Low temperatures behavior: S(T) > 0 for Ce ions Arrows mark - PowerPoint PPT Presentation

Theory of thermoelectricity in intermetallic compounds with Ce, Eu, and Yb ions V. Zlati ć , Institute of Physics, Zagreb • Motivation, introduction, problem setting • Microscopic description • Thermopower and entropy • Conclusions With R. Monnier, J. Freericks, K. Becker

Motivation • Thermoelectricity has been a fasinating subject for a long time. It unifies thermodynamics, electrodynamics, quantum mechanics. • Thermopower S(T) of Ce, Eu, and Yb ions could be quite large. Is there a potential for applications? • S(T) is often a non-monotonic and complicated function of temperature. • Functional form varies with pressure but only a few typical shapes appear. The shape of S(T) correlates with magnetic character of 4f ions.

Problem setting • Why does the temperature dependence of S(T) varies so much in different systems? What determines S(T)? • Is the low-temperature behavior really universal? • The thermopower slope S/T and specific heat coefficient ϒ =CV/T can differ in various systems by orders of magnitude. But the ratio q=(NAe)( S / ϒ T ) is almost constant (q ~ 1).

Schematic diagram of the thermopower of Ce compounds Theory should explain: • Large positive and negative values of S(T). p • Typical shapes (a) - (e) • Changes in shape with pressure or doping. P.Link et al., Physica B 225 , 207 (1996)

Some Ce examples • Large values of S(T) Type (d) • Typical shapes Type (a) and (b) systems Type (c) are magnetic. Type (d) is non-magnetic. Type (a)

Low temperatures behavior: S(T) > 0 for Ce ions Arrows mark specific heat coefficient. q is almost universal

S/T γ

Universal low-temperature behavior

Case study CeRu2Ge2 (d) (c) (b) Shape evolution of S(T): Type (a) (a) for P < 2 GPa (b) above 2 GPa, (c) above 4 GPa, (d) above 8 GPa.

Case study CeRu2Ge2 (d) Physical scales revealed by distinct points of S(T). Low-T maxima: TK(P) (c) (b) Shape evolution of S(T): Type (a) (a) for P < 2 GPa (b) above 2 GPa, (c) above 4 GPa, (d) above 8 GPa.

Case study CeRu2Ge2 (d) Physical scales revealed by distinct points of S(T). Low-T maxima: TK(P) High-T maxima: TV(P) (c) (b) Shape evolution of S(T): Type (a) (a) for P < 2 GPa (b) above 2 GPa, (c) above 4 GPa, (d) above 8 GPa. (AFM transition not shown)

Resistivity changes due to pressure in CeRu2Ge2 - characteristic energies are similar as before. 1.5 CeRu 2 Ge 2 100 ρ ⊥ c 1.0 2 ) p (GPa) A ( µ Ω cm/K 8 5.7 ρ mag ( µ Ω cm) 0.5 11 3.4 50 0.0 0 4 8 12 p (GPa) 0.9 T N T N T L T C 0 1 10 100 300 T (K)

Resistivity changes due to pressure in CeRu2Ge2 - characteristic energies are similar as before. 1.5 CeRu 2 Ge 2 100 ρ ⊥ c 1.0 2 ) p (GPa) A ( µ Ω cm/K 8 5.7 ρ mag ( µ Ω cm) 0.5 11 3.4 50 0.0 0 4 8 12 p (GPa) 0.9 T N T N T L T C 0 1 10 100 300 T (K)

Phase diagram derived from transport data 500 T S CeRu 2 Ge 2 100 T K -0.5 T K α A T (K) 10 T N T c T L 1 0 4 8 12 p (GPa)

Phase diagram derived from transport data 500 T S CeRu 2 Ge 2 100 T K -0.5 T K α A T (K) 10 T N AFM T c T L 1 0 4 8 12 p (GPa)

Phase diagram derived from transport data 500 T S CeRu 2 Ge 2 100 Kondo T K -0.5 T K α A VF T (K) 10 T N AFM T c T L 1 0 4 8 12 p (GPa)

Phase diagram derived from transport data 500 T S CeRu 2 Ge 2 100 Kondo T K -0.5 T K α A VF T (K) 10 T N AFM FL T c T L 1 0 4 8 12 p (GPa)

Phase diagram derived from transport data 500 T S CeRu 2 Ge 2 100 Kondo T K -0.5 T K α A VF T (K) 10 T N AFM FL T c T L afm SC I SC II 1 0 4 8 12 p (GPa) How to explain these features?

Modeling unstable 4f ions Configurational splitting is Ef (say, between 3+ and 2+) Configurational mixing is due to hybridization V. Mixing parameter is Γ =V2n(EF)/Ef Intra-configurational splittings are due to the CEF or H. 4f1 4f0 V Ef W Two possible local configurations.

Sequence of energy scales: 4f2 states states (in Ce) not admitted: Uff >> W Configurational splitting: Ef < W << Uff CF splitting: Δ << Ef f-d mixing: Γ << Ef but Γ < Δ or Γ > Δ Properties depend on g= Γ / π |Ef, ∆ / Γ But we are dealing with a many-body system, properties depend on the particle numbers nc(T) and nf(T)

Anderson lattice model H d = Σ ij, σ ( t ij − µ δ ij ) d † i σ d j σ H f = Σ l, η ( � f η − µ ) f † l η f l η − U Σ l, σ > η f † l σ f l σ f † l η f l η 1 Σ k ,l, σ ( V k c † H fd = k σ f l σ + h. c.) √ N Infinite correlation U → ∞ Fixed points of the periodic model not well understood.

Poor man’s solution • Neglect coherent scattering on 4f ions. • Impose local charge conservation at each f-site. ci=1 ntot = nc(T) + ci nf(T) Thermoelectric properties depend on g= Γ / π |Ef| and ∆ / Γ

What is needed? 1 Green’s function G f ( z ) = z − ( � f − µ ) − Γ ( z ) − Σ ( z ) A ( ω ) = − 1 Spectral function π Im G f ( ω + i 0 + ) � ∞ � f ( ω ) � − d Transport integrals τ ( ω ) ω i + j − 2 L ij = σ 0 d ω d ω −∞ 1 Transport relaxition time τ ( ω ) = c N π V 2 A ( ω )

Self-consistent NCA solution: � Hybridization parameter V 2 ( � ) ρ c ( � − ω ) Γ ( ω ) = 1 Bosonic Green’s function G 0 ( ω ) = ω − � 0 − Π ( ω ) 1 G ∆ Fermionic Green’s function f ( ω ) = ω − � ∆ f − Σ ( ω ) � Fermionic self energy Σ ( ω ) = d � G 0 ( ω + � ) Γ ( − � ) f ( � ) � Bosonic self energy � n ∆ d � G ∆ Π ( ω ) = f ( ω + � ) Γ ( � ) f ( � ) f ∆

Additional self-consistent loop for spectral functions: b ( � ) = e − β ( � − ω 0 ) B-spectral function Im G 0 ( � ) π Z a ∆ ( � ) = e − β ( � − ω 0 ) F-spectral function Im G ∆ ( � ) π Z � Self-consistency eqns. b ( ω ) = | G 0 | 2 d � a ∆ ( ω + � ) Γ ( − � ) f ( � ) � a ∆ ( ω ) = | G ∆ | 2 d � b ( ω + � ) Γ ( � ) f ( � ) � Partition function � Z = e − βω 0 d ω [ b ( ω ) + a ∆ ( ω )] ∆

NCA calculations for CeRu2Ge2 (initial parameters at ambient pressure) • Semielliptic conduction band of W=4 eV • Initial ground CF doublet at Ef = - 0.7 eV • Excited CF quartet at Ef + Δ =0.693 eV • Initial hibridization width Γ =0.06 eV • 0.93 particles per effective ‘spin’ channel • Chemical potential adjusted at each T and P • Pressure changes hybridization

Ce summary of calculations: Pressure increases Γ and reduces nf. E0 and Ef are measured with respect to μ P=0 P>0 E0 < 0 E0 > 0 W Ef - E0 Ef Ef - E0 Ef 4f1 For each Γ we shift μ so as to conserve ntot . E0 and Ef are shifted by δμ but Ef - E0 is unchanged. This procedure makes Ce less magnetic with applied pressure.

Changing the width of the f-state (with pressure) Type (c) Type (d) (b) Type (a) Low pressure High pressure

Changing the width of the f-state (with pressure) T S T S

Changing the width of the f-state (with pressure) T S TK T S

Changing the width of the f-state (with pressure) T S TK T S

Changing the width of the f-state (with pressure) T S TK T S

Comparing the NCA solution with CeRu2Ge2: low pressure data. Theory Experiment

Comparing the NCA solution with CeRu2Ge2: high pressure data. T T Experiment Theory

Eu2Cu2(SixGe1-x)2 example Eu ion is found in the 4f7 or 4f6 Hund’s rule state. Sz=7/2 Jz=7/2 Configurational splitting is Ef J=0 4f6 E f Configurational fluctuations give rise to Kondo effect Jz =7/2 4f7 conduction electron Eu 2+ Eu 3+

Thermopower of Eu2(Si1xGe1-x)2: comparison of NCA results with experiment. Experiment Theory

Summary of Yb calculations: Pressure shifts Ef and reduces nf. Γ is unchanged. P=0 4f13 Ef - E0 W Ef - E0 4f13.5 E0= 0 P>0 E0> 0 For each Ef(P) we shift μ so as to conserve ntot. Ef and E0 change with pressure. Yb gets additional f-holes and is more magnetic under pressure.

NCA results for Yb ions: Experiment Chemical pressure effects

Inadequacy of the single-ion description Electrical resistance f-particle number: Transport and thermodynamics should be related to the fixed points of the model!

Transport is defined by the spectrum of elementary excitation Δ > Γ (small hybridization) T= 700 K T ≈ Δ T= 200 K T ≈ Δ /2 T= 41 K T > T0 T ≈ T0 T= 2 K

Zooming in Δ > Γ case: Properties of A( ω ) in the Fermi window (± 2kBT) determine S(T) Δ > Γ T=700 K: T ≈ Δ T=700 K T ≈ Δ /2 T=200 K T>T0 T=41 K zoom x10 T ≈ T0 T=2 K T=200 K: The NCA Kondo scale T0 is defined zoom x 100 T0 by the low-temperature peak of A( ω ). T=2 K:

Spectrum of elementary excitation Δ > Γ Δ < Γ < 2 Δ T=700 K: T<T0 zoom x10 zoom x10 T>T0 T0 T=200 K: zoom x 100 T0 T=41 K T=2 K T=2 K: T=700 K T=200 K

Spectrum of elementary excitation Γ =2 Δ Γ >2 Δ larger hybridization very large hybridization T=2 K T0 T=41 K T ≈ Δ T=200 K T0 T ≈ Δ For Γ >2 Δ the low-energy CF peaks disappear. T0 jumps to new values.

Calculated phase diagram

Calculated phase diagram Kondo