Introduction to Graphene Physics

Size: px

Start display at page:

Download "Introduction to Graphene Physics"

Francine Davis
6 years ago
Views:

1 XIX VIETNAM SCHOOL OF PHYSICS (VSOP-19) Quy Nhon, August 3-18, 013 Introduction to Graphene Physics Gilles Montambaux Day 1 courses 1-1

2 Gilles Montambaux, Laboratoire de Physique des Solides, Orsay, France Saclay Palaiseau Orsay users.lps.u-psud.fr/montambaux

3 Graphene is... D crystal massless relativistic Fermions monolayer electrostatic doping nanoribbons bilayers 3

4 Graphene : from discovery to Nobel prize.. Graphene, the world s first -dimensional fabric Posted Oct 6, 004, 3:30 PM ET Researchers at The University of Manchester and Chernogolovka, Russia have created the first-ever single-atom-thick substance, a fabric they call graphene. The substance is stable, flexible, and highly conductive, and researchers believe it could be used to create computers made from a single molecule. Professor Andre Geim at The University of Manchester was able to extract a single plane of graphite crystal, resulting in the new fabric. The hope is that the fabric will be used in the future to create nanotubes, transistors for microscopic computers, that could result in some seriously small electronic gadgetry. The Nobel Prize in Physics 010 was awarded jointly to Andre Geim and Konstantin Novoselov "for groundbreaking experiments regarding the two-dimensional material graphene" 4

5 Number of papers on ArXiv with the key word «graphene» Total : about 6000 papers in 011!!! 5

6 Science, 34, 530 (009) Graphene and other D crystals BN, NbSe, BiSr CaCu Ox (005)

7 7

8 A few references Electric field effect in atomically thin carbon films,. K. Novoselov, A. Geim et al., Science 306, 666 (004) Two-dimensional gas of massless Dirac fermions in graphene, K. Novoselov, A. Geim et al., Nature 438, 197 (005) Experimental observation on the quantum Hall effect and Berry s phase in graphene, Y. Zhang, Y. Tan, H. Stormer and P. Kim, Nature 438, 01 (005) The rise of graphene, A. Geim and K. Novoselov, Nat. Mat. 6, 183 (007) The electronic properties of graphene, A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov and A.K. Geim, Rev. Mod. Phys. 81, 109 (009) Graphene: Status and prospects, A. Geim, Science, 34, 530 (009) Electronic transport in two dimensional graphene S. Das Sarma, S. Adams, A. Hwang and E. Rossi, Rev. Mod. Phys. (011) Epitaxial graphene electronic structure and transport W. A. de Heer, C. Berger et al., J. Phys. D: Appl. Phys. 43, (010), arxiv: See also ttp:// ( a list of review papers) also wikipedia

9 Tentative outline History, fabrication, first experiments Electronic structure, wave functions, DOS, new Landau levels Dirac spectrum Linearized Hamiltonian Berry Phase Absence of backscattering Klein tunnelling Classical transport Ribbons, edge states Magnetic field effect Landau levels, wave functions, QHE Edge states Bilayers Quantum transport, weak-antilocalization Graphene physics in other systems Engineering of Dirac points 9

10 Carbon 3d 1d 0d d Graphite Graphene Fullerene 1985 sp Nanotube Diamond Multi-wall 1991 Single-wall 1993 sp 3 10

11 graphene, fullerenes, nanotubes and graphite sp 11

12 Why is graphene interesting? Graphene = D honeycomb lattice Direct space atoms per unit cell Peculiar band structure: -The valence (VB) and conduction bands (CB) meet in points (K,K ) in reciprocal space [Dirac points]. -The dispersion relation close to the Dirac points is linear : electrons are massless! Energy spectrum CB -The VB is full, the CB is empty: the Fermi level is right at the Dirac points. 1Bz K K Graphene is a valleys (K,K'), D gapless semiconductor. Electrons are massless and chiral. VB 1

Short history 1564 Discovery of graphite (plumbago) 1779

Werner) Graphite: pencils, light bulbs, neutrons moderator,.

intercalation compounds 1985 Fullerenes [Kroto, Curl, Smalley]

13 Short history 1564 Discovery of graphite (plumbago) 1779 Graphite is carbon (C. Scheele) 1789 Named from greek grajein (A. Werner) Graphite: pencils, light bulbs, neutrons moderator, HOPG (graphite monocrystal) Graphite intercalation compounds 1985 Fullerenes [Kroto, Curl, Smalley] Carbon nanotubes [Iijima] few layers graphite on metal substrate 13

] 005 Graphene quantum Hall effect [N.,G.

14 Modern history 004 Exfoliated contacted (and gated) graphene on amorphous SiO substrate [Novoselov, Geim, Manchester] 004 Epitaxial graphene on SiC [Berger, de Heer, Georgia Tech.] 005 Graphene quantum Hall effect [N.,G.,Zhang,Stormer,Kim, Columbia] 006 Graphene bilayer QHE [N.,G.,McCann, Falko] 008 CVD fabrication 011 : over 6000 published papers since QHE

History: theory 1947 Graphene band structure

[Semenoff, DiVincenzo & Mele, Fradkin, Haldane, etc.

15 History: theory 1947 Graphene band structure [Wallace] 1956 Graphene Landau levels [McClure] 1985 Hofstadter butterfly [Rammal] P. Wallace ε Connection to +1 field theory [Semenoff, DiVincenzo & Mele, Fradkin, Haldane, etc.] ~90 s Theory of carbon nanotubes [Dresselhauss, Saito, Ando, Guinea, etc. 15 Φ/Φ 0

16 Experimental techniques a) Mechanically exfoliated graphene (the scotch trick) Take a (monocrystallite) graphite pencil. Draw on a Si/SiO substrate: you will deposit graphite flakes. A small fraction are monolayers. Detect them with an optical microscope (300nm SiO thickness). Contact the monolayer with metallic leads (gold) contacts (Au) graphene (~1 to 100 μm ) n-si 300nm of SiO Geim,Novoselov et al. (Manchester) P. Kim et al. (Columbia)

17 The Scotch trick Novoselov et al. (Manchester) 004 P. Kim et al. (Columbia) 005 SiO substrate

18 The Scotch trick tools: optical microscope, AFM, SEM, Raman, QHE 18

not for this simple yet effective way to scan substrates

19 Difficult to locate..! Substrate SiO : 1 cm² graphene sheet: 1 mm² --- (100 mm)² 1 m² (100) m² Paris: 10 km² One graphene monolayer is enough to modify the optical contrast 300nm of SiO If not for this simple yet effective way to scan substrates in search of graphene crystallites, they would probably remain undiscovered today (Geim, Novoselov) 19

20 Contact the monolayer with metallic leads (gold e.g.) Field effect : A gate voltage V g controls the density of electrons in the graphene sheet ( electrostatic doping ) Capacitor = graphene / insulator SiO /conductor n-si contacts (Au) Nc CgVg / e e n C 4e c 0 g Nc A e A / d a V g a 7.10 cm. V 10 1 n-si d 300nm SiO Novoselov et al., Science 004

21 (1/k) Electric Field Effect in Graphene conductivity Hall effect 3 1 T =10K V g (V) σ=n c (V g )eμ 1/ρ xy =ne/b B =T T =10K holes electrons V g (V) /r xy (1/k) Novoselov, Geim et al. Science (005) simple behaviour: practically constant μ, σ(n c 0) 0 1

Si-face Si C Si Si SiC (0001) C-face Graphene on SiC Berger et al., J. Phys. Chem.

22 b) Epitaxial graphene on SiC Epitaxial growth of graphene layers on top of a SiC monocristal A.Charrier et al., J. Applied Physics 9, 479 (00) Thermal decomposition of SiC at high temperature (~1400 o C) - high vacuum (0001) Si-face Si C Si Si SiC (0001) C-face Graphene on SiC Berger et al., J. Phys. Chem. 004 By controling temperature, growth of 1 to ~100 graphene layers C. Berger, W. De Heer et al. (Atlanta) Graphene layers behave as if they were isolated.

23 3

24 4

25 5

26 6

27 7

28 c) CVD : chemical vapor decomposition ~1000 K Ni Bae et al., Nature Nanotechnology 5, 574 (010) Massive production Yu et al., APL 93, (008) 8

29 9

30 d) Suspended graphene Bolotin et al., Columbia, SSC (008) X.Du et al., Rutgers (009) m cm /. V s terminals FQHE Bolotin et al., Columbia, Nature (009) Du et al., Rutgers, Nature (009) Shivaraman, et al. Nanoletters (009)

31 e) Graphene on Boron Nitride 31

Graphene electronic structure Graphene = D honeycomb carbon crystal Carbon: 6 electrons 1s, s p hybridation: 1 orbital s and orbitals p 3 orbitals sp -3

32 Graphene electronic structure Graphene = D honeycomb carbon crystal Carbon: 6 electrons 1s, s p hybridation: 1 orbital s and orbitals p 3 orbitals sp -3 coplanar σ bonds, angle 10 : honeycomb structure covalent bonding 1 orbital p z perpendicular to the plane 1 conduction e per C Half-filled band cf: benzène

33 «Honeycomb lattice» is not a Bravais lattice a=1.4 A a1 a t t t=.8ev a a a t =-0.eV a = C-C distance a 0 = lattice parameter Triangular Bravais lattice + atoms per unit cell

34 «Honeycomb lattice» is not a Bravais lattice a=1.4 A a1 a t t t=.8ev a a a t =-0.eV a = C-C distance a 0 = lattice parameter Triangular Bravais lattice + atoms per unit cell

35 Real Space Reciprocal space a 1 * a K * a 1 a a 1 3 K G M K Triangular Bravais lattice + identical atoms per unit cell a 0.14nm 1 * * K ( a1 a ) G 3 3a * 4 a i GK KK ' 4 3 3a

36 Nearest neighbor tight-binding model for the conduction electrons, Wallace 1947 Nearest neighbor hopping t 1 conduction electron per atom (next nearest hopping t ) j H t j j h. c j, j' A j' B t j j ' j j ' ' ja ja jb jb h. j, j ' c g1 t t + Bloch theorem... 36

37 Electronic spectrum Hopping between nearest neighbors ik R + Bloch theorem ( r R) e ( r ) 1 ik Rj j j e ca( k ) ja cb( k ) j B N cells a1 a H ik a e ika ec k t e e c k 1 B( ) (1 ) A( ) ika ika ec k t e e c k 1 A( ) (1 ) B( ) B A t t H k ca( k ) ca( k ) e cb( k) cb( k) with H k f A B * 0 f( k) ( k) 0 ika ika f k t e e 1 ( ) (1 )

38 x Hamiltonian H k f * 0 f( k) ( k) 0 ika ika f k t e e 1 ( ) (1 ) e ( k ) f ( k ) e( k ) 0? * * a a 1 Ka 1 3 K K Ka 3 1 * * K ( a1 a ) G 3 38

39 Next nearest neighbors a1 a gk ( ) f ( k ) ca( k ) ca( k ) E * f ( k ) g( k ) cb( k ) cb( k ) E( k ) f ( k ) gk ( ) ika ika f k t e e 1 ( ) (1 ) t t E( k ) f ( k ) t '[cos k. a cos k. a cos k.( a a )] 1 1 t t.8 ev ' 0. ev

40 Wave functions H k f 0 f( k) ( k) 0 * H k i k 0 e e ( k) i k e 0 k ( r ) e i k ae ik. r 1 k arg[ f( k)] a band index 40

41 Topological properties k ( r ) e ik. r 1 a i e k u e k ik. r k 1 B i u u. d k =. d k k k k k k C «Berry phase» C Berry phase effects on electronic properties, D. Xiao, M.C. Chang, Q. Niu, arxiv: Consequences on Landau levels spectrum Roth Wilkinson ( g ) / en n B a d g B 1

42 Topological properties k ( r ) e i k ae ik. r 1 No backscattering Diffusion from k to k : Fermi golden rule k k' i 1 k ' ( ) kk ' 1 i kk ' cos k P V e V e P( ) 0 4

43 Expansion near the Dirac points ika ika f k t e e 1 ( ) (1 ) H f ( K q) c( q iq ) K K H K K 0 qx iqy c qx iqy 0 p q A mc c( px ipy) c( px ipy) mc x y B K K Dirac eq. for a relativistic particle in D c v F 3 ta 10 m.s 6-1 H c p mc K z e( p) p c m c 4 e( q) vf q 43

44 How to write the effective linearized Hamiltonian 0 qx iqy H KK, ' c c( qxx qyy ) qx iqy 0 A B A B 0 qx iqy 0 0 qx iqy H c qx iqy 0 0 qx iqy 0 or A B B A 0 qx iqy 0 0 qx iqy H c c q. z qx iq y 0 0 qx iqy 0 4 x 4 effective Hamiltonian Two uncoupled valleys HK c p * H c p K HK c p H c p K 44

45 How to write the effective linearized Hamiltonian H KK, ' A B 0 qx iqy c qx iqy 0 q arctan q q y x H 0 cq e i q e i 0 q 1 e ( q) ac q k 1 1 ik. r iq. r ik '. r iq. r ( r ) e e i q iq ae ae The wave function is a linear combination of the contributions of the two valleys

46 Density of states 1953 e a ( px py) e cp re ( ) gg s v e v F Thermodynamics, transport g s = g v = p e m spin degeneracy valley degeneracy k e F n v n F F c c Remember : for e k a in d dimensions r( e) e d / a 1 46

47 Landau levels graphene gg s v re ( ) e c Onsager semi-classical quantization rule eb N( el) ( n g) g h re ( ) gs m DEG N( e) e d / a el c ( n g) e B g 0 e L c n B el n B a / [( g) ] d e L eb ( n g) m 1 e L ( n ) m g B 1

48 Beyond linearization «Trigonal warping» e 3t 8 ( q) vf q q a sin 3 q q arctan q q y x K K 48

49 ARPES experiments : Angular Resolved Photemission Spectroscopy measure the dispersion relation hn e e k k k f f i i e ( k ) i i K K Bostwick et al. Nat. Phys. 007

50 Cyclotron mass m * A e e e( k ) Novoselov, Geim et al., Nature 005 graphene m * e v F F e v n F F c m * v F n c 50

51 Boron Nitride : Dirac equation with a mass Blue and red atoms are now different : B and N site energies different H k f * ( k) f( k) Ek f ( k ) t.8 ev 5.9eV C-C B-N mdc qx iqy H c c q. mdc z qx iqy mdc

52 Graphene on Boron Nitride substrate P. Kim group, Nature Nanotechnology (010) FQHE Random crystallographic orientation to the substrate t.8 ev 50meV Giovannetti, et al. PRB (007) BN is a perfectly clean substrate 5

53 GaAs mobility, Pfeiffer et al. Suspended graphene (008) Graphene on h-bn (010) Graphene on SiO (004) 53

54 Transport 54

55 (1/k) Conductivity e v n F F c 3 n c a V g e ( ) F e kl F e h 1 T =10K V g (V) n c av g l ( e )??? e F ( T)??? 55

56 Physics of the neutrality point Thermally activated carriers kt B ne( T) nh( T) 7 ta E F =0 Specific heat 8 (3) kbt C( T) kb ta Conductivity e le e ( F 0, T) 4ln kbt h v F f ( T) ( e ) de e e e ( F, T 0) h k l F e 56

57 e le e ( F 0, T) 4ln kbt h v T classical plasma 400K F e v n F F c n c =av g T F (n c ) e e ( F, T 0) k l h degenerate gas F e n cm c T T F degenerate gas T T F classical plasma T T F degenerate gas 57

58 (1/k) Conductivity 3 σ=n(v g )eμ e n c v av n F F c g e ( ) F e kl F e h 1 T =10K V g (V) l ( e )??? e F Very small temperature dependence le Cte ( e ) e V F F g l e l e 1 k F e ( ) Cte kf ( e F) e F Vg F neutral impurities charged screened impurities Nomura, MacDonald, PRL 007 should depend on dielectric constant see also M.Monteverde et al. PRL (010) resonant impurities

59 Suspended graphene Bolotin et al., Columbia PRL 101, (008) e k F l e k F ( ) h 4 e k F L h diffusive ballistic 59

3 1 T =10K 0-100 -50 0 50 100 V g (V) Tan et al.

60 (1/k) conductivity minimum Should vanish at the neutrality point? 3 1 T =10K V g (V) Tan et al. PRL 99, (007) n c Explication : «flaques» de trous et d électrons 0 c 60 n 0 Yacoby et al., Nature Phys. 008

61 GaAs mobility, Pfeiffer et al. Suspended graphene (008) Graphene on h-bn (010) Graphene on SiO (004) 61

62 Strong dependence on the nature of substrate, impurities «Graphenes» Fabrication Substrate exfoliated SiC CVD SiO SiC Suspended BN 6

63 Disorder A B H K 0 px ipy ( r) 1 0 c V ( r) m( r) c * px ipy ( r) Scalar Static distorsions («ripples») Local A-B dissymetry Intervalley (K, K ) coupling and range of the disorder potential long range disorder does not couple valleys short range disorder couples valleys 63

64 Klein tunneling 64

65 Klein tunneling effect Oskar Klein ( ) 65

66 1 ) Potential step ) Potential barrier 66

67 E p x m E c p x V 0 Evanescent wave Transmission E p c m c 4 x Transmission V0 mc E Evanescent wave Transmission V mc E V mc 0 0 mc E V mc 0 67

68 E p x m E c p x 4 E pxc m c E c p p Evanescent mode x y Transmission V E mc 0 0 y V E cp 68

69 E p c m c 4 x E V mc 0 E Transmission 69

70 E p c m c 4 x E V mc 0 E Transmission 70

71 E p c m c 4 x E V mc 0 E Transmission 71

72 E p c m c 4 x V mc E V mc 0 0 E Evanescent wave 7

73 E p c m c 4 x V mc E V mc 0 0 E Evanescent wave 73

74 E p c m c 4 x V mc E V mc 0 0 E Evanescent wave 74

75 E p c m c 4 x V mc E V mc 0 0 E Evanescent wave 75

76 E p c m c 4 x V mc E V mc 0 0 E Evanescent wave 76

77 E p c m c 4 x E V E mc 0 77

78 E p c m c 4 x E V E mc 0 Transmission 78

79 E p x m E c p x 4 E pxc m c E c p p x y Evanescent mode Transmission E V mc 0 E V0 cpy 79

80 E c p p x y Transmission if : cp V E y 0 sin V 0 E E Perfect transmission if 0 Transmission( 0 ) depends on E V 0 80

81 Velocity E c p x E a vf k v 1 k E a v k F k Wave function k ( r ) e i k ae ik. r 1 81

82 Potential step coscost T ( ) t sin E V E sin ( 0 )sin t + Continuity equations

83 Potential step T() coscost T ( ) t sin Esin ( V0 E)sin t E/V 0 =0 E/V 0 =0.5 Evanesce nt waves E/V 0 =0.5 E/V 0= 0.75 E/V 0 T=0 E/V 0 =1 V sin transmission 0 E E When E approches V 0, more and more evanescent modes

84 Potential step T() E V 0 0 E V coscost T ( ) t sin Esin ( V0 E)sin t coscost T ( ) t cos Esin ( E V0 )sin t 84

85 Potential barrier + Continuity equations 85

86 J.N. Fuchs et P. Allain, Klein Tunneling in graphene : optics with massless electrons EPJB, arxiv:

87 Potential barrier T() E/V 0 =0.4, l=l/(hv F /V 0 )=4.85 E/V 0 =0.9, l=6.91 Katsnelson, Novoselov and Geim, Nat. Phys., 60 (006) E sin ( V E)sin q ( V E)cos x 0 0 t t T( ) cos cos cos cos cos sin (1 sin sin ) t qxl t qxl t 87

88 A very interesting case : E V 0 E V c q k 0 x y 0 q x i k y E c k k x y T( ) cosh cos klsin y 88

89 Propagation through ballistic graphene at the neutrality point E ' E V q k 0 0 x y E V 0 normal contacts 1 T( 0) k y cosh kl y n W Tworzydlo et al., PRL 96, 4680 (006) 89

90 T n G 1 L cosh n W e 4 Tn h n Called «pseudo-diffusive regime, because the distribution of the transmission coefficients is the same as in a diffusive disordered system W L G 4 e W h L min min 4 e h W / L Miao et al., Science, 317, 1530 (007) 90

Electronic properties of graphene. Jean-Noël Fuchs Laboratoire de Physique des Solides Université Paris-Sud (Orsay)

Electronic properties of graphene Jean-Noël Fuchs Laboratoire de Physique des Solides Université Paris-Sud (Orsay) Cargèse, September 2012 3 one-hour lectures in 2 x 1,5h on electronic properties of graphene