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, Paris VI+XI, Oxford PRB 74, (006)

2 Overview Multi-component quantum Hall systems 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 resolution of QH plateaux: SU(4) Skyrmions

3 Multi-Component Systems (Internal Degrees of Freedom) A physical spin: SU() (doubling of LLs) B bilayer: SU() isospin Landau levels +> > ν = 1/ + d exciton C graphene (D graphite) ν = 1/ ν = ν + ν = 1 T + τ τ 1 e 1 e 3 e two fold valley degeneracy SU() isospin : A sublattice : B sublattice spin + isospin : SU(4)

4 Ferromagnetism at ν = 1 Coulomb repulsion favours anti-symmetric orbital wavefunction... ν = 1 m 1 m m+1... exchange gap spin wavefunction: symmetric (ferromagnet) no interactions with repulsive interactions in QH systems : no kinetic energy cost! ( LL ~ flat band)

5 Ferromagnetism at ν = 1 Coulomb repulsion favours anti-symmetric orbital wavefunction... ν = 1 m 1 m m+1... exchange gap spin wavefunction: symmetric (ferromagnet) 0.8 energy spin wave dispersion activation gap z x E X y no interactions with repulsive interactions in QH systems : no kinetic energy cost! ( LL ~ flat band) topological excitations (skyrmions) z x y 0. Zeeman gap 1 wave vector 3 4 5

6 What s so special about graphene?

7 What s so special about graphene? Electrons travel through it so fast that their behaviour is governed by the theory of relativity rather than classical physics. Economist 006

8 What s so special about graphene? Electrons travel through it so fast that their behaviour is governed by the theory of relativity rather than classical physics. Economist 006 Inside every pencil, there is a neutron star waiting to get out. New Scientist 006

9 What s so special about graphene? Electrons travel through it so fast that their behaviour is governed by the theory of relativity rather than classical physics. Economist 006 Inside every pencil, there is a neutron star waiting to get out. New Scientist 006 We ll have to rewrite the theory of metals for this problem. Physics Today 006

10 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 (CB) ky (VB) Energy dispersion: v 3 u 3X ε k = ±tt4 cos(k e j ) 5 j=1 3 3X + 4 sin(k e j ) 5 j=1 k y kx electrons

11 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

12 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 )

13 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 Eigenvalues λ a a Landau level energies (degenerate in chirality 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

14 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] sqrt(b) L 0 L L L 1 L C L1 C 1 3 ( ) ( ) (C) B Energy [mev] 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) Grenoble high field group: Sadowski et al., cond mat/

15 Degeneracy of relativistic Landau level Guiding center R = (X, Y ) algebra same as non-relativistic: [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 with Ω(ǫ n ) = ρ(ε)dε ǫ κ+1 n nn φ ǫ n (nb) 1 κ+1 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

16 Wavefunctions States n, m; α are -spinors (entries: sublattice): ) n,m; + = ; n,m; = ( i sgn(n) n,m n 1,m ( n 1,m i sgn(n) n, m ) Wavefunctions n, m are those of non-relativistic case! both n,m, n 1,m occur in the spinor

17 Wavefunctions States n, m; α are -spinors (entries: sublattice): ) n,m; + = ; n,m; = ( i sgn(n) n,m n 1,m ( n 1,m i sgn(n) 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

18 Wavefunctions States n, m; α are -spinors (entries: sublattice): ) n,m; + = ; n,m; = ( i sgn(n) n,m n 1,m ( n 1,m i sgn(n) 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 Note: wavefunctions not orthogonal on lattice e.g. Landau gauge: n,k exp(iky)φ n (x k) for k 1 k = G (rec. lattice vector) n,k 1 n,k 0

19 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) n = cos π/3 + ( o 3/) [q y + (x + 1/4)B] g B (x + 1/) g B (x 1) Eg B (x) n = cos π/3 + ( o 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

20 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 Bernevig et al.

21 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]

22 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

23 Interaction model (II) Graphene form factors (l B 1): F ++ n (q) = 1 F + n (q) = F + n q «q L n + L n 1! i(q + q K K ) p n + (q) = ˆF ( q) n 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!!

24 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, α 1 ν 4, α ν 1, α1 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 )

25 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 )

26 Effective SU() interaction potentials FQHE effective interaction potential n=0 (relativistic and non relativistic) n=1 (relativistic) n=1 (non relativistic) n=5 (relativistic) n=5 (non relativistic) r/l Broadly similar to non-relativistic case Largest difference between rel. and non-rel. case in n = 1 B

27 Haldane s Pseudopotentials and Laughlin s Wavefunction Two-particle wavefunction in n = 0: z,z m,m (z + z ) M (z z ) m e ( z + z )/4 analyticity: integral M, m spin-polarised fermions: odd m Pseudopotentials: projection of V to relative angular momentum m Haldane, 1983 V n=0 m Laughlin s wavefunction: = m,m V m,m m,m m,m ψs L ({z j }) = (z i z j ) s+1 e P k z k /4 for ν = 1 s + 1 i<j Screens all pseudopotentials with m < s + 1

28 The Haldane pseudopotentials in Graphene 0.8 n m Pseudopotentials V n=0 n=1 (rel.) n=1 (non rel.) Relative angular momentum m Energies of two-particle eigenfunctions m odd (even) spin (un)polarised V (r) V (c m) Similar shape of rel. interaction in n = 0 and n = 1 (i) relativistic V0 n smaller for n = 1 (ii) Vm 1 smoother for relativistic case

29 FQHE physics in n = 1 Landau level (i) unpolarised states more stable than for n = 0 ν = /3 should be robustly (chirality) unpolarised SU(4) symmetry even better for unpolarised states (ii) far away from ν = 1/ charge-density wave instability CF liquid but: instability to Pfaffian weaker see numerics by Apalkov+Chakraborty; Töke et al.

30 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)

31 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; also: strong effect of onsite U Alicea+Fisher; CDW favoured by lattice distortion Fuchs+Lederer details of energetics depend on structure of wavefunction on lattice scale expect at least anisotropy of size a/l B breaking down SU() chirality symmetry

32 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)

33 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 )

34 Chirality quantum ferromagnetism Cf. exchange driven spin ferromagnetism at ν = 1 in GaAs Review: Moon et al., PRB 51, 5138 (1995) Ψ = Y m sin θ m e iφm/ c m,+ + cos θ «m eiφm/ c m, 0 n m = 0 sin θ m cos φ m sin θ m sin φ m cos θ m 1 C A Non-linear σ model H = ρn s d r[ n(r)] ρ 0 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

35 Exprimental evidence for chirality polarisation 5T spin 4T 45T 37T 30T 5T 9T (30mK) T=1.4K chirality ferromagnetism (??) E n=0 Plateaux at ν = 0, ±1, ±4 Zhang et al. PRL 06 some addt l Landau levels individually resolved Simplest consideration: gaps vs. broadening Γ sk 4meV> Γ > 1.8meV E n=1 sk ν = 4 plateau only at strong fields need E Z other scenarios exist...

36 Related work Ferromagnetism in graphene: MacDonald et al. Effective description: Alicea+Fisher Harper eq.: Bernevig et al. FQHE physics: Apalkov+Chakraborty; Töke et al. 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.)...

37 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 and FQH physics for n 0 peculiarities of the central Landau level n = 0 chirality polarisation exchange ferromagnetism and anisotropies new plateaux in experiment

38 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 and FQH physics for n 0 peculiarities of the central Landau level n = 0 chirality polarisation exchange ferromagnetism and anisotropies new plateaux in experiment Thank you for your attention!