STM spectra of graphene K. Sengupta Theoretical Physics Division, IACS, Kolkata. Collaborators G. Baskaran, I.M.Sc Chennai, K. Saha, IACS Kolkata I. Paul, Grenoble France H. Manoharan, Stanford USA Refs: L. Mattos, et al, (submitted to Nature); K. Saha, et al. arxiv:0906.2788 K. Sengupta and G. Baskaran PRB 2007
Overview 1. Introduction to Graphene 2. Introduction to Kondo effect 2. Kondo effect in graphene: old work 4. Experiments 4. STM spectra of graphene: Fano theory 5. Two channel Kondo physics: Qualitative explanation 6. Conclusion
Relevant Basics about graphene B A Honeycomb lattice Each unit cell has two electrons from 2p z orbital leading to delocalized p bond. Tight binding model for graphene with nearest neighbor hopping. Can in principle include next-nearest neighbor hopping: same low energy physics. Ref: arxiv:0709.1163
There are two energy bands (valence and conduction) corresponding to energies touching each other at the edge of the Brillouin zone Two of these points K and K are inequivalent; rest are related by translation of a lattice vector. Two inequivalent Fermi points rather than a Fermi-line. Dirac electrons in Graphene Dirac cone about the K and K points Finite doping lead to Fermi surface around K and K points
Kondo effect in conventional systems Metal + Magnetic Impurity Formation of a many-body correlated state below a crossover temperature T K, where the impurity spin is screened by the conduction electrons. Features of Kondo effect: 1. Appropriately described by the Kondo model: 2. The coupling J, in the RG sense, grows at low T and becomes weak at high T. Negative beta function and asymptotic freedom. Anderson J. Phys. C 3, 2436 2441 (1970). 3. For two or more channels of conduction electrons (multichannel) the resultatnt ground state is an exotic non-fermi liquid. For a single channel, the ground state is still a Fermi liquid. 4. All the results depend crucially on the existence of constant DOS at E F 5. Kondo state leads to a peak/dip in the conductance at zero bias. as measured by STM.
What s different for possible Kondo effect in graphene For undoped graphene, linearly varying DOS makes a Kondo screened phase impossible. At finite and large doping, an effectively constant DOS occurs and hence one should see a Kondo screened phase. One can tune into a Kondo screened phase by applying a gate voltage Also, two species of electrons from K and K points may act as two channels if the impurity radius is large enough so that large-momenta scatterings are suppressed. Possibility of two-channel Kondo effect and hence non-fermi liquid physics in graphene. Theoretical prediction: Sengupta and Baskaran PRB (2007)
Recent STM experiments on doped graphene Adding Cobalt impurity in graphene Constant current STM topography of pure graphene (100 nm 2 I=40pA) Typical parameters: E F =250meV and T=4K. There is no experimental control over the position of these cobalt atoms. The position of these atoms can be accurately determined by STM topography L. Mattos, et al, (submitted to Nature)
Two channel Kondo physics: Impurity at hexagon center Observation of Kondo peak in doped graphene sample With T K =16K Proof of two-channel character of the Kondo state: non-fermi liquid ground state in graphene.
Bimodal Spectra for the Conductance G Impurity at the center: peaked structure of G and 2CK effect Impurity on site: dip structure of G and 1CK effect No analog in conventional Kondo systems: property of Dirac electrons
Theory of STM spectra in graphene K. Saha, et al. arxiv:0906.2788 Model Hamiltonians for Graphene, Impurity and STM tip
Tunneling current Interaction between the tip, impurity and graphene: Anderson model Tunneling current is derived from the rate of change of number of tip electrons Obtain an expression for the current using Keldysh perturbation theory
Turn the crank and obtain a formula for the current Wingreen and Meir(1994) Contribution from undoped graphene Impurity contribution Shape of the spectra depends crucially on the Fano factor q and hence on W 0 /U 0 U 0 coupling of graphene to tip W 0 coupling of impurity to tip V 0 coupling of graphene to impurity What determines the coupling of Dirac electrons to the STM tip?
What determines U 0 Bardeen Tunneling formula ~ Tight-binding wave-function for graphene electrons Plane-wave part Localized p z orbital part Impurity on hexagon center 3 4 2 U 0 becomes small leading to large q. 5 6 1 Peaked spectra for all values of E F independent of the applied voltage Conductance spectra shows a peak for center impurities
Impurity on Graphene site 7 5 4 6 2 1 3 Asymmetric position: No cancelation and U 0 remains large G should exhibit a change from peak to a dip through an anti-resonance with change of E F Impurity on hexagon center Refs: Saha et al (2009) Wehling et al (2009) Uchoa et al (2009) Impurity on Graphene site Should be observed on surfaces of topological insulators
Kondo physics: Qualitative reason for two channel Kondo effect Consider an impuirty potential V(r) sitting at the center of thegraphene hexagon. Localized p orbital wavefunctions: Expect main contribution from terms with R 1 =R 2 Two valleys act as two independent channels leading to 2CK physics. For atom at center of the hexagon, the inter-valley scattering terms have no contribution from the neighboring sites.no such cancelation occurs when the impurity atom is on the top of a graphene site.
Preliminary Support from Experiments Center Site Both FFT spectra and Kondo physics shows lack of 2k F scattering when the Impurity is at the center of the hexagon
Conclusion and future work Developed a theory of STM spectra of Dirac liquid Should be applicable to surfaces of topological insulators such as HgTe [ work in progress] Predicted and explained reason for two-channel Kondo physics in Graphene. Need to develop more detailed theory to understand nature of Kondo ground state Need to address crossover phenomenon beyond large N Need to develop a theory for non-equilibrium transport in Kondo state