Matthias turned to the quest for realistic models for magnetic impurities. eg. its not so simple to figure out the Kondo model for Fe in gold. Not well known until very recently, because of the interplay of spin and orbital degrees of freedom. Is it spin 1/2, or two channel spin 1, three channel spin 3/2, four channel spin 2?
Alloul pointed out from the high temperature chi in the early days, it was known to be larger than S=1/2. Matthias argued that it did not really become clear until recently - Costi et al, PRL 102, 056802 (2009). Best fit to experimental dephasing rate suggests S=3/2, three channels.
The dephasing rate grows from T^2 up to a broad plateaux, and the subtle different fits favour S=3/2 for Fe in Au.
Outline of talk:
1 Impurities in Graphene, Dirac fermions, STM expts, orbital physics of d electron impurities.
2. Review: pseudogap Kondo model. Quantum phase transitions. Critical field theories.
3. Pseudogap Kondo model with voltage bias. (You can tune from linear density of states to a finite density of states. Maximal electron hole asymmetry. Spectral functions.
Turned to Dirac Fermions in graphene. Two atoms per unit cell. When you diagonalize the short-range hopping Hubbard model (U=0), you get two Dirac cones with Hamiltonian
H ~ vF (p-K).sigma_sublattice
E_k = v_F }| k - K|
Pauli matrix acts in "sublattice space". There are two copies of this Dirac Hamiltonian. The Dirac cones are "topologically protected". (Semenoff 1984, Haldane 1988). The Fermi points are robust against next nearest neighbour hopping etc.
By gateing, you can tune the Fermi surface to go from a Fermi point to a Fermi surface.
Dos (E) \propto |(E-E_F)| linear density of states.
Kondo effect in graphene: first observation
Manoharan group. STM shows the hexagonal structure, with a puckered, rippled surface. Schofield asked why there was a superstructure. There was no obvious answer from the croud. From dI/dV you can see the Co on the surface. Now you can see "blue dots" representing the Cobalt atoms. You can now see the dI/dV spectrum. You see a peak on some cobalt atoms, on others you see a dip. Can extract a width, or Kondo temperature of TK~ 15K. These pictures corresponded to an effective gate voltage of 200mV.
The point is, there are two different locations of the Cobalt atom. Site A corresponds to atom on top of a C atom (t-site, dip structure, pseudo-spin breaking) whereas site B (h-site) is in the middle of a hexagon (peak, pseudospin conserving). Add a field, the structures split, proving that the peak is of magnetic origin.
What is the correct Kondo model for Co on the graphene sheet? The symmetries are very important here. In the graphene you have band degeneracies - C3nu, C6nu - three and six fold degeneracies.
(Wehling et al, PRB 81, 115427 (2010)). DOS is strongly particle-hole asymmetric. J~ 2eV, bandwith from t=2.8eV.
Peter Hirschfeld asked where the spin is localized. Henri Alloul suggested that the GGA+U might not have enough correlations to locate the spin. The blogger thinks these methods are probably good enough to get the spin form factor.
So what happens for the Kondo effect in a non-magnetic host. If the DOS vanishes at the Fermi level, there is no Kondo screening at small J_K. (Fradkin and Withoff - though not referenced). Two possibilities
Hard gap - first order transition upon varying J_K at T=0.
Pseudogap - continuos transition upon variation of J_K. Non-trivial finite T behavior arising from quantum critical point. DOS ~ epsilon^r. r >0 gives phase transition. (d-wave, graphene r=1).
Two axes - Kondo axis J. Particle-hole asymmetry V.
For small r < r* = 0.3748, get simple Fradkin-Withoff behavior. Jc ~ r. Also an ASC, asymmetric strong coupling fixed point. beta (j) = rj - j^2.
For r* < r < 1/2 a new fixed point appears at finite Vc and Jc.
r=1 is upper critical dimension. r=0 is lower critical dimension. Hyperscaling is obeyed for r<1.
Chubukov asks can you do an expansion in epsilon = r-1? Matthias says yes - but to do it requires the Gaussian theory at r>1. The answer is a level crossing between a doublet of single impurity and a singlet of a screened impurity. Simple model with doublet of energy epsilon-0 hybridized to a singlet via an Anderson screening - it is a non-interacting pseudo-gap Anderson model. Can now do an epsilon expansion. Vojta Fritz PRB 70, 094502 (2004)
Pseudogap Kondo model with voltage bias Sofar, only neutral graphene. Next, mu>0. Now the moment will ultimately be screened at low T. But if the chemical potential is of order the TK, there will be critical physics. Chemical potential provides a fan of NFL physics. J=Jc, then predict TK = kappa * mu.
RG now done with chemical potential effect on flow equations. The leading effect is that one drives the impurity to the screened, or unscreened phase. (mu <0 epsilon = -infinity screened; mu > 0, epsilon = + infinity). Depends on sign of mu.
Ultimate results - TK as a function of gate voltage. (Vojta, Fritz, Bulla EPL (2010)).
Conclusions
- Magnetic impurities in graphene. Kondo criticality possible.
- Critical theory is not of Landau Ginzburg Wilson. but intrinsically fermionic
- TK(mu) extreme asymmetry between electron and hole doping, not only near criticality resulting from structure of critical fixed point.
- Systematic measuremets of Co impurities as function of gate voltage required.
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