pseudospin direction H (K) = vFs p = vF ps n is linked to an axis determined by electronic momentum. Helicity operator: h = s n for conduction band electrons, ˆ H (K) = vF ph s n = 1 Helicity is conserved s n = 1 valence band (‘holes’) E.Y.