Effects of Interactions in Suspended Graphene

Transcription

1 Effects of Interactions in Suspended Graphene Ben Feldman, Andrei Levin, Amir Yacoby, Harvard University Broken and unbroken symmetries in the lowest LL: spin and valley symmetries. FQHE Discussions with Bert Halperin and Dima Abanin In collaboration with Benjamin Krauss, Jurgen Smet, MPI Stuttgart

2 Degeneracy and Interactions Kinetic energy (Pauli exclusion) vs Interaction energy 2 2 r s E E e e = K = 2 e εr E F E e e E F e = εr 2 k 2m = α F * e ε n = n g ( ) d k F α -Disperssion 1-graphene 2 - bilayer g - Degeneracy: spin, layer, valley r s g m 1 1 α d n 2 d ε * d - Dimensionality (1d,2d,3d) We want: Large degeneracy, large mass, low density, low dielectric.

3 Bilayers - Band Structure B=0 Quadratic dispersion: α=2 m~0.03m e Near E=0; g =8 - Spin, valley and sublattice Low density Suspended ε~1 r s g m 1 1 α d n 2 d ε * Degenerate bands correspond to B1 and B2 sub-lattices

4 B=0 Generalization to Multi-layers For N layers: ABCABC stacking Dispersion k N α = N g - Spin, valley and sub-lattice Low density r s g m 1 1 d n 2 N d ε * Screening? Experiments: Lau, Tarucha, Jarillo-Herrero, Kim, Zaliznyak, Geim, Novoselov, Andrei, Heintz, Crommie, Schoenenberger, Ong Theory: Falko, McCann, Levitov, Katsnelson, Castro-Neto, Macdonald, Fogler, Fertig, Shimshoni Das Sarma, Polini, Guinea, Aleiner, Altshuler, Abanin, Sondhi, Kharitonov

5 QHE - Degeneracy s E n = ω n( n 1) c E n = n 2e v 2 F B Diverging mass McCann, Falko, 06

6 Quantum Hall Ferromagnetism For - ν=1 N=1 ν=2 N=0 ν=1 Spin φ θ Neglect Zeeman: SU(2) spontaneous symmetry breaking

7 Quantum Hall Ferromagnetism Isospin Top φ θ Bottom Simple example - ν=1 K. Moon, H. Mori, Kun Yang, S. M. Girvin, A. H. MacDonald, L. Zheng, D. Yoshioka, and Shou-Cheng Zhang, Phys. Rev. B 51, S. L. Sondhi, 95

8 Monolayer and Bilayers Interplay between valley and spin: Lowest LL occupied by electrons and holes 4 fold degenerate: Valley and spin Valley/ sub-lattice (pseudospin) spin Along B T Space of symmetries SU(4) ν=0 Partial spin ordering Partial valley ordering Abanin et al 07, Macdonald et al, M. Kharitonov 11, 12 AF CAF F Increasing Zeeman Kekule CDW PLP- Bilayers Determined by lattice scale anisotropy

9 Experimentally (Kim group): Increasing Zeeman reduces transport gap; suggests absence of strong spin polarization?? CAF?? Monolayers at the Edge Partial spin ordering AF CAF F Increasing Zeeman Analogous to QSH Kharitonov 12 Partial valley ordering F Kekule CDW PLP- Bilayers

10 Phase diagram for ν=0 in Bilayers CAF Zeeman Electric field PLP R. T. Weitz, AY, et al, Science Electric field From: Kharitonov, 2011 See also: Macdonald, Levitov

11 Fractional Quantum Hall Effect - GaAs Fractional QHE IQHE of composite Fermions (Jain) p ν = 2 p ±1 ν = N e N φ 0 Non Abelian phases at 5/2 and 12/5?? (Moore and Read) Edge reconstruction (Barak, AY et al) Neutral edge excitation modes (Heiblum et al, Venkatachalam, AY et al)

12 arxiv: v2 19 May 2012

13 arxiv: v2-5 Jul 2012

14 Background on FQHE - Suspended Andrei group: Nature 462, 192, (2009) (e/3)= 1/3 most developed Kim group Phys. Rev. Lett. 106, (2011)

15 Background on FQHE - hbn 4/3 most developed 5/3 absent Kim group, Nature Physics 7, 693 (2011)

16 n µ How to Measure Local Density of States? = Density of states µ n = Inverse (DOS ; Compressibility) Energy Electrochemical V Chemical µ = V - φ Need to measure δµ δµ = -δφ δv=0 Thermodynamic equilibrium V is constant in space Position δµ = -δφ If we can measure the local δφ we will immediately get the local δµ

17 Using a SET as a Local Electrostatic Probe I e QD e Current flow Coulomb Blockade N N + 1 N U e 2 = > kt c E V G E Use graphene as the gate. By monitoring the current we can extract the local electrostatic potential. N-1 N N+1 # Charge sensitivity: U N N+1 Voltage sensitivity: # 4 ~ 10 e / ~ 1µ V / Hz Hz Single Electron Transistor SET Current Spatial resolution: SiO2 ~ 100nm Electrostatic Potential T=300mK Si H. Hess, T. Fulton, M. Yoo, AY

18 Local Measurement of DOS Scanning Fixed T=300mK T=50mK Metal-Insulator Transition in 2D S. Ilani, et. al., PRL 84 (2000) Science 292 (2001) S. Ilani et al, Nature 328 (2004) Martin et al, Science (2004) Simultaneous Transport & Local Potential

19 Inverse compressibility Energy µ (n) Inverse compressibility dµ/dn DOS density density dn: AC voltage on backgate dµ: Single Electron Transistor 1µV, 100nm, B =[0 12 T], UHV, 300mK

20 Sample Geometry Transport 10µm Red: After first round of current annealing Blue, Green: New

21 Transport 1/3 2/3 1 R-2T 2 B [T] 2 Vbg [V] Red: After first round of current annealing Blue, Green: New Vbg [V]

22 Localized states Inverse Compressibility B Parallel to filling factor B. E. Feldman, AY, et al, arxiv:

23 Lowest Landau Level Localized states N=0 Landau Level 4 fold degeneracy spin and valley Valley ordered / CAF Valley and spin ordering 1/3 down to 1T B. E. Feldman, AY, et al, to appear in Science

24 Fractions in the Lowest Landau Level 7/5 11/7 5/3 No CF degeneracy s p ν = 2 p ±1

25 Filling Fraction vs Filling Factor ν=2 N=1 N=0 4-fold degeneracy ff=0 ν=0 ff=2-ν ff=1 ff=0 Filling fraction (ff) is analogous to the filling factor in GaAs

26 Missing Filling Fractions ff = p 2m = 2 p ± 1 4m ± 1 ν=4/3 ff=2/3 ν=10/7 ff=4/7 ν=14/9 ff=4/9 ν=8/5 ff=2/5 ν=2 ff=0 All fractions present Only even numerators Have CF degeneracy s ff=1+( 2/3, 3/5, 4/7, 5/9,, 4/9, 3/7, 2/5, 1/3 ) ff= 2/3, 4/7,, 4/9, 2/5

27 Filling Fraction vs Filling Factor ν=2 N=1 N=0 4-fold degeneracy ff=0 ν=0 ff=2-ν ff=1 ff=0 Filling fraction (ff) is analogous to the filling factor in GaAs No degeneracies Symmetries remain See also: Shayegan et al in AlAs Boebinger et al in double wells

28 Spatial Dependence B Fractions shift with position B = 12 T

29 Spatial Dependence B Fractions shift with position B = 6 T

30 Effects of Interactions in Suspended Graphene Ben Feldman, Andrei Levin, Amir Yacoby, Harvard University Broken and unbroken symmetries in the lowest LL: spin and valley symmetries. FQHE Discussions with Bert Halperin and Dima Abanin In collaboration with Benjamin Krauss, Jurgen Smet, MPI Stuttgart