1. KONDO EFFECT IN BILAYER GRAPHENE Diego Mastrogiuseppe, Sergio Ulloa & Nancy Sandler Department of Physics & Astronomy Ohio University, Athens, OH

2. Kondo effect in a simple metal Anderson model Schrieffer-Wolff transformation Kondo model Crossover between free spin behavior and entanglement between impurity spin and conduction electrons

3. Motivation D. Withoff and E. Fradkin, PRL 64, 1835 (1990) K. Ingersent, PRB 54, 11936 (1996) needed for Kondo regime Pseudogap Kondo effect in single layer graphene M. Hentschel M. and F. Guinea, PRB, 76 (2007) B. Dora and P. Thalmeier, PRB 76, 115435 (2007) K. Sengupta and G. Baskaran, PRB 77, 045417 (2008) P. Cornaglia et al, PRL 102, 046801 (2009) Z-G Zhu et al, EPL 90, 67001 (2010) T. O. Wehling et al, PRB 81, 115427 (2010) B. Uchoa et al, PRL 106, 016801 (2011) Kondo in graphene Image: A. Castro Neto et al, RMP 81,109 (2009)

4. Motivation (unpublished) Kondo effect in single layer graphene Kondo effect in bilayer graphene

5. Bilayer graphene – band structure Bernal stacking A1 B1 Feature in DOS at B2 A2

6. Addition of the impurity

7. Schrieffer-Wolff transformation With a suitable choice of S to first order in V Anderson model for bilayer Kondo model band electronsimpurity hybridization

8. Results Diagonalizing (and for ) Two effective channels coupled to the impurity, but no possibility of 2-channel Kondo two eigenvalues are 0 and We are going to concentrate in

9. Half filling Reduces to effective single layer problem For the impurity in the middle of the hexagon, J = 0 T. O. Wehling et al, PRB 81, 115427 (2010) B. Uchoa et al, PRL 106, 016801 (2011) The result is insensitive to the relationship in

10. Away from half-filling Problem with SW tranformation Possible issues (work in progress): We get closer to the jump in DOS Interband scattering induced by impurity (is charge well defined?) May higher order commutators in SW regularize the divergences?

11. Conclusions Two effective channels coupled to the impurity (FM, AFM). No possibility of two-channel Kondo. Quadratic bands at half filling, no need of a critical J for having Kondo. At half-filling the results reduce to an effective single layer problem. Away from half filling, problem with SW transformation (jump in DOS, interband scattering, higher order terms in SW).