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.
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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
??? To use ???
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