A BIT OF MATERIALS SCIENCE THEN PHYSICS

Transcription

1 GRAPHENE AND OTHER D ATOMIC CRYSTALS Andre Geim with many thanks to K. Novoselov, S. Morozov, D. Jiang, F. Schedin, I. Grigorieva, J. Meyer, M. Katsnelson

2 A BIT OF MATERIALS SCIENCE THEN PHYSICS

3 CARBON ALLOTROPES 3d d d d Graphite Buckyball Graphene 985 Mermin-Wagner theorem: strictly-d crystals are thermodynamically unstable Diamond Carbon Nanotube Multi-wall 99 Single-wall 993

4 RECENT EFFORTS Chemical Vapour Deposition up to µm in diameter thickness 3nm ( layers) Nature 388, 45 (998) grows epitaxially several-layers-thick graphite Carbon 4, 867 (4) nanographene Chem. Phys. Lett. 348, 7 ()

5 MOTIVATION 5 µm peeling off to nm thickness? how thin can you go before they segregate, decompose or scroll?

6 EARLIER EFFORTS CHEMICAL EXFOLIATION search for graphene in SLURRIES APL 84, 43 (4) MECHANICAL EXFOLIATION Ohashi et al, Tanso (997, ) from down to 5 layers

7 Extracting a Single Plane STRONGLY LAYERED MATERIALS PULL OUT ONE ATOMIC PLANE individual atomic sheets: do they exist?

8 Free-Standing Graphene Å 9Å 3Å AFM µm OPTICS µm layer of atoms visible by naked eye µm monolayers are stable under ambient conditions

9 Free- Hanging Graphene TEM IMAGE µm rotation images also prove that it is single layer

10 CLEAVAGE TO SINGLE LAYER Originally: Repeated Peeling of Mesoscopic Mesas Science 36, 666 (4) Now: Micromechanical Cleavage by Drawing PNAS, 45 (5) There are nine and sixty ways of constructing tribal lays, and every-single-one-of-them-is-right! (R Kipling)

11 Key: Visual Identification Å 9Å 3Å AFM µm OPTICS µm µm

12 Other D Atomic Crystals D boron nitride in AFM Å 9Å 6Å 3Å D NbSe in AFM Å 8Å 3Å µm.5µm µm also, can do,3,4 layers µm µm D Bi Sr CaCu O x in SEM D MoS in optics

13 Local Crystal Quality STM image of D NbSe HRTEM image of D Bi Sr CaCu O x NO RECONSTRUCTION (except for BISCCO) electron diffraction from single-layer graphene electron diffraction from D Bi Sr CaCu O x

14 Macroscopic Quality & Homogeneity Electric Field Effect Au contacts 3 3 SiO µm Si D crystal σ (/kω) graphene D NbSe D MoS σ (/MΩ) σ =neµ =εε µv g /d D NbSe & MoS :.5 to 3 cm /Vs as in bulk at 3K -8-4 V gate 4 (V) 8 graphene: up to 6, cm /V s at 3K up to 5, cm /V s at 4K

15 D ATOMIC CRYSTALS new class of crystalline materials wide choice of materials properties (electronic, mechanical, chemical, etc.) they exist, therefore they can be made en masse sublimation of Si from surface of SiC C Berger et al, J. Phys. Chem. B 8 (5)

16 FROM MATERIALS SCIENCE TO PHYSICS

17 Electric Field Effect in Graphene resistivity Au contacts 6 ρ (kω) 4 T =K SiO Si GRAPHENE V g (V) peak around zero can be shifted by chemical doping (exposure to NO, NH 3, CO, etc) µm SEM in false colour

18 Electric Field Effect in Graphene conductivity Hall effect σ (/kω) V g (V) T =K σ=n(v g )eµ /ρ xy =ne/b B =T T =K holes electrons V g (V) - /ρ xy (/kω) simple behaviour; practically constant µ; no trapped carriers; σ(n )

19 Electric Field Effect in Graphene conductivity Hall effect σ (/kω) V g (V) T =K σ=n(v g )eµ /ρ xy =ne/b B =T T =K holes electrons V g (V) - /ρ xy (/kω) mobilities up to 6, cm /V s at 3K ballistic transport already on submicron scale! 5, cm /V s (below 3K)

20 Quantum Oscillations in Graphene ρ xx (kω).6.4 V g = -6V B (T) 4 8 K SdH oscillations /B (/T). B F (T) B F =(h/fe)n / one type of electrons & one type of holes degeneracy f =4 two spins & two valleys 5 integer N at minima of ρ xx N n ( cm - )

21 Quantum Oscillations in Graphene ρ xx (kω).6.4 V g = -6V B (T) 4 8 K SdH oscillations /B (/T). thin graphite graphene ν = 6 ν = 5 ν = 4. / 5 integer N at minima of ρ xx N SdH oscillations shifted by π odd phase is the Berry phase

22 Quantum Oscillations in Graphene 6 ω c τ =µb =const n =4B/φ.6 σ xx (/kω) T 4K 8K K ρ xx (kω) V g (V).4. σ xx (/kω) σ (a.u.) V g (V) +V +9V T (K) 5 ShdH oscillations as a function of carrier concentration

23 Band Structure of Graphene cyclotron mass strongly depends concentration.6 E=ħkc * m c /m.4 S k y n ( cm - ) k x B F =(ħ/πe)s and m c =(ħ /π) S/ E experimental dependences B F ~ n and m c ~ n / necessitates S ~ E(k) or E ~k Einstein s mass m c = E F /c * c * 6 m/s

24 Half-Integer Quantum Hall Effect ρ xx (kω) 4 3 3K 4 8 B (T) QHE occurs at half-integer filling factors related to the odd, Berry phase 5 5 h/e ρ xy (kω) h/6e σ xy (4e /h) ρ xx (kω) 7 / 5 / 3 / / - / - 3 / - 5 / - 7 / T =4K B =4T -4-4 n ( cm - ) σ xy (4e /h)

25 Half-Integer Quantum Hall Effect quantization at ν = N + / σ xy (4e /h) 7 / 5 / 3 / / - / - 3 / - 5 / - 7 / graphene σ xy (4e /h) σ xy (4e /h) bi-layer graphene quantization at ν = N σ xy (4e /h) ρ xx (kω) 6 4 T =4K B =T -4-4 n ( cm - ) half-integer QHE is exclusive to graphene ρ xx (kω) 6 4 non-zero hole mass -4-4 n ( cm - ) T =4K B =T

26 Half-Integer Quantum Hall Effect E =hc k rotation of spin is the origin of Berry phase pseudospin E =

27 Half-Integer Quantum Hall Effect E =hc k E N =[ehc B(N + ½ ± ½)] / pseudospin E = N = hω C E = N = N = n =4B/φ N =3 N =4 the lowest Landau level is at ZERO energy and shared equally by electrons and holes

28 Half-Integer Quantum Hall Effect σ xy (4e /h) 7 / 5 / 3 / / - / - 3 / - 5 / - 7 / graphene -4-4 n ( cm - ) both plateaux ν =+½ and -½ are ODDLY at the same LL with E = single-particle but relativistic effect: integer QHE at half-integer filling factors Nature (appear 5)

29 Quantum-Limited Resistivity E = ρ (kω) 6 4 ρ max K no temperature dependence in the peak between 3 and 8K zero-gap semiconductor ρ max (h/4e ) V g (V) 5 devices µ (cm /Vs) 4, 8,

30 Quantum-Limited Resistivity ρ max (h/4e ) µ (cm /Vs) 4, 8, quantized resistivity h/e (or conductivity) NOT the resistance or conductance Mott s argument: σ = neµ = (e /h)(k F l) in the absence of localization l λ F requires σ e /πh BUT never expected to be accurate

31 CONCLUSIONS D FULLERENE EXISTS APPLICATIONS ballistic transport at (sub)micron distances under ambient conditions PHYSICS DIRIC equation rather than SCHRÖDINGER Equation model QED system