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

Transport and optical properties

• f 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• T. A. C., N. Manini, JLTP 2002 & unpublished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

H =

• k,σ

ǫk c†

k,σck,σ + J S0 · s0

• One low energy scale:

– TK = f(J/D) – Fermi liquid scale T0 = TK

• universal scaling functions:

– ρ(T, J/D) ⇒ fρ(T/TK) – A(ω, T, J/D) ⇒ fA(ω/TK, T/TK)

10-3 10-2 10-1 100 101 102 103 104 105

T/T0 0.2 0.4 0.6 0.8 1 ρ(T)

T0: χ(0)=1/4T0

Motivation:experiments, e.g. Andres et al 1975

• Fermi liquid coherence scale T0 ≈ 3K.
• Tmax = 35K ≈ 10T0. However, Tmax = TK !
• In fact TK generally absent in ρ(T), σ(ω, T).

Kondo Lattice Model

H =

• k,σ

ǫk c†

k,σck,σ +

• j

J Sj · sj Solve for paramagnetic solutions by DMFT(NRG) on a Bethe-Lattice : N0(εk) = 2 πD2

• D2 − εk2

Consider c-electron fillings 0 < nc < 1 (hole dopings 0 ≤ δ ≤ 1). Calculate :

• Σσ(ω, T), c-electron self-energy
• ρc(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)

10-4 10-3 10-2 10-1 100 101 ω/D 10 20 30 40 Im[χ’’(ω,T=0)] T0 T* J=0.4

nc=0.6 (δ=40%) nc=0.65 (δ=35%) nc=0.7 (δ=30%) nc=0.75 (δ=25%) nc=0.8 (δ=20%) nc=0.85 (δ=15%) nc=0.9 (δ=10%) nc=0.95 (δ=5%) nc=0.98 (δ=2%) nc=0.991 (δ=0.9%) nc=0.995(δ=0.5%)

• Fermi lquid scale T0, discernible in χ for all δ > 0.
• Single-ion Kondo scale T ∗ = TK, descernible in χ for δ < 20%

Low energy scales in spectra

50

ω/T0

0.2 0.4 0.6

T/T0=10.23 T/T0=4.53 T/T0=2.08 T/T0=0.93 T/T0=0.41 T/T0=0.18

ρc A

nc=0.8

T

*

−50 50 100

ω/T0

0.2 0.4 0.6

T/T0=12.82 T/T0=5.71 T/T0=2.55 T/T0=1.14 T/T0=0.51 T/T0=0.22

ρc A

nc=0.4

T

*

• T ∗ = TK discernible in f-electron spectrum A(ω) only for δ < 20%
• T ∗ = TK allways discernible in c-electron spectrum ρc(ω).

Suggests tunneling measurement to obtain T ∗ = TK from local c-electron DOS.

• T0 sets T-dependence of A(ω = 0, T), ρc(ω = 0, T).
• T0/TK → 0, nc → 0 (Pruschke et al. Anderson Lattice, PRB 1999)

Comparison of Spectra with Photoemission

−0.2 −0.1 0.1 E [eV] Intensity [arb.units]

photoemission (Moore et al.) Kondo lattice

YbInCu4 T=12 K T0=400 K

−0.2 −0.1 0.1 E [eV] Intensity [arb. units]

T= 50K T= 90K T=130K T=150K

YbInCu4 T0=30 K

• YbInCu4: volume collapse at Tv = 42K. High T0 = 400K phase for T < Tv.

Low T0 = 30K phase for T > Tv. Kondo system with 14 − nf = 0.85 − 0.96.

• Single crystals, ∆E = 25meV FWHM resolution
• Lineshape and T-dependent intensity consistent with KL scenario.

Optical conductivity: Kondo insulator

10

• 2

10

• 1

10 10

1

10

2

ω/∆g 0.0 2.0 4.0 6.0 8.0 σ(ω,T) 5.4 2.5 1.1 0.49 0.20 0.08 ∆ind ∆dir nc=1.0 (δ=0%)

µ ∆ind=∆g Ek

+

Ek

• ∆dir
• No Drude peak as T → 0.
• T = 0 threshold set by indirect gap ∆ind = ∆g = TK (see Logan’s talk).
• T-dependence set by T ∗ = TK = ∆ind
• mid-infrared peak; transitions across quasiparticle bands

Optical conductivity: δ > 0

10

• 1

10 10

1

10

2

10

3

ω/T0 0.0 1.0 2.0 3.0 4.0 σ(ω,T)

T/T0 = 21.5 9.5 4.2 1.8 1.2 0.6 0.2 ∆dir nc = 0.95, (δ=5%), ∆dir/T0 = 110

10

• 1

10 10

1

10

2

10

3

10

4

ω/T0 0.0 1.0 2.0 3.0 4.0 σ(ω,T) T/T0 = 36.9

16.5 7.4 3.3 0.3 0.7 1.5 ∆dir nc = 0.5 (δ=50%), ∆dir/T0 = 765

• Drude peak; transitions within E−

k . Develops for T < T0.

• Scale for T-dependence set by T0
• 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 T0 = T0(nc) seen for all δ > 0 in all quantities – single-ion Kondo scale TK = TK(nc) 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 nc and lattice type (N0(εk)):

– χ(T, J/D) ⇒ fχ,nc(T/T0) – ρ(T, J/D) ⇒ fρ,nc(T/T0) – A(ω, T, J/D) ⇒ fA,nc(ω/T0, T/T0)

• numerically, scaling found to persist up to at least 100T0

Resistivity scaling: Kondo insulator

10

• 2

10

• 1

10 10

1

T/∆g 10

• 1

10 10

1

10

2

10

3

10

4

ρ(T)

ρ(T<<∆g)=A exp(∆g/T)

J=0.275 J=0.30 J=0.325 J=0.35 J=0.375 J=0.40 nc=1.0 (δ=0%)

• Temperature scale: TK = ∆g = ∆ind

Resistivity scaling: 5% doped Kondo insulator

20 40 60 80 100

T/T0 0.0 1.0 2.0 3.0 ρ(T) J=0.225 J=0.25 J=0.275 J=0.375 J=0.3 J=0.4

nc=0.95 (δ=5%)

• Scaling w.r.t. T/T0 up to T ≈ 100T0
• Incoherent metal region with linear T resistivity for T ≈ T0.

Resistivity scaling: 50% doping: heavy fermion metal

20 40 60 80 100

T/T0 0.0 0.2 0.4 0.6 0.8 1.0 ρ(T) J=0.3 J=0.4 nc=0.50 (δ=50%)

• Scaling w.r.t. T/T0 up to T ≈ 100T0
• Typical paramagnetic heavy fermion metal , e.g. CeAl3
• Tmax ≈ 5 − 10T0 ≪ TK 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

100 200 300 400 500 600 700 800 10 20 30 40 50 60 70 80

A.F insulator

Semiconductor Bad metal Fermi liquid Mott insulator

P (bar)

T (K)

P

(T) P

(T) T

ρ(Τ)

T

(dσ/dP)

50 100 150 200 250 300 0.0 0.1 0.2 0.3

U=4000 K ρ (Ω.cm) T (K) 300 bar W=3543 K 600 bar W=3657 K 700 bar W=3691 K 1500 bar W=3943 K 10 kbar W= 5000 K

0.0005 0.001 0.0015 T

2/W 2

0.2 0.4 ρ(T)

U/W=0.6 0.7 0.8 0.9 1.0 1.1

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 T0 = zD (HWHM of QP peak)
• ρ(T) ∼ A T 2, A ∼ 1/(T0)2 for T ≪ T0 = zD
• Collapse of QP peak on scale T ∼ T0, loss of FL coherence
• Large ρ for U > W and T > T0 (scattering from local moments)
• Small ρ for U ≪ W and T > T0 (no local moments)

Interpretation

• Far from Kondo insulating state, three temperature ranges:

– T ≫ TK ≫ T0: single-ion Kondo behaviour – T0 ≪ T ≤ TK: lattice coherence sets in, TK relevant scale; protracted or two-stage screening of moments (Jarrell) ? – T ≪ T0: Fermi liquid coherence sets in (lattice Kondo scale)

• For fixed nc, scaling of ρ(T) w.r.t. T/T0 up to T ≈ 100T0

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 T0 and T ∗ = TK
• 3. No clear signature of TK in most quantities.
• 4. Clear signature of T0 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 ≫ T0 and Fermi liquid coherence resulting from formation of singlets (Kondo effect) at T ≪ T0.