Kondo satellites in photoemission spectra of heavy fermion compounds

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1 Kondo satellites in photoemission spectra of heavy fermion compounds P. Wölfle, Universität Karlsruhe J. Kroha, Universität Bonn

2 Outline Introduction: Kondo resonances in the photoemission spectra of Ce systems Single ion Anderson model of 4f multiplets: slave boson mean field theory Spectral function from perturbative RG method Spectral function in non-crossing approximation Comparison with experiment and conclusion

3 Collaborators: Theory Experiment S. Kirchner, Universität Karlsruhe (till 2004) G. Sellier, S. Hüfner, Universität Saarbrücken D. Ehm S. Schmidt F. Reinert Universität Würzburg O. Stockert, MPI-CPS, Dresden C. Geibel H. v. Löhneysen, Universität Karlsruhe

4 Modelling Ce compounds: Periodic Anderson model Localized 4f electrons hybridizing with conduction electron band Single ion physics expected to dominate local properties observed in photoemission spectra Lattice effects leading to coherent transport less important for PES

5 Multiorbital single impurity Anderson model 4f-states m, σ > : 7 spin-degenerate levels split by octet spin-orbit interaction crystal field sextet four doublets three doublets conduction band

6 Kondo resonances in the single particle spectra Model system: triplet of localized states, occupation n=1 Kondo resonances main resonance at Fermi level above Fermi level : rigid shift of multiplet below Fermi level: mirror image of excited levels width of excited levels of order a few T K

7 Methods: Pseudoparticle representation Limit of infinite Coulomb interaction U: multiple occupancy of f-states excluded b + Pseudoparticle operators: (empty level), (singly occupied level) f-electron operator:, f = f b + + el m m f + m = + m m = m + + Constraint: Q b b f f 1 Implementation of constraint H H λq, + λ

8 Methods: Slave boson mean field theory Replace Bose operators by mean field plus fluctuations: ˆ ˆ ˆ b= r+ aˆ, r = b, λ = λ + λ 0 Mean field Hamiltonian Mean field equations Level shifts: Kondo temperature: ε 0 = ε0 + λ0 = αt K, α 1 ε = ε + ( ε ε ) T m K 0 m 0 M 1 mm D Γ = { ( ) } De ε ε m= 1 m 0 Γ π ε0 2Γ 00 00

9 Electron spectral function: sb MFT + fluctuations dm, where G σ ( ω) = [( ω ε ) δ iγ ( ω)] f,, m, m' d, m m, m' m, m' 1 ε = ε + ( ε ε ) and at ε dm, = ( εdm, εd,0) Resonance peaks at dm, d,0 dm, d,0 Resonance widths: Γ = Γ = 2 mm, r mm, O( TK) weak dependence on ε bare level spacings

10 Methods: Mapping on to Kondo model Elimination of empty and multiply occupied states yields Kondo model : where H = H + J f f c c + ( ε ε ) f f K 0 m, m' m', σ ' m, σ k, σ k', σ' m 0 m, σ m, σ σσ, ', mm, ' kk, ' m J = 2 V V /( ε + ε ) Constraint: mm, ' m m' m m' m, σ f f + m, σ m, σ Perturbative second order correction to exchange constants ( ε = ε ε ): m = 1 m 0 D (2) f ( ε ) mm ' ml m ' l l ε + ω + ε D m l J ( ω) = N(0) J J { dε + ( ω ω)} ε N( J J / D ) ml m ' l l n{([ εm εl ] ω ) } l Logarithmically divergent terms at ω = ± [ ε ε ] m l

11 Methods: Renormalization group equations Poor man s scaling (Anderson, 1970) remove high energy states at ±D and absorb change into coupling constant g(d), depending on running cutoff D, take D 0. Extend renormalization group meth. to energy dependent coupling functions dg dg mm, ' dln D mm, ' ( ω) dln D ( ω) = 2 g g Θ( D ω) l ml, lm, ' = 2 g g Θ( D ω) l ml, lm, ' ω = ω ε + ε Θ-(step)-functions account for absence of renormalization if energy is outside the bandwidth D Two levels, splitting ε: 01 g 1 ( ω) =Θ( ω ε ε) 2ln[ ω ε / T ] 1 +Θ( ε ω ε ) 2ln( ε / T K ) l K ε ω ε m K 1/ln( ε / T )

12 Methods: Decoherence stops RG flow Logarithmic divergencies involving excited states are cut off by the finite spin relaxation rate (even at T=0): γ ε [ln( ε / )] T K 2 To account for this effect, we replace the conduction electron energy in the RG equations by 2 2 ω + γ Width of Kondo satellite peaks γ, increasing with ε Systematic and controlled method, provided γ T K, or ε / T K 1 ε T 2 ln ( / K )

13 Methods: Non-crossing approximation Conserving approximation derived from generating functional Pseudoparticle self energies: Σ ( ) f, σ, m, m' ω = = Γmm, ' dε f( ε) Ac, σ ( ε) Gb( ε + ω) Conduction electron DOS = Γ dε f( ε) A ( ε) G ( ε + ω) mm, ', σ mm, ' c, σ f, σ, mm, ' Γ mm Ac, σ ( ε ) Bare level broadening, ' ( ω) = [( ω λ ε ) δ Σ ( ω)] G f, σ, m, m' 0 f, m m, m' f, σ, m, m' G ( ω) = [ ω λ Σ ( ω)] b 0 b 1 1 el G ( ), σ,, ' ω = f m m

14 Results: Theoretical 4f spectrum in NCA

15 Results: Temperature dependence of 4f spectrum

16 Results: PES of CeCu 6 D. Ehm et al. (2002)

17 Results: Temperature dependence of PES of CeCu 6

18 Results: PES of CeCu 2 Si 2 F. Reinert et al.(2001)

19 Results: PES of CeRu 2 Si 2

20 Results: PES of CeNi 2 Ge 2 D. Ehm et al.(2005)

21 Results: PES of CeSi 2 D. Ehm et al. (2005)

22 Results: Kondo temperature and crystal field splittings D. Ehm et al.(2005)

23 Conclusion Photoemission spectra of Ce compounds may be modelled within single ion Anderson model of spin-orbit and crystal field split ionic states Excited crystal field split 4f-states lead to Kondo-type satellite resonance peaks in the single particle spectral function The peak positions are given by a rigid shift of the multiplet up to the Fermi level, as obtained by slave boson mean field theory The Kondo character of the peaks is apparent from logarithmically divergent terms in perturbation theory Summing the leading logarithms by renormalization group methods, observing the effect of phase decoherence for excited states yields resonance peaks of width increasing with level splitting Quantitative results were obtained within NCA, in excellent agreement with experiment

24 Methods: Diagrams of one loop RG equation Main contribution from Keldysh comp. of conduction electron G and real part of pseudofermion G:

PG5295 Muitos Corpos 1 Electronic Transport in Quantum dots 2 Kondo effect: Intro/theory. 3 Kondo effect in nanostructures

PG5295 Muitos Corpos 1 Electronic Transport in Quantum dots 2 Kondo effect: Intro/theory. 3 Kondo effect in nanostructures Prof. Luis Gregório Dias DFMT PG5295 Muitos Corpos 1 Electronic Transport in Quantum