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, 218
Graphene: a theorists invention 216
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 long range order in 2D No Magnets.No superfluids No superconductors... No 2D crystals
First 2D materials Andre Geim Konstantin Novoselov Graphene, hbn, MoS 2, NbSe 2, Bi 2 Sr 2 CaCu 2 O x,
Can We Cheat Nature? ANY LAYERED MATERIAL SLICE DOWN TO ONE ATOMIC PLANE? A. Geim
Making graphene P. Blake et.al, 27 D. S. L. Abergel 27 S. Roddaro et. al Nano Letters 27 Novoselev et al (25) Micromechanical cleavage by drawing 3 nm SiO 2 Si 3nm oxide White light 3nm oxide 56nm green light 2nm oxide White light K. Novoselov Andre Geim
Properties: Mechanical Unsupported graphene with Cu particles.5 µm Young s modulus ~ 2 TPa graphene Young s modulus E ~ 2TPa Stiffest materal ~5m ~1nm 1 11 nano-particles S. Bunch et al,nano Letters 8 T. Booth et al,nano Letters 8
Properties: Optical, Transmittance at Dirac point: T1-ap 97.2% R.R. Nair et al, Science (28). P. Blake et al, Nano Letters, 8
Properties: Chemical Single molecule detection NO 2, NH 3, CO F Schedin et al,nature Materials 7
2D Building Blocks Graphene family Transition Metal Dichalcogenides (TMDCs) Phosphorene family E.g.: MoS 2,WSe 2, NbS 2, TaS 2 E.g.: Phosphorene, SnS, GeS, SnSe, GeSe E.g.: Graphene, hbn, Silicene, Germanene, Stanene. Group 13 Monochalcogenides Group 2 Monochalcogenides About 4 2D materials are currently known. About half of them have been isolated. Others shown to be stable using simulations. E.g.: GaS, InSe E.g.: BaS, CaS, MgS 1
Van der Waals heterostructures Nature 499, 419 (213)
Stacking 2D Layers
The most important element in nature
Carbon chemical bonds Carbon: Z6 4 valence electrons 2s 2 2p 2 sp 3 2S + 2p x + 2p y + 2p z hybridize 4 sp 3 orbitals: tetrahedron Diamond
Carbon chemical bonds Carbon: Z6 4 valence electrons 2s 2 2p 2 sp 2 2S + 2p x + 2p y hybridize 3 planar s orbitals: tetragon 1 Out of plane 2p z [ p orbital] p electrons allow conduction s bonds: 2D and exceptional rigidity
Carbon chemical bonds Carbon: Z6 4 valence electrons 2s 2 2p 2 sp 2 2S + 2p x + 2p y hybridize 3 planar s orbitals: tetragon 1 Out of plane 2p z [ p orbital] Graphite
sp 3 Carbon allotropes sp 2 p s Diamond Graphite Diamond is Metastable in ambient conditions!! break
Carbon Allotropes sp 2 3D 21 2D 1D 1996 D Graphite ~16 century Graphene Single -layer 25 Carbon nanotube Multi-wall1991 Single wall 1993 Buckyball 1985
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
Tight binding model Periodic potential Atomic orbitals Eigenstates of the isolated atom: H at a E a a Overlaps between orbitals a corrections to system Hamiltonian H ( r) H at ( r - R) DU ( r R H k E k Tight binding a small overlaps a small corrections DU k
Tight binding model Build Bloch waves out of atomic orbitals (Bravais lattice!) ka ( r) j e a ( r R j ) Solutions have to satisfy: 1. Normalization R j ik R Lattice vector k k 1 2. Eigenstate of the Hamiltonian k H k E k 3. Algebra afind Solutions in terms of: Nearest neighbor transfer (hopping) integral Nearest neighbour overlap integral H t i i j j s ij ij
Band Structure E Simple metal Bravais lattice Parabolic dispersion 2 p p x E 2 m* How is graphene different? honeycomb lattice Not Bravais strong diffraction by lattice z unconventional dispersion Wallace, 1947
Graphene honeycomb lattice 1. 2D 2. Honeycomb structure (non-bravais) 3. 2 identical atoms/cell A B 2 interpenetrating Bravais (triangular) lattices
oneycomb lattice two sets of Bloch functions 1. 2D 2. Honeycomb structure (non-bravais) 3. 2 identical atoms/cell A B N 1 A A A N ik. R k, r e r R R N 1 B B B N A ik. R k, r e r R sum over all type B atomic sites in N unit cells R B A B atomic wavefunction K 4p, 3a
oneycomb lattice two sets of Bloch functions 1. 2D 2. Honeycomb structure (non-bravais) 3. 2 identical atoms/cell A B 4p K (1,) 3a 2 interpenetrating Bravais (triangular) lattices k ( r ) c A A c B B New degree of freedom like spin up and spin down - but instead sublattice A or B a pseudo-spin
Tight binding model of monolayer graphene : Energy solutions 1 1 ; * * k sf k sf S k tf k tf H B B det H ES Secular equation gives the eigenvalues: det 2 2 2 * k f Es t E E k f Es t k f Es t E k f s k f t E 1 2 3 2 / 3 / 3 1. cos 2 a k a ik a ik k i x y y j j e e e k f B A B A B B B A A A atoms identical H t H H : ; Note: If atoms on the two sublattices are different: B A k tf k tf H * Solutions more complicated Edward McCann arxiv:125.4849
E Graphene tight binding band structure t f ik ya / 3 ik ya / 2 3 kxa f k e 2e cos 2 1 s f Typical parameter values : k k, t 3.33eV, s.129 Antibonding Bonding
Band Structure k ( r ) c A A c B B New degree of Freedom Pseudospin Low energy excitations Dirac cones E K G K k x k y Dirac point 6 Dirac 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, C 3
Low energy band structure Conduction band a electrons Dirac cones Fermi energy E F Fermi surface 2 points K K Valence band a holes
Low energy band structure 1 Linear expansion near the K point p Linear expansion in small momentum: Consider two non-equivalent K points: 4p K, K ',; 1 3a and small momentum near them: f k ik a / 3 ik a / 2 3 2 cos k a k e e y 4p, 3a f y p 3a 2 x 2 k p ip Opa / 2 x p x i ; p x y y i y H * tf k tf k v F p x ip y p x ip y 3 6 at v F 1 m / s 2
2. Dirac-like equation with Pseudospin B A 1 1 ; ; 1 1 ; z y x i i s s s p v p p v ip p ip p v H F y y x x F y x y x F. s s s For one K point (e.g. +1) we have a 2 component wave function, with the following effective Hamiltonian (Dirac Weyl Hamiltonian): Bloch function amplitudes on the AB sites ( pseudospin ) mimic spin components of a relativistic Dirac fermion. Low energy band structure Pauli Matrices Operate on sublattice degree of freedom
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: H H v F p x ip y p x H(K) ip y p x ip y p H(K ) x ip v s. p * H v s p K F K ' F H K and H K are related by time reversal symmetry. y
Massless Dirac fermions 3. Massless Dirac Fermions Hamiltonian near K point: E Dirac Cone H i x y v F vf i x x i y i s s x y y P y P x E sv F p s band arctan( p / p ) y index x i 1 1 k r e 2 i se polar angle Pseudospin vector Is parallel to the wave vector k in upper band Antiparallel to k in lower band (s-1) Massless particle : m..photons, neutrinos.. But 3 times slower E cp v F 1 6 m / s
No backscattering within Dirac cone 4. Absence of backscattering within a Dirac cone i 1 1 k r e 2 i se angular scattering probability: 2 2 cos / 2 under pseudospin conservation, backscattering within one valley is suppressed
Helicity (Chirality) 5. Helicity Hamiltonian at K point: H ( K) Helicity operator: v s p v ps n F h H ( K) Helicity is conserved F s n v F phˆ Helical electrons pseudospin direction is linked to an axis determined by electronic momentum. for conduction band electrons, s n s n 1 1 valence band ( holes )
Helicity (Chirality) 5. Helicity E p y v F p p x Note: Helicity is reversed in the K cone p p Helical electrons pseudospin direction is linked to an axis determined by electronic momentum. for conduction band electrons, s n s n 1 1 valence band ( holes ) Helicity conserved a No backscattering between cones
No Backscattering X A K X K B p k K No backscattering between cones No backscattering within cones Dirac Weyl Hamiltonian H vf s *. p conserved Helicity s. p s. p h No backscattering High carrier mobility
Pristine graphene Pseudospin + chirality a Klein tunneling a no backscattering a large mobility Klein Tunneling No electrostatic confinement Katsnelson, Novoselov, Geim (26) No quantum dots No switching No guiding Katsnelson, Novoselov, Geim (26)
Klein tunneling Angular dependence of transmission Katsnelson et al Nature physics (26) Klein paradox Transmission of relativistic particles unimpeded even by highest barriers Physical picture: particle/hole conversion Carrier density outside the barrier is.5 x 1 12 cm- 2. Inside the barrier, hole concentrationn is 1 x 1 12 and 3 x 1 12 cm -2 for the red and blue curves.
The Berry phase
Berry phase Examples Curved space Mobius strip Twisted optical fiber Bohm Aharonov phase Dirac belt Berry Phase Tutorial: https://www.physics.rutgers.edu/grad/682/textbook/ch-3.pdf
Berry phase ( ) 1 2 e ikr 1 i e,2p,1 2p e 1 ikr cos(2p) 1 e 2 i2p d phase d d Berry Im p Homework: Prove this relationship
Graphene Relatives Many natural materials have Dirac cones in their bandstructure (e.g. hightc compnds). But the Fermi surface of these materials contains many states that are not on a Dirac cone. As a result the electronic properties are controlled by the normal electrons and the Dirac electrons are invisible. Graphene is the ONLY naturally occurring material where the Fermi surface consists of Dirac points alone. As a result all its electronic properties are controlled by the Dirac electrons.
Ingredients Graphene bandstructure Graphene offshoots Graphene (25) Silicene (21) 1. 2D 2. Honeycomb structure 3. 2 identical atoms/cell Germanine (214) Stanene (213) Phosphorene (214) Borophene B 36 (214)
Artificial Graphene A B Ingredients: 1. 2D 2. Honeycomb structure 3. Identical populations Also: a ~ 2nm 2 t 12meV 2 m* a 3a 5 vf t 31 m / s 2 Cold atom graphene 87 Rb
Graphene bandstructure A B Can one control/manipulate the 1. 2D bandstructure? 2. Honeycomb structure 3. 2 identical atoms/cell 4. t 1 t 2 t 3 t Relax the condition: t 1 t 2 t 3 Strain Impose external potential Hybridization Electrostatic potential Defects
Summary 1. 2D 2. Honeycomb structure (non-bravais) 3. 2 identical atoms/cell H v F s. p E s sv 1 F p ; band 1 2 index Pseudospin ; Helicity Chiral quasiparticles No backscattering e ikr 1 se i K X p k K K Berry phase p Density of states 2 L g( E) 2 p v F 2 E