Graphene and Carbon Nanotubes 1 atom thick films of graphite atomic chicken wire Novoselov et al - Science 306, 666 (004) 100μm Geim s group at Manchester Novoselov et al - Nature 438, 197 (005) Kim-Stormer group at Columbia University Zhang et al - PRL 94, 176803 (005) Zhang et al - Nature 438, 01 (005) One atom thick layer of graphite = graphene
Scanning Transmission Electron micrograph of GRAPHENE Thanks to Dr Nicolosi (Oxford Materials) Tight binding Calculation: Graphene antibonding
Tight binding Calculation: Graphene antibonding Graphene bandstructure N N N N NB, Two C atoms per unit cell (ie 1 electrons, 8 of which are valence) Non-hybridized (pi) electrons play key role in graphene conductivity
c Tight Binding Theory 3 a * 0 0 K c c K * = 1 x 10 6 ms -1 * * 1 ck 1 E Typical values of are in the region of 3 ev, β = ~0.1, giving: c / c 1.005 3 million k.p.h. Minimum conductance at K (Dirac)-point Conductance minimum as Fermi energy passes through zero density of states Manchester and Columbia groups 0 ne e h ne l mv F X 4 e h k F l Mott criterion: k F l 1
A graphene based future Graphene predicted to lead to lots of new physics + New fast transistors, super strength materials, transparent electrodes, chemical sensors... Single walled Carbon nanotubes: Discovered in 1993!
Carbon nanotubes: rolled up graphene! (10,10) Armchair Carbon Nanotubes http://www.photon.t.u-tokyo.ac.jp/~maruyama/agallery/agallery.html Wrapping (10,5) SWNT (0,0) C h = (10,5) a a 1 y x
Carbon Nanotubes (CNTs) Chiral vector for CNT Tube diameter (5,) CNT translation vector Greatest common devisor of Number of hexagons in nanotube unit cell: 1 st lattice point reached! Chiral vectors are used to label CNTs: (n,0) are called zigzag nanotubes (n,n) are called armchair nanotubes Chiral angle (definition) Carbon Nanotubes Cyclic boundary conditions give allowed k-states for CNT: Where reciprocal lattice vectors for CNT unit cell: Number of CNT translation vectors along full length of tube (circumference recip. lattice vector) (CNT axis recip. lattice vector) If allowed k-state coincides with graphene 1st BZ K point = metallic!
Allowed k-states for a (3,1) CNT Extended Brillouin Zone Scheme Zone Folded Scheme Tight binding Calculation: Graphene antibonding
Tight binding Calculation: Graphene antibonding Carbon Nanotubes (examples) (7,4) (7,6) metallic semiconducting
Carbon Nanotube Bandgaps Carbon Nanotubes (examples) Armchair CNT Zigzag CNT Zigzag CNT METAL METAL SEMICONDUCTOR
Carbon nanotubes: metallic if n 1 n is a multiple of 3 zigzag 1,0,0 4,0 5,0 7,0 8,0,1 3,1 5,1 6,1 8,1 3, 4, 6, 7, 4,3 5,3 7,3 5,4 6,4 6,5 Metallic armchair Semiconducting 10,011,0 9,1 9, 8,3 8,4 7,5 7,6 11,1 10, 10,3 9,4 9,5 8,6 8,7 CNT (1-D) Bandstructure NB 1-D CNT Brillouin Zone
CNT (1-D) Density of States (8,8) metallic (8,0) semiconducting Photo-Luminescence Excitation Mapping Scan E excitation energy and measure emission from E11. Luminescence (fluorescence)
Low Dimensional Structures and Materials Artificial layered structures - Quantum Wells and Superlattices Electric or Magnetic Fields applied in one direction. Layers may be only a few atoms thick Heterojunctions Energy levels for different semiconductors Energy line up at junction of two (undoped) materials
Reduced Dimensionality Quantum Well removes 1 Dimension by quantization Electron is bound in well and can only move in plane -D system - motion in x, y plane Quantum Mechanical Engineering quantum wells give levels (symmetric and antisymmetric combinations) Superlattice generates a (mini)band E 0 k /L
Quantum Well - Type I Typical Materials: 1: GaAs : (Al 0.35 Ga 0.65 )As (E g = 1.5 ev) (E g =.0 ev) Energy levels are quantized in z-direction with values E n for both electrons and holes E = E n + k /m* 1-D -D Infinite well - Particle in a box 1-D Motion in z-direction d E m* dz nz na sin L L n = 3 n = E n n m* L k m* n = 1 Typical values L = 10 nm, m e * = 0.07 m e E n = 54 n mev System is Two-Dimensional when: E -E 1 > kt 16 mev 5 mev at 300 K
Finite Well even parity odd parity n (z) Acos kz Asin kz z < L/ Bexp[-(z - L/)] Bexp[-(z - L/ z > L/ Bexp[+(z + L/)] -Bexp[+(z + L/)] z < L/ where: n k V0 m * m * 1 assume m 1 =m k + =k 0 =m*v 0 / boundary conditions: wavefunction 1 = probability current A cos (kl/) = B ka sin (kl/) = B 1 1 1 m z m z 1 A sin (kl/) = B ka cos (kl/) = -B k tan (kl/) = k cot (kl/) = k sec (kl/) = k 0 k cosec (kl/) = k 0 cos (kl/) = k/k 0 sin (kl/) = k/k 0
Graphical solution of finite Quantum Well Well depth determines value of slope k 0-1
Optical Properties 3-D Absorption coefficient is proportional to the density of states: ~ 1/ Modified close to the band gap due to excitons -D - Big Changes Multiple Band gaps - Band gap shift - Sharper edge For wide wells the sum of many -D absorptions becomes equivalent to the 3-D absorption shape ( 1/ ) Correspondence principle.
GaAs/Al 0.35 Ga 0.65 As Quantum Well absorption Sharp peaks due to excitons peaks doubled due to heavy and light holes Semiconductor lasers Forward biased p-n junction Quantum Well laser Fibre Optic Communications, CD players, laser pointers
Molecular Beam Epitaxy (MBE) Ultra High Vacuum evaporation of molecular species of elements (Molecular Beam) Epitaxy - maintaining crystal structure of the substrate - which is a single crystal Metal Organic Vapour Phase Epitaxy (MOVPE) Chemical reaction of elements bonded in volatile organic compounds e.g. (CH 3 ) 3 Ga + AsH 3 GaAs + 3CH 4 Reaction takes place on a heated substrate and growth is also epitaxial
Heterojunctions and Modulation Doping New idea for superlattices and heterojunctions: Separate the dopant impurities from the electrons Gated structures Place metallic electrode on surface to apply variable electric field gives a variable potential and surface charge density Basis of MOS transistors as well as controllable -D systems.
Negative electrode potential repels electrons underneath leaving only a narrow 1-D channel of conducting electrons Quantized Conductance in 1-D - the Quantum Point Contact For a short 1-D structure there is no scattering - Ballistic transport Calculate the current carried by the electrons by adding up the contributions from all carriers travelling in one direction. For flow in one direction only (k > 0) the density of states is: 1 L m 1 * / 1 / d i.e. half the usual 1-D density of states, but with a factor to account for spin degeneracy.
The electrons have velocity v = (/m*) 1/. Therefore the current in the positive direction is: J 0 e v g( ) d e h Apply a voltage V along the 1-D channel to give a difference in chemical potential. This causes a net current to flow: J Tot. J J e h 0 d e h ev 0 d e h V Therefore the conductance is: When there are p 1-D subbands occupied (e.g. if the 1-D wire is wider) each contributes one unit of conductance Total J Tot. V e h e p h B.J. van Wees et al, Phys. Rev. Lett. 60, 848 (1988)