Kondo Effect Ljubljana, 20.1.2016 Author: Lara Ulčakar Mentor: prof. dr. Anton Ramšak

Introduction History and properties of metal at low temperatures Models Kondo effect in quantum dots Calculation methods Conclusion

Discovered in 1934. Unusual scattering process of conduction electrons: Electrical resistivity increases with lowering temperature, Kondo resonance in the density of states. It took 30 years before first theoretical framework appeared: Kondo's and Anderson's perturbative calculation in early 1960s, Wilson's nonperturbative approach in 1974 and exact solution in 1980. Today: testing ground for other many-body problems, quantum dots.

Properties of metal at low temperatures Electric resistivity: Normal metal: drops with lowering temperature and saturates at very low. Superconductor: at critical temperature suddenly drops to zero. Normal metal with magnetic impurities: drops with temperature but at low temperatures starts to rise logarithmically: KONDO EFFECT!

Properties of metal at low temperatures First observed in 1934, then took 30 years to link this with magnetic impurities. Anderson model in 1961: The key ingredient is the Coulomb interaction between electrons on impurity site. explains formation of magnetic moments in metals. Kondo model: Antiferromagnetic coupling of impurity and spin density of surrounding electrons: By applying third order perturbation, Kondo showed resistivity growth: Incorrect at T = 0, because resistivity does not really diverge! Wilson's RG theory: Below conduction electrons form a singlet state with impurity divergence vanishes.

Properties of metal at low temperatures Magnetic susceptibility is a sum of: Pauli term: of electrons in host metal, temperature independent. Curie-Weiss term: from magnetic impurities: Low T: But instead of following the Curie-Weiss law, it is constant!

Kondo model In 1964 Jun Kondo proposed the Kondo model: Large reservoir of noninteracting electrons is coupled to the impurity by Heisenberg-like interaction term: / creates/anihilates a straight wave with and spin . This model in equivalent to Anderson model at lower temperature. Impurity spin Kondo interaction term Electron spin density Electron eigenenergy Band Hamiltonian

Anderson impurity model In 1961 P. W. Anderson proposes the Anderson impurity model: Large reservoir of noninteracting electrons, coupled by to impurity site with energy and on site Coulomb repusion . : energy of localized electron on impurity. / creates/anihilates an electron spin , bound to impurity. Hybridization Hamiltonian Overlap of atomic potential Hamiltonian of isolated impurity Coulomb repulsion

Anderson impurity model Only in presence of local magnetic moments form: (Energy of the system grows for if 2 electrons per impurity.) Eigenstates of an isolated impurity are : Density of states: Isolated atom: Delta functions at and . Atom, coupled to metal leads: Two resonances of width:

Magnetic moment formation Anderson model: Magnetic moments appear in parameter regime: Virtual exchange processes: Leads to spin-flip processes. Effectively works as antiferromagnetic interaction.

Magnetic moment formation Criterion for local moment formation: obtained by Hartree-Fock approximation: Equivalent to noninteracting model with: Solutions are found self-consistently:

Kondo effect in quantum dots Quantum dot: isolated island of electrons, connected by electrodes to two electron baths: Parameters like and can be tuned very easily. Conduction through quantum dot: : low, because no empty states near on the dot. : high, because of the resonance state at on the dot. Vg+U Vg+U Vg Vg

Kondo effect in quantum dots Density of states: Conductance through the dot:

Cheminal potential of electron bath

Calculation methods Poor Man's Scaling: Anderson, 1970. Sets stage for Wilson's RG approach. Numerical renormalization group: Wilson, 1974. Gives first explanations for finite resistivity and constant susceptibility at . Nobel prize in 1982. Fermi liquid theory: Nozieres, Yamada, 1975, 1/N expansions: 1980s, Exact analytical solution by Bethe Ansatz: Andrei, Wiegmann, 1980.

Exact methods: renormalization group approach Renormalizable systems: self-similar when observed from different scales. RG idea: transforms the system to larger scales: After one RG transformation Anderson impurity model is transformed to the Kondo model: Less degrees of freedom More complex couplings

Conclusion Kondo effect occurs in: Metals with magnetic impurities: resistivity's growth with lowering temperature. Quantum dots: Almost total electrical conductance because of the Kondo resonance in density of states. Models: Kondo model, Anderson impurity model. Methods to solving the problem: Nonperturbative renormalization group theory, Many others. Still actual because of development of nanotechnology. Its numerical approaches are now used to solve other heavy fermion problems like high-temperature superconductors.

Thank you for your attention!

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