From Kondo and Spin Glasses to Heavy Fermions, Hidden Order and Quantum Phase Transitions A Series of Ten Lectures at XVI Training Course on Strongly Correlated Systems, October 2011 J. A. Mydosh Kamerlingh Onnes Laboratory and Institute Lorentz Leiden University The Netherlands

Lecture schedule October 3 – 7, 2011  #1 Kondo effect  #2 Spin glasses  #3 Giant magnetoresistance  #4 Magnetoelectrics and multiferroics  #5 High temperature superconductivity  #6 Applications of superconductivity  #7 Heavy fermions  #8 Hidden order in URu 2 Si 2  #9 Modern experimental methods in correlated electron systems  #10 Quantum phase transitions Present basic experimental phenomena of the above topics

Lecture schedule October 3 – 7, 2011  #1 Kondo effect  #2 Spin glasses  #3 Giant magnetoresistance  #4 Magnetoelectrics and multiferroics  #5 High temperature superconductivity  #6 Applications of superconductivity  #7 Heavy fermions  #8 Hidden order in URu 2 Si 2  #9 Modern experimental methods in correlated electron systems  #10 Quantum phase transitions Present basic experimental phenomena of the above topics

#1] The Kondo Effect: Experimentally Driven 1930/34; Theoretically Explained 1965 as magnetic impurities in non-magnetic metals. Low temperature resistivity minimum in AuFe and CuFe alloys. Increased scattering. Strange decrease of low temperature susceptibility, deviation from Curie-Weiss law. Disappearance of magnetism. Broad maximum in specific heat. Accumulation of entropy. Not a phase transition but a crossover behavior! Virtual bond state of impurity in metal. Magnetic or non-magnetic? s – d exchange model for Ĥ sd = Σ J s · S Kondo’s calculation (1965) using perturbation theory for ρ. Wilson’s renormalization group method (1974) and χ(T )/C(T) ratio. Bethe ansatz theory (1981) for χ, M and C: thermodynamics. Modern Kondo behavior: Quantum dots, Kondo resonance & lattice.

Interaction between localized impurity spin and conduction electrons – temperature dependent. Many body physics, strongly correlated electron phenomena yet Landau Fermi liquid. Not a phase transition but crossover in temperature

Kondo effect: scattering of conduction electron on a magnetic imputity via a spin-flip (many-body) process. Kondo cloud

Magnetic resistivity Δρ(T) = ρ mag (T) + ρ 0 = ρ total (T) - ρ phon (T) AuFe alloys. Note increasing ρ 0 and ρ(max) as concentration is increased

Concentration scaled magnetic resistivity Δρ(T)/c vs lnT CuAuFe alloys. Note lnT dependences (Kondo) and deviations from Matthiessen’s rule.

Now Δρ spin /c vs ln(T/T K ) corrected for DM’sR Note decades of logarithmic behavior in T/T K and low T  0 Δρ spin /c = ρ un [1 – (T/T K ) 2 ], i.e., Fermi liquid behavior of Kondo effect

Quantum dots – mesoscopically fabricated, tunneling of single electrons from contact reservoir controlled by gate voltage This is Kondo!

Schematic energy diagram of a dot with one spin-degenerate energy level Ɛ 0 occupied by a single electron; U is the single-electron charging energy, and Γ L and Γ R give the tunnel couplings to the left and right leads. S M Cronenwett et al., Science 281(1998) 540.

Quantized conductance vs temperature Gate voltage is used to tune T K ; measurements at 50 to 1000 mK.

Kondo – quantum dot universality when scaled with T K

Inverse susceptibility ( χ = M/H) scaled with the concentration for CuMn with T K = K

Inverse susceptibility and concentration scaled inverse susceptibility (c/ χ i ) for CuFe with T K = 30K CuFe XXXX

Excess specific heat ΔC/c on logarithmic scale CuCr alloys with T K = 1K

Place a 3d (4f) impurity in a noble (non-magnetic) metal Virtual bound state (vbs) model-See V.Shenoy lecture notes

e e

 - U -  up-spin down-spin

U splits the up and down vbs’, note different DOS’ Net magnetic moment of non-half integral spin  U 

transition”

( J = V 2 /U; antiferromagnetic)

1 st order perturbation theory processes ● S(S+1) Spin disorder scattering

2 nd order perturbation non-spin flip

Spin flip 2 nd order perturbation

Calculation of the logarithmic – T resistivity behavior

Calculation of the resistivity minimum with phonons added

Clean resistivity experiments on known concentrations of magnetic impurities, AuFe with T K = 0.5 K.

Collection of Kondo temperatures

Wilson renormalization group method (1974): scale transformation of Kondo Hamiltonian to be diagonalized Spherical wave packets localized around impurity Shell parameter Λ > 1; E ~ Λ -n/2 for n states Calculate via numerical iteration χ (T) as a universal function and C(T) over entire T- range Lim(T  0): χ (T)/[C(T)/T] =3R(gµ B ) 2 /(2 ∏ k B ) 2 Wilson ratio R = 2 for Kondo, 1 for heavy fermions Determination of Kondo temperature T K = D|2Jρ| 1/2 exp{-1/2Jρ} where J is exchange coupling and ρ the host metal density of states K. Wilson, RMP 47(1975)773.

Bethe Ansatz (1980’s) - Andrei et al., RMP 55, 331(1983). “Bethe ansatz” method for finding exact solution of quantum many-body Kondo Hamiltonian in 1D. Many body wave function is symmetrized product of one-body wave functions. Eigenvalue problem. Allows for exact (diagonalization) solution of thermodynamic propertries: χ, M and C as fct(T,H). Does not give the transport properties, e.g. ρ(T,H). “1D” Fermi surface T K << D

Impurity susceptibility χ i (T) Agrees with experiment Low T χ i is constant : Fermi liquid; C-W law at high T with T o ≈ T K

Impurity magnetization as fct(H) Agrees with experiment M ~ H at low H; M  free moment at large H (Kondo effect broken)

Specific heat vs log(T/T K ) for different spin values Agrees with experiment Note reduced C i V as the impurity spin increases.

Kondo cloud - wave packet but what happens with a Kondo lattice? Never unambiguously found!

Kondo resonance - how to detect? Photoemission spectroscopy (PES) Still controversial

Kondo effect (  Kondo lattice) gives an introduction to forthcoming topics, e.g., SG, GMR, HF; QPT.  #1 Kondo effect  #2 Spin glasses  #3 Giant magnetoresistance  #4 Magnetoelectrics and multiferroics  #5 High temperature superconductivity  #6 Applications of superconductivity  #7 Heavy fermions  #8 Hidden order in URu 2 Si 2  #9 Modern experimental methods in correlated electron systems  #10 Quantum phase transitions

Kondo resonance to be measured via PES

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