Physics of graphene. Hideo Aoki Univ Tokyo, Japan. Yasuhiro Hatsugai Univ Tokyo / Tsukuba, Japan Takahiro Fukui Ibaraki Univ, Japan

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1 Physics of graphene Hideo Aoki Univ Tokyo, Japan Yasuhiro Hatsugai Univ Tokyo / Tsukuba, Japan Takahiro Fukui Ibaraki Univ, Japan

2 Purpose Graphene a atomically clean monolayer system with unusual ( massless Dirac ) dispersion Band structure, group theory a anomalous integer QHE (one-body problem) topological quantum # bulk vs edge states Many-body states a

3 Graphene - monolayer graphite * Band structure: Wallece 1947 * Group theory: Lomer 1955 A B K K K K K π band K Honeycomb = Non-Bravais Mechanical exfoliation (Geim) SiC decomposition (de Heer) Benzene (Saiki) 10 μm (courtesy of Geim) (Novoselov et al, Science 2006, suppl info)

4 Graphene s band dispersion extended zone two massless Dirac points K K K K K K T-reversal: K K H K = v F (σ x p x + σ y p y ) ARPES, Bostwick et al, nature phys 2007 Effective-mass formalism H K = v F (- σ x p x + σ y p y ) = v F = v F - +

5 Massless Dirac eqn for graphene (Lomer, Proc Roy Soc 1955) K K K K + B [McClure, PR 104, 666 (1956)]

6 How does the massless Dirac appear on on honeycomb (1) extended zone A R 3 B R 2 R 1 H = 0 H AB H AB 0

7 How does the massless Dirac appear on on honeycomb (2) K K K K K K K K * Group theory (Lomer, Proc Roy Soc 1955) 2-dim representation at K and K * k p Hamiltonian [Wallece, PR 71, 622 (1947)] H= v F (±σ x p x + σ y p y ) +: K, -: K

8 Ordinary QHE systems Graphene Landau levels Graphene Landau levels (McClure 1956) E n = 2 E ε n = (n+1/2)hω c n = 1 ω c = eb/mc k n = 0 n = 0 level at E = 0 k n = -1 n = -2

9 Massless Dirac a variety of of anomalous phenomena observed / predicted * QHE (Novoselov et al, Nature 2005; Zhang et al, Nature 2005) * Spin Hall effect (spin-orbit; Kane & Mele PRL 2005) * Ferromagnetism (B =0; Peres et al, PRB 2005) * FQHE (Apalkov & Chakraborty, PRL 2006) * Superconductivity (Uchoa & Castro Neto, PRL 2007) * Negative refractive index (Cheianov et al, Science 07) * Klein paradox (Katsnelson et al, nature phys 2006) * Gapped state (Nomura & MacDonald PRL 2006) * Bond-ordered state (Hatsugai et al, 2007) * Landau-level laser (Morimoto et al, 2007) > 400 preprints on graphene in cond-mat in (A review, incl. a brief history of graphene: Andre Geim & Kostya Novoselov, nature materials 2007)

10 B = 0

11 Edge states in in graphene

12 Band F in nonmagnetic materials? Criterion (Stoner s) for band F: UD(E F ) > too crude a criterion Flat-band ferromagnetism (Lieb 1989; Mielke 1991; Tasaki 1992) (a) Flat one-electron band (b) Connectivity condition (``Wannier orbits overlap) i.e., band ferromagnetism

13 Hubbard model on on flat-band systems 1D version (Lieb, 1989) E F (Ovchinnikov rule guaranteed for 0 < U < ) (Mielke, 1991) Ferromagnetism E F (Tasaki, 1992) E F generalised Hund s coupling (Kusakabe & Aoki, 1992)

14 Edge states in graphene Nakata et al, PRB 54, (1996) armchair zigzag cf. Potential confinement impossible (Klein paradox)

15 Long-period graphene (Shima & Aoki, PRL 1993)

16 B > 0

17 Graphene Landau levels Graphene Landau levels (McClure 1956) n = 2 E n = 1 n = 0 n = 0 level at E = 0 k n = -1 n = -2

18 Landau levels for massive Dirac particles Dirac eqn (MacDonald, PRB 1983) Energy σ z σ x non-relativistic massless Dirac leading term in hω c /mc 2 expansion? ω c = eh/mc

19 QHE can reflect band structures q2d organic metal (TMTSF) 2 PF 6 (Chaikin et al) T SDW 2D FISDW p SC metal B Lab. de Physique des Solides Incomplete nesting Landau levels in Fermi pockets IQHE

20 What s so special about graphene Landau level Quantisation in B Onsager s semicl Landau tube? n = 0

21 QHE in honeycomb lattice (SCBA; Zheng & Ando, PRB 2002; Gusynin & Sharapov, PRB, PRL 2005) B Hall conductivity (-e 2 /h) σ xy = (2n+1)(-e 2 /h) steps

22 QHE in graphene (Novoselov et al, Nature 2005; Nature Phys 2006; Zhang et al, Nature 2005) σ xy = (2n+1)(-e 2 /h) 2 (n+1/2)(-e 2 /h) (half-integer) for massless Dirac Valley degeneracy ρ xx 7 5 σ xy 3 σ xy /(e 2 /h) =

23 Integer quantum Hall effect (Aoki & Ando 1980) (Aoki, Rep Prog Phys 1987) (Paalanen et al, 1982)

24 QHE in ordinary valence / conduction bands σ xy /(-e 2 /h) vb k cb E

25 QHE for massless Dirac Can we interpret this in terms of topoogical quantum #? σ xy /(-e 2 /h) : K : K k E

26 Hall conductivity = a topological number Linear response (Thouless, Kohmoto, Nightingale & den Nijs, 1982) clean, periodic systems Berry s curvature disordered systems (Aoki & Ando, 1986) Gauss-Bonnet distribution of topological # in disordered systems (Aoki & Ando, 1986; Huo & Bhatt, 1992; Yang & Bhatt, 1999) = n band (Chern #)n (Avron et al, 2003)

27 Questions about graphene IQHE (A) How does the topological nature appear? (B) How do the edge states look like? (C) Is specific to honeycomb? For these, effective-mass original lattice

28 Whole spectrum for honeycomb lattice E E =0 B Rammal 1985 E=0 Landau B level outside Onsager s semiclassical quantisation scheme

29 QHE # over the whole spectrum (Hatsugai, Fukui & Aoki, PRB 2006) D(E) Hall conductivity (-e 2 /h) E anomalous (2n+1) QHE: lattice gauge (Fukui & Hatsugai) E/t t ~1 ev for graphene

30 Massless Dirac for a sequence of t t t =-1: π-flux lattice t =0: honeycomb t =+1: square

31 E k y k x Not specific to graphene but shared by a class of non-bravais lattices? cf. recent organic system * Dirac cones seem to always appear in pairs --- Nielsen-Ninomiya

32 Adiabatic continuity for QHE Persistent gap topological σ xy protected E=0 φ B

33 σ bulk xy xy =σ edge xy in xy honeycomb honeycomb Ordinary QHE: σ xy bulk =σ xy edge Hatsugai, 1993 with Laughlin s argument E=0

34 k space Edge Landau states real space Edge Landau states much more robust than B=0 edge states

35 Boundary states with STM * STM for graphite: Edge(Niimi et al, PRB 2006) Point defect in B (Niimi et al, PRL 2006)

36 2D crystals should not exist J Meyer, UC Berkeley

37 Related Related structures: structures: GIC, GIC, MgB MgB22

38 MgB 2 bandstructure (Kortus et al, PRL 2001) cf. p wave SC becomes p + ip at K point in honeycomb (Uchoa & Castro Neto, PRL 2007) d 1 +id 2 (Onari et al, PRL 2002)

39 Why the bands stick together at Bz edges 1D (Se, Te) 2D (graphite) 3D (diamond) helical symmetry band sticking (see, eg, Heine 1960) (gyroid system: [Koshino & Aoki, PRB 2005])

40 Band for diamond (s orbits)

41 Optical spectroscopy for graphene Landau levels (Sadowski et al, PRL 2006) Graphene Landau levels n = 2 n = 1 E 0 1 * Uneven Landau level spacings * Peculiar selection rule: n n +1 (usually, n n+1) n = 0 n = -1 n = -2 k

42 Observed splitting of Landau levels (Zhang et al, PRL 2006) T = 1.4K B = T T = 30mK

43 Correlated electron systems FQHE systems

44 Many-body effects in in graphene Landau levels n = 2 E n = 1 n = 0 + electron-electron interaction, electron-lattice interaction? k n = -1 n = -2

45 Various phases in the quantum Hall system DMRG result: Shibata & Yoshioka 2003 Bubble N = 2 N =2 2k F N =1 Stripe 1 N = ν Ν Wigner crystal Laughlin state Compressible liquid 0 (Onoda et al, 2003

46 As strongly-correlated systems: HTC FQHE graphene Coulomb anisotropic gauge field isotropic Reminds us of Kohn-Luttinger 1965 every metal normal states become instable T 0

47 Observed splitting of Landau levels (Zhang et al, PRL 2006) B = 9 T = 1.4K T = 30mK T spin(2) x valley(2)=4 ν= 6 ν= 4 ν= 2 ν= 1 n=0 Landau level splits ν= 0 B ν= - 1 ν= - 2 ν= - 4

48 Proposed mechanisms for split Landau levels * Excitonic gap (Gusynin et al, PRB 2006) * SU(4)-breaking (Nomura & MacDonald, PRL 2007) * FQHE (Apalkov & Chakraborty, PRL 2006) * Peierls distortion (Fuchs & Lederer, PRL 2007) Bond order (Hatsugai et al, 2007)

49 Bond ordering (Hatsugai, Fukui & Aoki, 2007) bond-order parameter (Affleck-Marston 1988) stabilised due to D(E)= Graphene Landau levels E 2 1 n = 0-1 k -2 Kekulean pattern but the chiral symm preserved 3-fold degenerate

50 Gap opening with the bond ordering Landau levels with bond order Energy without bond order split IDOS 18 x 18, =1/324

51 Domain structures in bond ordering Charge Density of the in-gap states + many others quantum liquid?

52 Summary Graphene: one-body massless Dirac + B peculiar QHE QHE: topological bulk-edge correspondence Graphene: many-body Landau level + interaction various instabilities expected Future problems re-doing the condensed-matt phys picking up anomalies on the way