Transport and optical properties of heavy fermions Theo Costi - PowerPoint PPT Presentation

Transport and optical properties of heavy fermions Theo Costi Institute for Solid State Research, Research Centre J¨ ulich, Germany October 5, 2005 • What are the low energy scales in (paramagnetic) heavy fermions ? • How are these manifested in physical properties such as – spectra, – dynamical susceptibilities, – resistivities, – optical conductivities ? • In what sense is there universality and scaling in heavy fermions ? T. A. C., N. Manini, JLTP 2002 & unpublished

Motivation:single Kondo impurity � ǫ k c † H = k ,σ c k ,σ + J S 0 · s 0 k ,σ • One low energy scale: – T K = f ( J/D ) – Fermi liquid scale T 0 = T K • universal scaling functions: – ρ ( T, J/D ) ⇒ f ρ ( T/T K ) – A ( ω, T, J/D ) ⇒ f A ( ω/T K , T/T K ) 1 0.8 T 0 : χ (0)=1/4T 0 0.6 ρ (T) 0.4 0.2 0 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 T/T 0

Motivation:experiments, e.g. Andres et al 1975 • Fermi liquid coherence scale T 0 ≈ 3 K . • T max = 35 K ≈ 10 T 0 . However, T max � = T K ! • In fact T K generally absent in ρ ( T ) , σ ( ω, T ) .

Kondo Lattice Model � ǫ k c † � H = k ,σ c k ,σ + J S j · s j k ,σ j Solve for paramagnetic solutions by DMFT(NRG) on a Bethe-Lattice : 2 � D 2 − ε k 2 N 0 ( ε k ) = πD 2 Consider c-electron fillings 0 < n c < 1 (hole dopings 0 ≤ δ ≤ 1 ). Calculate : • Σ σ ( ω, T ) , c-electron self-energy • ρ c ( ω, T ) , c-electron DOS • A ( ω, T ) , f-electron DOS • χ ( ω, T ) , dynamical susceptibility • σ ( ω, T ) , optical conductivity • ρ ( T ) , resistivity

Low energy scales in χ ( ω, T ) 40 n c =0.6 ( δ =40%) T 0 n c =0.65 ( δ =35%) J=0.4 n c =0.7 ( δ =30%) n c =0.75 ( δ =25%) 30 n c =0.8 ( δ =20%) Im[ χ ’’( ω ,T=0)] n c =0.85 ( δ =15%) n c =0.9 ( δ =10%) 20 n c =0.95 ( δ =5%) n c =0.98 ( δ =2%) n c =0.991 ( δ =0.9%) T * 10 n c =0.995( δ =0.5%) 0 10 -4 10 -3 10 -2 10 -1 10 0 10 1 ω /D • Fermi lquid scale T 0 , discernible in χ for all δ > 0 . • Single-ion Kondo scale T ∗ = T K , descernible in χ for δ < 20%

Low energy scales in spectra 0.6 n c =0.8 n c =0.4 0.6 * T 0.4 ρ c T/T 0 =10.23 ρ c 0.4 T/T 0 =4.53 * T/T 0 =12.82 T T/T 0 =2.08 T/T 0 =5.71 A T/T 0 =0.93 T/T 0 =2.55 0.2 0.2 A T/T 0 =0.41 T/T 0 =1.14 T/T 0 =0.18 T/T 0 =0.51 T/T 0 =0.22 0 0 0 50 −50 0 50 100 ω /T 0 ω /T 0 • T ∗ = T K discernible in f-electron spectrum A ( ω ) only for δ < 20% • T ∗ = T K allways discernible in c-electron spectrum ρ c ( ω ) . Suggests tunneling measurement to obtain T ∗ = T K from local c-electron DOS. • T 0 sets T-dependence of A ( ω = 0 , T ) , ρ c ( ω = 0 , T ) . • T 0 /T K → 0 , n c → 0 (Pruschke et al. Anderson Lattice, PRB 1999)

Comparison of Spectra with Photoemission T 0 =30 K T 0 =400 K YbInCu 4 Intensity [arb.units] YbInCu 4 T=12 K Intensity [arb. units] T= 50K T= 90K T=130K photoemission (Moore et al. ) T=150K Kondo lattice −0.2 −0.1 0 0.1 −0.2 −0.1 0 0.1 E [eV] E [eV] • YbInCu 4 : volume collapse at T v = 42 K. High T 0 = 400 K phase for T < T v . Low T 0 = 30 K phase for T > T v . Kondo system with 14 − n f = 0 . 85 − 0 . 96 . • Single crystals, ∆ E = 25 meV FWHM resolution • Lineshape and T-dependent intensity consistent with KL scenario.

Optical conductivity: Kondo insulator n c =1.0 ( δ =0%) 8.0 5.4 ∆ dir 6.0 + E k σ ( ω ,T) 2.5 4.0 ∆ dir ∆ ind = ∆ g 1.1 2.0 ∆ ind 0.49 µ 0.20 0.08 0.0 -2 -1 0 1 2 10 10 10 10 10 - ω / ∆ g E k • No Drude peak as T → 0 . • T = 0 threshold set by indirect gap ∆ ind = ∆ g = T K (see Logan’s talk). • T-dependence set by T ∗ = T K = ∆ ind • mid-infrared peak; transitions across quasiparticle bands

Optical conductivity: δ > 0 n c = 0.95, ( δ =5%), ∆ dir /T 0 = 110 n c = 0.5 ( δ =50%), ∆ dir /T 0 = 765 ∆ dir ∆ dir 0.3 4.0 4.0 T/T 0 = 21.5 0.2 3.0 3.0 σ ( ω ,T) T/T 0 = 36.9 σ ( ω ,T) 9.5 0.7 2.0 2.0 1.5 0.6 16.5 4.2 7.4 1.0 1.0 1.2 1.8 3.3 0.0 0.0 -1 0 1 2 3 4 -1 0 1 2 3 10 10 10 10 10 10 10 10 10 10 10 ω /T 0 ω /T 0 • Drude peak; transitions within E − k . Develops for T < T 0 . • Scale for T-dependence set by T 0 • mid-infrared peak; transitions across quasiparticle bands

Summary of low energy scales; scaling • Two low energy scales (Pruschke et al 1999, Burdin et al 2000, TAC et al 2001): – Fermi liquid coherence scale T 0 = T 0 ( n c ) seen for all δ > 0 in all quantities – single-ion Kondo scale T K = T K ( n c ) present for δ < 20% in χ ( ω, T ) , A ( ω, T ) (and for all δ > 0 in local c-electron DOS) • universal scaling functions for fixed n c and lattice type ( N 0 ( ε k ) ): – χ ( T, J/D ) ⇒ f χ,nc ( T/T 0 ) – ρ ( T, J/D ) ⇒ f ρ,nc ( T/T 0 ) – A ( ω, T, J/D ) ⇒ f A,nc ( ω/T 0 , T/T 0 ) • numerically, scaling found to persist up to at least 100 T 0

Resistivity scaling: Kondo insulator n c =1.0 ( δ =0%) 4 10 3 10 J=0.275 J=0.30 J=0.325 2 J=0.35 10 J=0.375 ρ (T) J=0.40 1 10 ρ (T<< ∆ g )=A exp( ∆ g /T) 0 10 -1 10 -2 -1 0 1 10 10 10 10 T/ ∆ g • Temperature scale: T K = ∆ g = ∆ ind

Resistivity scaling: 5% doped Kondo insulator n c =0.95 ( δ =5%) 3.0 J=0.225 2.0 J=0.25 J=0.275 J=0.375 ρ (T) J=0.3 J=0.4 1.0 0.0 0 20 40 60 80 100 T/T 0 • Scaling w.r.t. T/T 0 up to T ≈ 100 T 0 • Incoherent metal region with linear T resistivity for T ≈ T 0 .

Resistivity scaling: 50% doping: heavy fermion metal n c =0.50 ( δ =50%) 1.0 0.8 J=0.3 J=0.4 0.6 ρ (T) 0.4 0.2 0.0 0 20 40 60 80 100 T/T 0 • Scaling w.r.t. T/T 0 up to T ≈ 100 T 0 • Typical paramagnetic heavy fermion metal , e.g. CeAl 3 • T max ≈ 5 − 10 T 0 ≪ T K is not a low energy scale. It is temperature at which lattice Kondo resonance vanishes on increasing T (cf. Hubbard model)

Related work: DMFT(NRG): transport crossovers in organic conductors 80 0.3 300 bar W=3543 K 1.1 0.9 1.0 600 bar W=3657 K U=4000 K 0.8 c c P (T) P (T) 70 1 2 700 bar W=3691 K 0.4 * ρ(Τ) T 1500 bar W=3943 K Met max * (d σ /dP) 10 kbar W= 5000 K T Semiconductor 60 0.7 Ins max 0.2 Bad metal ρ (T) 50 ρ ( Ω .cm) T (K) 0.2 40 Mott U/W=0.6 insulator 0.1 30 20 Fermi liquid A.F 0 insulator 0 0.0005 0.001 0.0015 10 0.0 0 100 200 300 400 500 600 700 800 2 /W 2 T 0 50 100 150 200 250 300 P (bar) T (K) Experiments: Limelette et al. cond-mat/0301478. Theory: A. Georges, S. Florens, T.A.C. cond-mat/0301478. DMFT(NRG) results, T.A.C. cond-mat/0301478 & upublished. • Low energy Fermi liquid scale T 0 = zD (HWHM of QP peak) • ρ ( T ) ∼ A T 2 , A ∼ 1 / ( T 0 ) 2 for T ≪ T 0 = zD • Collapse of QP peak on scale T ∼ T 0 , loss of FL coherence • Large ρ for U > W and T > T 0 (scattering from local moments) • Small ρ for U ≪ W and T > T 0 (no local moments)

Interpretation • Far from Kondo insulating state, three temperature ranges: – T ≫ T K ≫ T 0 : single-ion Kondo behaviour – T 0 ≪ T ≤ T K : lattice coherence sets in, T K relevant scale; protracted or two-stage screening of moments (Jarrell) ? – T ≪ T 0 : Fermi liquid coherence sets in (lattice Kondo scale) • For fixed n c , scaling of ρ ( T ) w.r.t. T/T 0 up to T ≈ 100 T 0

Conclusions and open questions 1. DMFT(NRG): allows calculation of photoemission spectra, optical conductivities, resistivities (thermodynamics ?) 2. Spectra of Kondo Lattice show two low energy scales T 0 and T ∗ = T K 3. No clear signature of T K in most quantities. 4. Clear signature of T 0 in all quantities ( e.g. ρ ( T ) ). 5. common features in ρ ( T ) for Kondo Lattice and U ∼ W Hubbard models stems from similar physics: incoherent scattering from “local” moments at T ≫ T 0 and Fermi liquid coherence resulting from formation of singlets (Kondo effect) at T ≪ T 0 .