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Unconventional QHE and Berry Phase of 2 π in Bilayer Graphene

Unconventional QHE and Berry Phase of 2 π in Bilayer Graphene. Robert Niffenegger 3/30/09. V xx / I = R xx. B. Integer quantum Hall effect Von Klitzing, PRL, 1980 (Nobel prize 1985). V xy / I = R H. I. h/e 2 = 25 812.807 449 . ( from a relatively poor 2DES). n≈3x10 11 cm -2. cl.

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Unconventional QHE and Berry Phase of 2 π in Bilayer Graphene

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  1. Unconventional QHE and Berry Phase of 2π in Bilayer Graphene Robert Niffenegger 3/30/09

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

  5. Room Temperature QHE in Graphene • Large LL Energy Gap • High Carrier Concentration • Mobility ~ independent of Temperature

  6. Berry Phase • Phase acquired under Adiabatic shift of parameter (B field or E field)

  7. Berry Phase in Graphene • Temperature dependence of the Shubnikov-de Haas oscillations at Vg = - 2.5V • Gives VF = 1.1 *106 m/s

  8. Berry Phase in Graphene • BF = Shubnikov-de Haas Oscillation Frequency in 1/B • β = Berry Phase • Aquired when quasiparticle moves between sublattices

  9. SdH oscillations at different VG • The location of 1/B for the nth minimum (maximum) of Rxx, counting from B=BF, plotted against n(n +1/2) • Slope (lower inset) =BF • Intercept (upper inset) = Berry’s phase

  10. Bilayer Graphene • Double Degenerate Lowest LL • g=2*4 • Nearest Neighbor Interactions with opposite sub-lattice of other layer • Interaction Energy • γ0 = Intra-Layer • γ1 = Inter-Layer

  11. BiLayer Graphene • RXY Steps of 4e2/h • RXX zero at Plateaus

  12. Double Broad RXX Peak at Low n • Implies Double Degeneracy of Lowest LL • Berry Phase Double Graphene • 2π

  13. Berry Phase of BiLayer Graphene • Double Degeneracy of Lowest LL

  14. Review • 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 = 1, Φ = 2π) • Berry Phase 2π • εN =+- vf(2ehBN)1/2

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