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. MydoshKamerlingh Onnes Laboratory and Institute LorentzLeiden UniversityThe 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 URu2Si2 #9 Modern experimental methods in correlated electron systems #10 Quantum phase transitions

Present basic experimental phenomena of the above topicsPresent 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 URu2Si2 #9 Modern experimental methods in correlated electron systems #10 Quantum phase transitions

Present basic experimental phenomena of the above topicsPresent 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 Ĥsd = 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) = ρmag(T) + ρ0 = ρtotal(T) - ρphon(T) AuFe alloys. Note increasing ρ0 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 Δρspin/c vs ln(T/TK) corrected for DM’sR Note decades of logarithmic behavior in T/TK and low T 0 Δρspin/c = ρun[1 – (T/TK)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 Ɛ0 occupied by a single electron; U is the single-electron charging energy, and ΓL

and ΓR 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 TK; measurements at 50 to 1000 mK.

Kondo – quantum dot universality when scaled with TK

Inverse susceptibility (χ = M/H) scaled with the concentration for CuMn with TK = 10-3K

Inverse susceptibility and concentration scaled inverse susceptibility (c/χi) for CuFe with TK = 30K

CuFe

XXXX

Excess specific heat ΔC/c on logarithmic scaleCuCr alloys with TK = 1K

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

ee

- 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 = V2/U; antiferromagnetic)

1st order perturbation theory processes

● S(S+1)

Spin disorder scattering

2nd order perturbation non-spin flip

Spin flip 2nd 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 TK = 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(T0): χ(T)/[C(T)/T] =3R(gµB)2/(2∏kB)2

Wilson ratio R = 2 for Kondo, 1 for heavy fermions

Determination of Kondo temperature

TK = D|2Jρ|1/2exp{-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

TK << D

Impurity susceptibility χi(T) Agrees with experiment

Low T χi is constant: Fermi liquid; C-W law at high T with To ≈ TK

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/TK) for different spin values Agrees with experiment

Note reduced CiV 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 URu2Si2 #9 Modern experimental methods in correlated electron systems #10 Quantum phase transitions

Kondo resonance to be measured via PES

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