Electron interactions in graphene in a strong magnetic field

Transcription

1 Electron interactions in graphene in a strong magnetic field Benoit Douçot Mark O. Goerbig Roderich Moessner K = K K CNRS and ENS Paris VI+XI cond-mat/

2 Overview Recent experiments: integer QHE in graphene Graphene background band structure length/energy scales Harper equation and continuum theory Interactions and SU(4) (spin chirality) symmetry what s special about n = 0 pseudopotentials: low and high Landau levels easy-plane anisotropy due to backscattering effective stiffness Outlook

3 What is graphene? Graphene = D graphite Graphite = stack of weakly coupled graphene sheets Honeycomb lattice = triangular lattice with two-atom basis τ τ 1 e 1 : A sublattice : B sublattice e 3 e Low-tech sample preparation: Scotchtape Technical difficulty: good contacts

4 ν IQHE in graphene Novoselov et al., Nature 438, 197 (005) Zhang et al., Nature 438, 01 (005) Density of states V =15V T=30mK 1/ν B=9T T=1.6K Plateaux corresp. to ν = 4n +

5 Graphene band structure papers from the 1950ies Band-structure calculation in the tight-binding model H 0 = t i A 3 ( ) b R i +e j a Ri + H.c. τ τ 1 e 1 e 3 e j=1 : A sublattice : B sublattice Energy K K K K K K kx holes ky electrons Energy dispersion: 3 ε k = ±t cos(k e j ) j=1 3 + sin(k e j ) j=1 k y kx

6 Continuum theory with no magnetic field Zero-energy states: ε ± K = 0 j=1cos(k e j ) = 3 j=1 sin(k e j ) = 0 Energy K K K K K K kx holes ky electrons at K and K points of the 1st BZ k y kx Continuum limit k = K ± + κ with κ 1/a: H ± (κ) = 3 ( ) ta 0 κ 1 iκ ( = v κ 1 ± iκ 0 F κ1 σ 1 ± κ σ ) Energy dispersion (two-fold degenerate, chirality α = ±): ε α=± κ = ± v F κ

7 Zero-energy states at Dirac points Wavefunction ψ = i f i i Zero-energy states: H ψ = 0 j:i f j = 0 Can choose wavefunction amplitudes f i = f i exp(iφ i ) to be non-zero on B sublattice only For f j constant, require j:i exp(iφ j) = 0 GS of triangular XY model These are distinguished by irrel. global phase chirality = ± K, K Location of Dirac points K K

8 Naïve continuum theory with magnetic field (I) Usual route: Peierls substitution + minimal coupling: k p 1 (p + ea) Π Non-commuting momenta (magnetic length: l B = /eb): [x µ,p ν ] = i δ µ,ν [Π x, Π y ] = i /l B Ladder operators [a,a ] = 1 as usual: a = l B (Π y + iπ x ), a = l B (Π y iπ x )

9 Naïve continuum theory with magnetic field (II) matrix Hamiltonians for each chirality (at K and K ): H K = v ( ) F 0 a, H l B a 0 K = v ( ) F 0 a l B a 0 Energy dispersion (degenerate in chirality quantum number α): ǫ n = ± v F l B n B n [Relativistic Landau levels (LLs)] Energy 0 Relativistic Landau Levels n=4 n=3 n= n=0 n=1 Magnetic Field B n= 1 n= n= 3 n= 4

10 Transition energy (mev) Relative transmission Relative transmission Infrared transmission spectroscopy relative transmission Transmission energy [mev] (A) 0.4 T 1.9 K (B) E Energy (mev) L 3 L ( D) L L3 ( D) L L L L 3 L 1 E 1 c ~ e B sqrt(b) L 0 L L L 1 L C L1 C 1 3 ( ) ( ) (C) B Energy [mev] Sqrt[B] E 1 E 1 L L A B D (D) C 0 L1 B 1 L0 B ( ) ( ) L1 L ( A) relative transmission T 0.T transition C 0.3T 0.5T 0.7T T 0.90 transition B 0.88 T 4T Energy [mev] Energy (mev) Sadowski et al., cond mat/

11 Degeneracy of relativistic Landau level Guiding center R = (X,Y ) algebra as in non-relativistic case: [H,R] = 0, [X,Y ] = il B, Degeneracy generated by nd set of ladder operators b,b Degeneracy: N φ = A/πl B as per usual; E(n) nb Filling factor: ν = N el /N φ measured from particle-hole symmetric point Chirality α = ± and spin σ =, internal SU() SU() space 4 copies of each Landau level possibility of SU(4) symmetric Hamiltonian

12 Wavefunctions States n, m; α are -spinors (entries: sublattice): ( ) n, m n,m; + = sgn(n) n 1,m n,m; = ( sgn(n) n 1,m n, m Wavefunctions n, m are those of non-relativistic case! both n,m, n 1,m occur in the spinor Special case n = 0: electrons at K (K ) live on A (B) sublattice only chirality=sublattice index )

13 Continuum limit via Harper equation Define wavefunctions g {A/B} for sublattices A/B Near K (l B /a 1, a 1): Eg A (x) { = cos π/3 + ( } 3/)[q y + (x + 1/4)B] g B (x + 1/) g B (x 1) Eg B (x) { = cos π/3 + ( } 3/)[q y + (x 1/4)B] g A (x 1/) g A (x + 1) In Landau gauge A = Bxe y ; q y is good quantum number Problem: Bx is unbounded, no matter how small B For given q y 1 st B.Z., define auxiliary q.n. m such that 3/(qy + Bx m ) = πm and write x = x m + δx Strategy solve for given {m,q y } assuming δx small check consistency of solution with assumption

14 From Harper to Dirac Expansion near K,K gives Dirac equation: Eg α (x) = (3/)(d/dx ± Bδx)g β (x) Eg β (x) = (3/)( d/dx ± Bδx)g α (x) Tunnelling between solutions m m suppressed? Require: R L = nl B n/b 1/B x n 1/ 1/B m m+1 1/B Equivalent to E t and ρ 1/a cm

15 Length and energy scales in graphene Length scales Distance between neighbouring carbon atoms: a = 0.14 nm Magnetic length: l B = 6nm/ B[T] Larmor radius: R L = nl B Energy scales Band width: t =.7 ev Landau level spacing : v F /l B = 3ta/l B 0 B[T] mev Landau level dispersion: exp( a/r L ) Zeeman splitting: z = gµ B B 0.1B[T] mev Interaction energy: e /ǫl B B[T] mev Lattice effects (anisotropies, etc.): a/l B B[T]

16 Interaction model densities Electrons in a single relativistic LL at ν n (no spin): H = 1 q V (q)ρ n ( q)ρ n (q), V (q) = πe ǫq ρ n (q) = ρ n A(q) + ρ n B(q) ρ n τ (r) = α,α ψ n;α;τ (r)ψ n;α ;τ(r) with τ = A/B, α = ± (chirality) Projected densities ρ n (q) = α,α F n αα (q) ρ αα (q): relativistic Landau levels n = 0 l ρ αα (q) = m,m m e i[q+(α α )K] R m c n,m,αc n,m,α

17 Interaction model (II) Graphene form factors (l B 1): F ++ ( q ) ( q )] [L n + L n 1 n (q) = 1 ( ) F n + i(q + q K K ) (q) = n n (q) = [ F n + ( q)] F + L 1 n 1 e q /4 = Fn (q) F n(q) ( q K ) e q K /4 Model: H = 1 with interaction vertex: α 1,...,α 4 q vα 1,...,α 4 n (q) ρ α 1α 3 ( q) ρ α α 4 (q) v α 1,...,α 4 n (q) = πe ǫ q F α 1α 3 n ( q)f α α 4 n (q), No SU() chirality symmetry so far!!

18 Interaction vertices Terms of the form Fn α,α ( q)f α, α n (±q) : exp. suppressed exp( K /8) Umklapp terms Fn α, α ( q)fn α, α (q) : exp. suppressed exp( K /) Backscattering terms ( q)fn α,α (q) : F α, α n ν 3, α1 ν 4, α ν 1, α 1 ν, α ν 3, α ν 4, α ν 1, α ν, α ν 3, α ν 4, α alg. small 1/ K a/l B 1 ν, α ν, α H n SU() = 1 α,α q πe ǫ q [F n(q)] ρ α,α ( q) ρ α,α (q) + O(a/l B )

19 Leading [SU() invariant] interaction Hamiltonian H n SU() = 1 q vg n (q) ρ( q) ρ(q); with total projected density ρ(q) = ρ ++ (q) + ρ (q) and v G n (q) = πe ǫq F n(q) F 0(q) = exp( q /) F n(q) = e q / ( ) ( )] q q [L n + L n 1 Magnetic translation algebra for projected densities: [ ρ(q), ρ(q )] = i sin (q q /) ρ(q + q )

20 Effective SU() interaction potentials FQHE effective interaction potential n=5 (non relativistic) n=0 (relativistic and non relativistic) n=1 (relativistic) n=1 (non relativistic) n=5 (relativistic) r/l B 0.8 n m Pseudopotentials V Broadly similar to non-relativistic case n=0 n=1 (rel.) n=1 (non rel.) Relative angular momentum m Largest difference between rel. and non-rel. case in n = 1 Similar behaviour of rel. interaction in n = 0 and n = 1: absence of non-rel. n = 1 physics: Pfaffian at ν = 5/? unpolarised (chirality vs. spin) states

21 Non-relativistic limit n 1 For large n, can approximate envelope of Fn(q) as Fn (q) 1 [ J 0 (q n 1) + J 0 (q n + 1) 4 ] J 0(q n) ( ) q [L n ] e q /4 Same as non-relativstic interactions Charge-density wave states in high Landau levels (stripes, bubbles)

22 What s special about the central Landau level Effective interaction looks the same as in non-relativistic LLL Only one sublattice occupied for each chirality no backscattering term unlike other LL, can easily arrange for CDW with λ = a Hartree energy details of energetics depend on structure of wavefunction on lattice scale expect anisotropy of size a/l B breaking down SU() chirality symmetry

23 SU() symmetry-breaking terms for n 0 Backscattering terms in n 0: H bs = 1 α q vn α, α (q) ρ α, α ( q) ρ α,α (q) with interaction v + n (q) = v + ( q) v + n (q) = πe Re(q K) ǫ q n [ L 1 n 1 ( ) q K ] e q K /4 peaked near q = ±K: v + n (q) e /ǫ K l B (e /ǫl B )(a/l B )

24 Chirality quantum ferromagnetism Cf. exchange driven spin ferromagnetism at ν = 1 in GaAs Review: Moon et al., PRB 51, 5138 (1995) Ψ = m ( sin θ m e iφm/ c m,+ + cos θ ) m eiφm/ c m, 0 n m = sin θ m cos φ m sin θ m sin φ m cos θ m Non-linear σ model (in coherent states: m r): H = ρ s d r[ n(r)] ρ s = 1 16 π (e /ǫl B ) Easy-plane anisotropy due to backscattering terms in n 0: H Z = z d r πl B [n z (r)] z = π 3 (e /ǫl B )(a/l B ) Kosterlitz-Thouless physics

25 Graphene as a multi-component system Graphene is one of several multi-component systems Spin; valley (Si); double layer;... Symmetry-breaking terms are quite weak (lattice effects a/l B ) highly isotropic hard to probe? need to investigate fate of unpolarised states Zeeman splitting much bigger than in GaAs not particularly close to SU(4)

26 Exprimental evidence for chirality polarisation 5T T=1.4K spin 4T 45T 37T chirality ferromagnetism (??) Plateau transitions at ν = 0, ±1 Zhang et al. PRL 06 Landau levels individually resolved 30T 5T 9T (30mK)

27 Related work Ferromagnetism in graphene: MacDonald et al. Effective description: Alicea+Fisher FQHE physics: Apalkov+Chakraborty Disorder and interaction effects (Guinea et al.) Edge states (Peres, Castro Neto, Guinea; Brey, Fertig;...) Minimal conductance e /h (Beenaker s group, Katsnelson,...) Disorder and weak (anti-)localisation (Altshuler et al, Guinea et al., Khveshchenko,...) Dielectric breakdown (Baskaran et al.)... We ll have to rewrite the theory of metals for this problem. (Physics Today, January 006, p. 1)

28 Summary Graphene in a strong magnetic field continuum limit via Harper equation interaction Hamiltonian chirality SU() and its breaking by lattice effects graphene form factor pseudopotentials the central Landau level large-n limit chirality polarisation exchange ferromagnetism different routes to anisotropies

29 Summary Graphene in a strong magnetic field continuum limit via Harper equation interaction Hamiltonian chirality SU() and its breaking by lattice effects graphene form factor pseudopotentials the central Landau level large-n limit chirality polarisation exchange ferromagnetism different routes to anisotropies Thank you for your attention!