Search for FQHE in graphene

1. Search for FQHE in graphene John Watson, SumitMondal, Robert Niffenegger 4/29/09

2. Vxx/I=Rxx B Integer quantum Hall effect Von Klitzing, PRL, 1980 (Nobel prize 1985) Vxy/I=RH I h/e2 = 25 812.807 449  ( from a relatively poor 2DES) n≈3x1011cm-2 cl. B (T) REVIEW FROM CLASS LECTURE Quantum Hall Effect --- Integer (IQHE)

3. Unconventional QHE in Graphene Semiconductor g=2 • Dirac Fermions allow Zero Energy solutions for Landu Levels (LL) • LL are doubly Degenerate (Spin and Sub-Lattice) Graphene g=4

4. Unconventional QHE in Graphene • Steps of 4e2/h • No Plateau at n=0

5. Integer QHE • Free-fermion QHE • No Berry’s phase • εN = hωc(N + 1/2) • Graphene (J = 1, Φ = π) • Berry Phase π • εN =+- vf(2ehBN)1/2 • BiLayer Graphene (J = 2, Φ = 2π) • Berry Phase 2π • E • Doubly Degenerate Lowest LL

6. Spin Hall Effect • Dirac Relativistic Electrons • Spin-Orbit Interaction • Higher T • Not observed in Graphite bismuth-antimony Dr. ZahidHasan, Princeton

7. Theoretical Expectations of FQHE in Graphene • In contrast to conventional 2DEG, electrons in single layer graphene are massless, relativistic Dirac quasiparticles • Papic et al. theoretically investigated that electrons in graphene lose their relativistic character associated with the Lorentz invariance once they are restricted to a single LL • Electrons have 4-fold degeneracy because of the additional pseudo-spin degree of freedom [ SU(4) spin-valley symmetry] • when the Zeeman energy is sufficiently high that only the two-fold pseudospin degeneracy remains [SU(2) symmetry] the graphene FQHE problem maps into the problem of FQHE in GaAs in the zero Zeeman energy limit

8. SU(4) symmetry • The most salient difference between the conventional 2DEG and graphene arises from its valley symmetry • slightly different effective interaction potentials in the n-th LL in strong B field

9. the polarised FQHE states in the zero-energy LL are expected to be the same in graphene as in the 2DEG because there is no difference in the effective interaction potential • the unpolarised FQHE states in n = 0 will be affected due to larger internal symmetry of graphene • the n = 1 LL in graphene is much more similar to the n = 0 LL than of the n = 1 LL in the 2DEG • odd-ℓ pseudopotentials are systematically larger in n = 1 than in n = 0 • the n = 1 FQHE states are more stable than those in n = 0, for the same B field value (opposite of that of the nonrelativistic system, where the excitation gap decreases monotonically with increasing Landau-level index)

10. Effect of valley polarization • If the inter-valley splitting is large, the electrons will be valley polarized independent of the filling factor • small inter-valley splittings will affect which Landau level is the most stable (for example, in the ν = 2/3 case the valley unpolarized state is favored over the valley polarized states for both n = 0 and n = 1)

11. Practical possibilities • Disorder plays a detrimental role • Cleaner sample, Higher mobility? Suspended Graphene • Graphene on graphite? The close lattice matching would introduce a mass term into the Dirac Hamiltonian which would lift the valley degeneracy of the zero energy Landau level

12. Summary of observed Hall effects • 1Stormer, H. Nobel Lecture: The fractional quantum Hall effect. Reviews of Modern Physics71, 875 (1999). • 2Zhang, Y. et al. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature438, 201-204 (2005). • 3Novoselov, K.S. et al. Room temperature quantum Hall effect in graphene. Science315, 1379 (2007).

13. Fabrication Cr/Au E-beam lithography graphene 90 s buffered oxide etch Si02 Si Bolotin, K.I. et al. Ultrahigh electron mobility in suspended graphene. Solid State Communications146, 351-355 (2008).

14. Reported results • Mobility increase from 28,000 cm2/Vs to 230,000 cm2/Vs b: AFM of device c: AFM after O2 plasma High mobility sample on substrate Before annealing After annealing

15. Conclusions on mobility • Mean free path ~ 1.2 μm • Remaining scattering due to edges and electrodes • Fabricate larger devices • Annealing works because defects present above and below graphene

16. Secondary approach Li, G. et al. Scanning tunneling spectroscopy of graphene on graphite. Physical Review Letters102 (accepted April 2, 2009, unpublished as of April 28, 2009). Larger separation between top & second layer

17. How is this graphene? • Linear density of states ~ Dirac point • LL energy En ~ Differential tunneling conductance (proportional to DOS)

18. Summary of graphene on graphite • Layer spacing, DOS, and LL indicate decoupled graphene layer • 1Mobilities reported up to 107 cm2/Vs • No work reported on FQHE 1Neugebauer, P. et al. How perfect can graphene be? arXiv 0903.1612v1 (2009).