Spring College on Computational Nanoscience May Electric Transport in Carbon Nanotubes and Graphene

Transcription

1 145-8 Spring College on Computational Nanoscience 17-8 May 1 Electric Transport in Carbon Nanotubes and Graphene Philip KIM Dept. of Physics, Columbia University New York U.S.A.

2 Electric Transport in Nanotubes and Graphene Philip Kim Department of Physics Columbia University

3 SP Carbon: -Dimension to 3-Dimension Atomic orbital sp D 1D D 3D Fullerenes (C 6 ) Carbon Nanotubes Graphene Graphite

4 Electronic Band Structure of Graphene Band structure of graphene (Wallace 1947) E(k D ) E E v k F empty filled k y ' k x ' k y K x K k x k y D Brillouin Zone k x Zero effective mass particles moving with a constant speed v F = c/3

5 Graphene Lattice Symmetry: Pseudo Spinor Graphene Lattice Structures A sublattice: p z orbitals Spinor Representation A B sublattice : p z orbitals B Superposition: A e i B Two inequivalent lattice sites! Pseudo spin

6 Dirac Fermions in Graphene : Helicity E E E momentum pseudo spin K y k y K y x k x x H eff Effective Dirac Equations v F kx iky kx iky v k DiVincenzo and Mele, PRB (1984); T. Ando, JPSJ (1998);McEuen at al, PRL (1999) F k = e ikr. 1 k = tan -1 (k y / k x ) 1 e i

7 Single Wall Carbon Nanotube. since 1991

8 Extremely Long Mean Free Path in Nanotubes Multi-terminal Device with Pd contact See also S. Frank, P. Poncharal, Z. L. Wang, and W. A. de Heer, Science 8, (1998) R (k) Resistance (k) R (k) 4 T = 5 K 4 L (m) l e ~ 1 m 6 T = 5 K = 8 k/m R ~ L * Scaling behavior of resistance: R(L) M. Purewall, B. Hong, A. Ravi, B. Chnadra, J. Hone and P. Kim, PRL (7) R ~ R Q 4 Length (m) h ( L) Ne L (m) 6 h Ne R L l e Room temperature mean free path >.5 m

9 Extremely Long Mean Free Path of Nanotubes: Role of Pseudo Spin 1D band structure of nanotubes E Low energy band structure of graphene k 1D E F Pseudo spin right moving left moving 1 k pseudo V ( r ) k pseudo Small momentum transfer backward scattering becomes inefficient since it requires pseudo spin flipping.

10 Electron Mean Free Path of Nanotube R (k) T = 5 K L (m) K 11 K 6 K 3 K L (m) R (k) T = 5 K 175 K 11 K 6 K 3 K mean free path (m) Lines are fit to h ( ) 4e h 4e 6.45 k h 4e R L Rc L l Non-ideal contact resistance R c < k l e ~.5 RT!! Temperature (K) e

11 Characterization of Nanotube Structures by Rayleigh Light Scattering Nanotube Growth over trenches Rayleigh Scattering Characterization Brus, Heinz, Hone, O Brien groups, Science 36,154 (4)

12 Temperature Dependent Resistivity B. Chandra M. Purewal, P, Kim and J. Hone 5.k (m).k 15.k 1.k (6, 11) Super linear behaviors for T > 15 K Scattering due to acoustic phonons 5.k ac kbt 1 v hv s F T(K) Enhanced scattering activated T > 15 K

13 Carbon Electronics: Challenges Pros: High mobility High on-off ratio (nanotubes) High critical current density Con: Controlled growth graphene IBM, Avouris group Nanotube Ring Oscillators Artistic dream (DELFT)

14 Graphene Sample Preparation Epitaxial graphene on SiC GTech, IBM, NRL, HRL, Purdue, Chemical Vapor Deposition Geim & Novoselov (4) SKKU, MIT, Austin,

15 Field Effective Transport in Single Layer Graphene Single layer graphene device Resistivity vs Gate Voltage 5 4 ~h/4e Cleaved graphite crystallite m () 3 E 1 N D (E) V g (V) -1 = e v F l e N D

16 D Gas in Quantum Limit : Conventional Case Density of States E * m e Landau Levels in Magnetic Field Quantum Hall Effect in GaAs DEG c eb / m* E F DOS * m h s s Graphene Vanishing carrier mass near Dirac point Strict electron hole symmetry eb Electron hole degeneracy c * m

17 Quantum Hall Effect in Graphene Zhang et al (5), Novoselov et al. (5) Hall Resistance (k) h e 1 h 6 e 1 h 1 e 4 B (T) Magnetoresistance (k) T=5 mk V g =- V Hall Resistance (k) h e 1 h 6 e 1 h 1 e 1 h 14 e 1 h -14 e 1 h -1 e 1 h -6 e Quantization: R xy = 4 (n + _ 1 ) e h V g (V) 1 h - e T= 1.5K, B= 9T

18 Berry s Phase and Magneto Oscillations Landau orbit near the Fermi level k y Magnetic flux in cyclotron orbit B = o B F /B Graphene: H eff v F k k F k x o = h/e F = o k F /4 E Pseudo Spin ~ exp[( B / o )] B= R xx DEG Graphene k x ' k y ' S = h/ = Quantum Hall States 1/B F 1/B ~ exp[( B / o ) S/h]

19 Room Temperature Quantum Hall Effect +_ E 1 ~ 1 5 T Typical sample on SiO mobility: ~ 15, cm /Vsec Novoselov, Jiang, Kim and Geim et al. Science (7)

20 Quantum Hall Effect in Suspended Graphene Suspended graphene samples Resistance () μ FE ~, cm /Vs Cleaning: current annealing Bachtold et al. (7) Mechanical stability J. Lau et al. (9) Bolotin et al., (8); Du et al., (8) Vg (V).5 =1/3 FQHE is observed!! 1T Condutance (e /h) Low Field QHE 1T.5T.3T 1.6 K Increasing B Conductance (e /h) T 14T Gate voltage (V) Gate Voltage (V)

21 Creation of Energy Gap in Graphene Confinement of Dirac Particles: Nanoribbons, Quantum Dot (Columbia, IBM, ) (Manchester, ETH, ) Breaking Symmetry: Biased Bilayer Graphene (Manchester, DELFT, Berkeley, Columbia, ) Chemical Treatment: Graphane, Graphene Oxide (Manchester, Rutgers, )

22 Graphene Nanoribbons: Confined Dirac Particles Gold electrode Graphene Dirac Particle Confinement W W k y 3 W 1 m 1 nm < W < 1 nm Graphene nanoribbon theory partial list y x k y k y W 1 W W k y W Zigzag ribbons E v F k x ( n / W ) E gap ~ hv F k ~ hv F /W

23 Scaling of Energy Gaps in Graphene Nanoribbons 1 E g (mev) 1 1 P1 P P3 P4 D1 D W (nm) E g = E /(W-W ) Han, Oezyilmaz, Zhang and Kim PRL (7) Chen, Lin, Rooks, and Avouris, Physica E. (7)

24 Graphene Nanoribbons Edge Effect Crystallographic Directional Dependence Son, et al, PRL. 97, 1683 (6) 4 E g (mev) m (degree) Rough Graphene Edge Structures

25 Localization of Edge Disordered Graphene Nanoribbons Querlioz et al., Appl. Phys. Lett. 9, 418 (8) See also: Gunlycke et al, Appl. Phys. Lett. 9 (14), 1414 (7). Areshkin et al, Nano Lett. 7 (1), 4 (7) Lherbier et al, PRL (8) Transport gap

26 Variable Range Hopping in Graphene Nanoribbons T E E F G max G T exp T 1 1 d: dimensionality d Conductance (S) K 15K 1K K 3K W = 37 nm x.1 4 V g (V) 6 ln(r) nm nm D VRH 31 nm 37 nm 48 nm ln(r) nm nm 1D VRH 31 nm 37 nm 48 nm ln(r) nm Arrhenius plot nm 31 nm 37 nm 48 nm -1-7 nm -1-7 nm -1-7 nm...4 T -1/3.6.. T -1/.4..1 T -1.

27 Nature of Transport Gap in Graphene Nanoribbons Transport Characterization: nanoribbons Temperature Dependent Transprot Log (G) (A.U.) Energy Gap Scaling in Nanoribbons Graphene nanoribbons exhibits three different energy scales for energy gap formation due to the quantum confinement and edge disorders: m : mobility edges induced by edge disorder E a : Activation energy from the quantum confinement origin k B T * : Hoping length to the localized states in gapped regime Han et al., Phys. Rev. Lett. (1)

28 Top Gated Graphene Nanoribbon FET I d (A) K Vd (V) -8 V G (e /h) V tg = V -1 V - V -3 V -4 V -5 V -6 V -7 V V LG (V) Good Bad OFF I off / I on ~ K I d (na) High saturation velocity v Fermi = 1x1 8 cm/s Silicon: 1x1 7 cm/s GaAs:.7x1 7 cm/s Operation current density > 1 ma/m Mobility: ~ 5-1 cm /Vsec On-off ratio: ~ V sd = 1 mv 3 K I off / I on ~1 V top_gate (V)

29 Graphene Electronics Conventional Devices FET Band gap engineered Graphene nanoribbons Graphene quantum dot Nonconventional Devices (Manchester group) Graphene Veselago lense Cheianov et al. Science (7) Graphene Spintronics Son et al. Nature (7) Graphene psedospintronics Trauzettel et al. Nature Phys. (7)

30 Transport Ballistic Graphene Heterojunction electrode V BG = 9 V graphene nnn ppp Young and Kim (8) 1 1 m PN junction resistance Cheianov and Fal ko (6) See also Shavchenko et al and Goldhaber-Gordon s recent PRL Conductance Oscillation: Fabry-Perot n 1,, k 1, T T npn V BG = -9 V pnp Conductance (ms) n,, k R* R T L 18 V -18 V 4 n 1,, k 1, k 1 /k = sin /sin = L /cos V TG (V)

31 Fabry-Perot Oscillations in Ballistic Graphene Heterojunction Derivative Map 5 - dg/dn L T T R density: n 1 density: n T 1 R 1 n 1 (1 1 cm - ) Theory with L m = 5 nm n (1 1 cm - ) e 4 L / L m

32 Pseudo Spin Control with Magnetic Field G osc (e /h) Young and Kim, Nat. Phys. (9) Theory with L m = 5 nm G osc (e /h) B (T) -1 n 1 =3.6x1 1 cm - B k y / el n (1 1 cm - ) n (1 1 cm - ) With increasing Magnetic field: * Oscillations phase shift * Abrupt phase at a certain magnetic field Berry phase associated with pseudo spin Levitov et al. (8) B k y / el

33 Temperature Dependence T (K) V TG (V) 8 K 6 K 43 K 3 K 16 K 4 K FP resonance to ~8K Energy scale G (e /h) A(T)/A(4K) v F T L 1K Better mobility and better electrostatic will make the energy scale larger!

34 Nanowire Top Gate: Ultimate Electrostatic V NW (V) 3 K Temperature Dependence Quantum Oscillations at elevated temperatures

35 Nanotube on SiO substrate 5.k.k (6, 11) Super linear behaviors for T > 15 K (m) 15.k 1.k Scattering due to acoustic phonons 5.k ac kbt 1 v hv s F T(K) Enhanced scattering activated T > 15 K

36 Optical phonons in substrates Transport in graphene Chen et al. Nature Phys (8) Optical phonons SiO substrate is much softer! (~ 3-5 mev)

37 Substrate Phonon Scattering Analysis Perebeinos et al. Nano Lett. 9, 31 (9) Total static acoustic T SPP ( nb ( w ) nb ( w 4,5 )) F wso, v v 4 F 5 ( F F ) n B ( w v ) ( e w v k 1 B T 1) 5.k.k (6,11) Total static acoustic T SPP ( nb ( w ) nb ( w 4,5 )) (m) 15.k 1.k 5.k Total static acoustic T T(K)

38 hexa-boron Nitride: Ideal Dielectric graphene Boron Nitride Comparison of h-bn and SiO Band Gap Dielectric Constant Optical Phonon Energy Structure BN ev ~4 >15 mev Layered crystal SiO 8.9 ev mev Amorphous

39 Mechanical Exfoliation of BN 14nm 8nm 1nm SiO Mechanically cleavable 1. SiO.8 BN 7.8 BN BN 1..8 Atomically flat frequency.6.4 roughness [nm].6.4 SiO.. BN surface height [nm] BN thickness [nm]

40 Precision Transfer Technique 1. spin coat substrate with PMMA and scratch onto top surface. Lift off PMMA; graphene comes with it! Hone Columbia 3. Align graphene over target using a micro-manipulator. PMMA is brought into contact with target and annealed.

41 Graphene on h-bn is Flat! graphene BN.5 um 1. Roughness um frequency [a.u.].8.4. nm Graphene BN height [nm]

42 Transport in Graphene/h-BN Resistance () 1 Mobility ~, cm /Vs -4-4 Vg (V) ~1 V R xx [k] V g = 1 V, T = 1.6 K =4 = B = 14 T, T = 1.6 K B [Tesla] = Vg=-1V R xx [k] Enhanced mobility on BN versus SiO for the same flake Very narrow DP peak: reduced inhomogeneity. Reduced chemical reactivity (no appreciable doping by H/Ar annealing) xx [e /h] 1..5 = Vg [V] =1 = xy [e /h]

43 Bilayer graphene on Boron Nitride 8 resistance [k] K 3 K ~1. V x Bilayer graphene transferred over BN n [x1 1 cm - ] RT mobility > 4, cm /sec mobility [cm /Vs] K 3K R xx [k] B [Tesla] = R xx [k] =5 =4 Vg=-17.6V =3 1 5 R xy [k] SdH Oscillation =7 n [x1 1 cm - ]. 5 1 B [Tesla]

44 Conclusions Relativistic QM: High Energy Physics Quasi Relativistic QM: Low Energy Physics CERN Electro-Positron Collider Kim Columbia in City of New York. ~ H * c p Dirac Equation:

45 Acknowledgement Special Thanks to: Melinda Han Yuri Zuev Yue Zhao Andrea Young Mitsuhide Takekoshi Dmitri Efetov Fereshte Ghahari Young Jun Yu Namdong Kim Kirill Bolotin Vikram Deshpande Paul Cadden-Zimansky Collaboration: Stormer, Pinczuk, Heinz, Hone, Brus, Nuckolls, Flynn, KS Kim, GC Yi, BH Hong, A Chen Funding:

46 FP resonances and Berry Phase ref k y / el ebl ebl ref ( k y ) ( k y ) k y / el Real Space trajectories Momentum Space Trajectories Critical field Bc k y / el B k y (nm -1 ) k x (nm -1 ) B k y / el B k y / el

47 Wave Function Collimation in Diffusive Transport Diffusive transport outside of heterojunction regime Collimated beam direction T R 4 T 4 R Probability

48 Rayleigh Light Scattering of Individual Tubes Intensity (A.U.) Semiconducting Metallic photon energy (ev) DOS photon energy (ev) DOS Structure can be analyzed from the Rayleigh Scattering Spectrum Intensity (a.u.) a (13,1) (13, 1) ( (1, 1) Intensity (a.u.) a (1,1) Tube 1 (1,1) Tube photon energy (ev) photon energy (ev).5 Energy

49 Resistivity of (6, 11) Nanotubes Temperature Dependence 3 K 1-dimensional resistance scaling h R( L) 1 Ne d L Measurement of resistivity of known chirality in 4-5 K