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some things you might be interested in knowing about Graphene

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FEW EXAMPLES OF MOST RECENT MANCHESTER

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Physics at the Dirac Point (Lifshitz transition in bilayer) Serge Morozov unpublished

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-55 concentration (10 10 cm -2 ) resistivity (k ) K 200 K Temperature suspended devices 2 m 2 K level degeneracy lifted < 0.1T SdH oscillations start < 100G quantum mobilities > 1,000,000 cm 2 /V·s transport mobilities > 1,000,000 cm 2 /V·s remnant doping < 10 9 cm -2

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suspended devices 2 m 1 n (10 11 cm -2 ) R (k ) B=0.5T zero B T = 2K GAP IS OPEN BY MAGNETIC FIELD for some devices <1T (VALLEY GAP?) -55 concentration (10 10 cm -2 ) resistivity (k ) K 200 K Temperature

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suspended devices 2 m fractional QHE Manchester, unpublished first reported by Andreis group, Nature 09 Kims group, Nature 09; Laus group, arxiv 2010 million mobilities but the quality of quantization remains really bad need 4-probe devices ! -55 concentration (10 10 cm -2 ) resistivity (k ) K 200 K Temperature

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continues sharpening below 1K charge inhomogeneity < 10 8 cm -2 suspended devices 2 m 1 n (10 11 cm -2 ) (k ) zero B T = 2K 12 can smoothly pass from one electron to one hole Fermi energy scale < 1 meV (k ) -5 n (10 8 cm -2 ) 0 ~T~T remnant doping < 10 9 cm concentration (10 10 cm -2 ) resistivity (k ) K 200 K Temperature

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-55 concentration (10 10 cm -2 ) resistivity (k ) K 200 K Temperature Dirac point PROBING DIRAC POINT NO ENERGY GAP NO ENERGY GAP NO METAL-INSULATOR NO METAL-INSULATORTRANSITION even with one electron per device monolayer: min moves closer to 4e 2 / h bilayer: min remains > 4e 2 /h T T approaches T=0 approx. linearly

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PROBING DIRAC POINT MONOLAYER: gradual gap opening BILAYER: more complex behavior

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PROBING DIRAC POINT BILAYER: more complex behavior magnetoresistance at Dirac point (k ) 5 15 (k ) 128 B =0.25 T ~3x10 9 cm -2 ~3x10 9 cm -2 energy gap induced by gate is less than T =4 =-4

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-1010 gate-induced concentration (10 9 cm -2 ) min / e (10 9 cm -2 ) K 50 K 100 K 150 K PROBING DIRAC POINT -55 n (10 10 cm -2 ) (k ) K 200 K Temperature thermal broadening near Dirac point, curves collapse on a universal dependence totalconcentration measure concentration of thermally excited carriers first, let us analyze monolayer

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PROBING DIRAC POINT -55 n (10 10 cm -2 ) (k ) K 200 K Temperature thermal broadening 0 thermal carriers (10 10 cm -2 ) T (K) T 2 T 2monolayer first, let us analyze monolayer number of carriers at energy ~kT

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PROBING DIRAC POINT -55 n (10 10 cm -2 ) K 200 K Temperature T Tbilayer 0 thermal carriers (10 10 cm -2 ) T (K) T 2 T 2monolayer thermal broadening (k ) number of carriers at energy ~kT

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PROBING DIRAC POINT -55 n (10 10 cm -2 ) K 200 K Temperature thermal broadening (k ) 0 thermal carriers (10 10 cm -2 ) T (K) 030 3bilayer ~3x10 9 cm -2 ~3x10 9 cm -2DENSITYDIMINISHES like a gap ~1 meV at the Dirac point but with a finite T- dependent DOS within the gap number of carriers at energy ~kT

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PROBING DIRAC POINT -55 n (10 10 cm -2 ) K 200 K Temperature (k ) thermal broadening ~3x10 9 cm -2 ~3x10 9 cm -2DENSITYDIMINISHES like a gap ~1 meV at the Dirac point but with a finite T- dependent DOS within the gap number of carriers at energy ~kT 0 thermal carriers (10 10 cm -2 ) T (K) 030 3bilayer

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PROBING DIRAC POINT -55 n (10 10 cm -2 ) K 200 K Temperature (k ) thermal broadening like a gap ~1 meV at the Dirac point but with a finite T- dependent DOS within the gap number of carriers at energy ~kT ~3x10 9 cm -2 ~3x10 9 cm -2 ~1 meV 0 thermal carriers (10 10 cm -2 ) T (K) 030 3bilayer

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PROBING DIRAC POINT one of many possibilities: symmetry-breaking e-e phase transition Falkos group arxiv 2010 OPEN FOR INTERPRETATION cyclotron gaps between different Landau levels different Landau levels 0.026m 0 gap between zero and first LL does not want to close with decreasing B down to 500G with decreasing B down to 500G

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MESSAGE TO TAKE AWAY LIFSHITZ TRANSITION MODIFIED BY SOMETHING(?) DIRAC POINT PHYSICS IS ACCESSIBLE TO STUDIES IN ALL DETAILS million mobilities in 4-probe geometry should bring a lot more of new physics

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Leaving the Carbon Flatland Peter Blake unpublished vertical transport through one-atom-thick crystals

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Many Other 2D Materials Possible 1 m 0Å 9Å 16Å 23Å 0.5 m 2D boron nitride in AFM 2D MoS 2 in optics 1 m 0Å 8Å 23Å 2D NbSe 2 in AFM 1 m 2D Bi 2 Sr 2 CaCu 2 O x in SEM SOME ARE INSULATORS Manchester, PNAS 2005

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spin tunneling devices resonant tunneling devices SS superconducting junctions one-atom-thick barriers? atomically smooth and continuous (impossible by MBE)

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2D CRYSTALS AS TUNNEL BARRIER 1 layer BN can now find BN monolayers in an optical microscope top Au contact Aucontact boron nitride AuAu 2 m

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7-layer BN breakdown 2 V/nm -0.4 tunnel current ( A) voltage (V) -505 I ( A) voltage exponential dependence monolayer resistivity ~1 M m 2 tunnel current ( A) voltage (V) voltage (1/M ) 2 resistivity ~1 k m tunnel current ( A) voltage (V) voltage (1/k ) trilayer NO temperature dependence except for Zero Bias Anomaly MONOLAYER BARRIER height: gap ~5eV effective thickness ~5-6Å NO pin holes AuAu

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MESSAGE TO TAKE AWAY LAYER-BY-LAYER CONSTRUCTION OF VARIOUS TUNNELING DEVICES & QUANTUM WELLS NEW VENUE: ATOMICALLY SMOOTH, CONTINUOUS ONE-, TWO, FEW-ATOM-THICK TUNNEL BARRIERS (beyond MBE; any surface)

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Pseudo-Magnetic Fields by Strain by Strain Paco Guinea, M. Katsnelson & AKG Nature Phys 2010

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Electronic Properties under Strain elastically stretched by > 15% Manchester+Cambridge, PRB 2009; Small 2009 Hones group, PNAS 2009 band structure changes little no gap expected even at 25% stretch Castro Neto, PRB 2009

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practically always rippled non-uniform strain causes pseudo-magnetic field Manchester, PRL 2006 Manchester, PRL 2006 B+ B- Non-Uniform Strain

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[100] [010] [001] Creating Uniform Pseudo-Magnetic Field graphenedisk graphenerectangular UNIFORM FIELD K K insulatingbulk counter propagating edge currents STRAIN ONLY Nature Phys 2010; PRB 2010 field of 10T: 10% strain in m samples

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equivalent to magnetic fields of ~400T Giant Pseudo-Magnetic Fields M. Crommies group, Science 2010 strained graphene bubbles on Pt surface

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MESSAGE TO TAKE AWAY Strain Engineering Can Open Really Large Gaps Pseudo-Magnetic field can be UNIFORM: Landau quantization and QHE in zero magnetic field

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Magneto Oscillations in Quantum Capacitance Leonid Ponomarenko arxiv & PRL 2010

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Capacitance Measurements 100 m SiO 2 /insulating Si Au/Ti Al (top gate) 10 nm aluminium oxide graphene QUALITATIVE OBSERVATIONS: Chen & Appenzeller 2008 Xia et al Nature Nano 2009 Giannazzo at al NanoLett 2009 ~10,000 cm 2 /Vs ~10,000 cm 2 /Vs

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Capacitance Measurements saturates to classical value C oxide sharpness of the dip is determined by v F 100 m SiO 2 /insulating Si Au/Ti Al (top gate) 10 nm aluminium oxide graphene ~10,000 cm 2 /Vs ~10,000 cm 2 /Vs QUALITATIVE OBSERVATIONS: Chen & Appenzeller 2008 Xia et al Nature Nano 2009 Giannazzo at al NanoLett 2009

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Capacitance Varies with Concentration v F 1.1(±0.1)·10 6 m/s 100 m SiO 2 /insulating Si Au/Ti Al (top gate) 10 nm aluminium oxide graphene smearing n 5·10 11 cm -2 n 5·10 11 cm -2 saturates to classical value C oxide sharpness of the dip is determined by v F QUANTITATIVEAGREEMENT ~10,000 cm 2 /Vs ~10,000 cm 2 /Vs

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Magneto Capacitance Oscillations 250K 150K 100K 200K B =16 T C ( F/cm 2 ) V top gate (V) 30K 12T 8T 4T B =0T T =10 K C ( F/cm 2 ) V top gate (V) pronouncedmagneto-oscillations easily survive to room T

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MESSAGE TO TAKE AWAY Quantum Capacitance is a Huge Effect in Graphene Landau Quantization Survives at Room T in Modest Fields (unlike transport, this does not require >30T)

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CONCLUSION GRAPHENE IS A GOLD MINE FOR NEW SCIENCE & APPLICATIONS does NOT feel at all like a mature research area MUCH MORE TO COME

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Misha Katsnelson (Nijmegen) Rahul Nair Sergey Morozov (Chernogolovka) F. Schedin P. Blake Nuno Peres (Porto), Paco Guinea (Madrid), Leonid Levitov (Boston), Rui Yang, Volodya Falko (Lancaster), Soeren Neubeck, Ernie Hill, Sasha Grigorenko graphene reviews: Nature Mat 07 ; RMP 09; Science 09 Kostya Novoselov D. Elias A. Ferrari (Cambridge) L.Ponomarenko A. Castro Neto (Boston) Irina Grigorieva

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graphene dreams: substitute for Si Manchester, Science 2004 de Heer et al, J.Phys.Chem see also Dresselhaus 1996

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GRAPHENE ELECTRONICS ballistic transport on submicron scale, high velocity, great electrostatics, scales to nm sizes BUT no pinch off V g (V) (k ) SiO 2 Si graphene

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GRAPHENE NANO-CIRCUITS E = v F h/2D E = v F h/2D not 1/D 2 as for electrons but much larger 1/D as for slow photons 10 nm e-b lithography

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E = v F h/2D E = v F h/2D gate (V) ( S) few nm few nm 300 K 10 nm stable and robust down to a few nm in size sustains large (~1 A per atom) currents GRAPHENE NANO-CIRCUITS Manchester, Science 08 also, Dai et al, Science 08 e-b lithography

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10 nm stable and robust down to a few nm in size sustains large (~1 A per atom) currents gate (V) ( S) nm 1-10 nm 300 K PROBLEM: no tools to sculpture at true nm scale (same for any other nanoelectronics approach) GRAPHENE NANO-CIRCUITS top-down molecular electronics molecular electronics

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