PowerPoint Presentation

Kondo effect in bilayer grapheneDiego Mastrogiuseppe, Sergio Ulloa & Nancy Sandler

Department of Physics & AstronomyOhio University, Athens, OHIm going to present a theoretical work about the kondo effect in bilayer graphene. This work has been done in colaboration with Sergio Ulloa and Nancy Sandler at Ohio University.1Kondo effect in a simple metal

Anderson modelSchrieffer-Wolff transformationKondo model

Crossover between free spin behavior and entanglement betweenimpurity spin and conduction electrons

So let me start introducing the Kondo effect for a simple metal. And with a simple metal I mean a metal described by one band at some filling in which one can consider the density of states constant around the Fermi level.So the usual way to describe a magnetic impurity embedded in a metal is through the Anderson model which consists basically in three terms.The first one, H_0 is the hamiltonian describing the band electrons, H_imp has one term for the energy of putting one electron in the impurity orbital and U is the coulomb repulsion to pay for double ocuppancy in the impurity. Then, H_{hyb} describes the hybridization between the orbital of the localized level and the Bloch states of the conduction electrons.Through a canonical transformation, the SW transformation, its possible to eliminate perturbatively high energy states (empty o double ocuppied impurity) and arrive to the Kondo model which is an effective model describing an interaction between the localized spin and the spin of the conduction electron. The exchange coupling is given by J which depends on the parameters of the Anderson model. For electrons around the Fermi level its possible to show that the effective coupling is antiferromagnetic, and the image at low temperatures is a many body state in which the conduction electrons scatter with the spin of the impurity and produce spin flips. Its possible to define a temperature called Kondo temperature which is exponential in J and in the density of states at the Fermi level, and determines a crossover between a free spin behaviour (at high temperatures) and the low temperature entangled state of the impurity spin with the conduction electrons.2Motivation

D. Withoff and E. Fradkin, PRL 64, 1835 (1990)K. Ingersent, PRB 54, 11936 (1996)

needed for Kondo regime

PseudogapKondo effect in single layer grapheneM. 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 grapheneImage: A. Castro Neto et al, RMP 81,109 (2009)Our motivation starts with the Kondo effect in single layer graphene. Here we can see that the density of states in graphene vanishes lineary around the Dirac points, giving rise to a pseudogap at those points. In early papers it was shown that one needs a coupling J larger than a critical J to have Kondo screening when there is a feature of this kind in the DOS.More recently, and now directly related to graphene, people has been studying the formation of localized magnetic states and the Kondo effect in this material. These are some references on the subject.3Motivation

(unpublished)Kondo effect in single layer grapheneKondo effect in bilayer graphene

There are also a few recent experiments which measure data compatible with the Kondo effect.It has been predicted theoretically that vacancies and cobalt adatoms in graphene behave as spin impurities. Although the appearance of Kondo with vacancies seems to be clear, with Co adatom theres still controversy. So its natural to go a step beyond and study the properties of the Kondo effect in bilayer graphene.In this respect, there are two theoretical works that address the formation of a local magnetic moment in bilayer graphene. To my knowledge, there are no experimental results on the subject.4Bilayer graphene band structure

Bernal stacking

A1B1

Feature in DOS at

B2A2So, lets go to see the band structure of bilayer graphene. The layers usually arrange in the so called Bernal stacking, in which one of the layers is displaced so that atom A in layer 1 is connected to atom B in layer 2 with a hopping amplitude t_p, and atom A in layer 2 lies in the middle of the hexagon of the lower one. We need 4 atoms per unit cell to describe the bilayer structure. The hamiltonian can be written in spinorial notation, and diagonalizing the 4x4 matrix one obtains the four bands that are quadratic around the K point. The density of states has a jump at E=t_perp.5Addition of the impurity

We are going to consider intercalated impurities. In this figures I show a espherically symmetric impurity for simplicity in the middle of the layers. There are two inquivalent positions for it. The first one is the impurity lying in between a carbon atom of each layer, and we consider the posibility of varying the hybridization with each atom. The second case is the impurity below one atom and above the middle of the hexagon. In this case the impurity can hybridize with the six sorrounding atoms and we allow for different hybridizations with atom A and B. So, for the case for instance, I show the hybridization hamiltonian in real space, which will become a k dependent coupling with a Fourier transform.6Schrieffer-Wolff transformation

With a suitable choice of S to first order in V

Anderson model for bilayerKondo modelband electronsimpurityhybridization

Here I show the anderson hamiltonian for our model. These functions V contain the information of the configurations of the impurity. So we perform the SW transformation and we get the effective kondo hamiltonian. The spin operator of the conduction electrons and the J coupling have information of the bands of bilayer through the indices alpha and beta. Besides the exchange term in the effective hamiltonian, there are more terms generated by the transformation. One is a potential scattering in the bands induced by the impurity and terms which renormalize Ed and U.7Results

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 inHere I show the formula for J, which depends on the hybridization functions and these propagators generated in the transformation. Now its posible to write a J matrix on the band indexes and diagonalize it, which gives rise to two zero eigenvalues, and two non-zero eigenvalues, one ferromagnetic and the other antiferromagnetic. So we end up with 2 effective channels coupled to the impurity. In an RG analysis, JF goes to weak coupling and JAF to strong coupling. Its Jaf the one that determines Tk.8Half filling

Reduces to effective single layer problemFor the impurity in the middleof the hexagon, J = 0T. O. Wehling et al, PRB 81, 115427 (2010)B. Uchoa et al, PRL 106, 016801 (2011)

The result is insensitive tothe relationship in

Now I show results for the 2 impurity configurations with the system at half filling. Im going to fix t_perp at 0.2 t in for all the results. For the first case, I show the variation of J with respect to the level of the impurity. If I write this combination of V1 and V2, and I fix V to some value, when I do this plot for different V1 or V2, all the curves lie in top of each other so one recovers the case of and impurity on top of a carbon atom in single layer graphene.For the second case, we fix the level of the impurity and want to see the variation of J with this coupling V1 . The results are insensitive to the relation between V2a and V2b. Moreover, when V1 =0, we see that J goes to 0, which means that at half filling the impurity decouples from the band electrons if its in the middle of the hexagon.9Away from half-filling

Problem with SW tranformationPossible 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?

Now I show some results away from half filling. Lers consider this case and see the variation of J with Ed. We see the appearance of divergences for fillings different from zero, the black curve show the case at half filling for reference. So it seem that there is a problem with the SW transformation. There are some possible issues which we are working on. First, when we move away from hal filling we approach the jump in the density of states, so we wonder if theres some effect of that feature in J. Second, the impurity induces interband transitions in the electron gas, and we are studying the possibility that in these cases there can be charge fluctuations in the impurity. And the thrid possibility is that one needs to include higher order commutators in the transformation to regularize the divergences.10ConclusionsTwo 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).