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

• Motivation, introduction, problem setting
• Microscopic description
• Thermopower and entropy
• Conclusions
• V. Zlatić, Institute of Physics,

Zagreb With R. Monnier, J. Freericks, K. Becker

• 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. Motivation

• 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/ϒT) is almost constant (q ~ 1). Problem setting

p

P.Link et al., Physica B 225, 207 (1996)

Schematic diagram of the thermopower of Ce compounds Theory should explain:

• Large positive and

negative values of S(T).

• Typical shapes (a) - (e)
• Changes in shape with

pressure or doping.

Type (d) Type (c) Type (a)

• Large values of S(T)
• Typical shapes

Some Ce examples Type (a) and (b) systems are magnetic. Type (d) is non-magnetic.

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

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

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

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

4 8 12 0.0 0.5 1.0 1.5

1 10 100 300 50 100

A (µΩ cm/K

2)

p (GPa)

TN TC TL TN 5.7 p (GPa) 11 8 0.9 3.4

CeRu2Ge2

ρ ⊥ c ρmag (µΩcm) T (K)

Resistivity changes due to pressure in CeRu2Ge2

• characteristic energies are similar as before.

4 8 12 0.0 0.5 1.0 1.5

1 10 100 300 50 100

A (µΩ cm/K

2)

p (GPa)

TN TC TL TN 5.7 p (GPa) 11 8 0.9 3.4

CeRu2Ge2

ρ ⊥ c ρmag (µΩcm) T (K)

Resistivity changes due to pressure in CeRu2Ge2

• characteristic energies are similar as before.

4 8 12 1 10 100 500 TK TS TN Tc TL TK α A

• 0.5

CeRu2Ge2

T (K) p (GPa)

Phase diagram derived from transport data

4 8 12 1 10 100 500 TK TS TN Tc TL TK α A

• 0.5

CeRu2Ge2

T (K) p (GPa)

AFM Phase diagram derived from transport data

4 8 12 1 10 100 500 TK TS TN Tc TL TK α A

• 0.5

CeRu2Ge2

T (K) p (GPa)

AFM Kondo

VF

Phase diagram derived from transport data

4 8 12 1 10 100 500 TK TS TN Tc TL TK α A

• 0.5

CeRu2Ge2

T (K) p (GPa)

AFM Kondo

FL VF

Phase diagram derived from transport data

SC I

4 8 12 1 10 100 500 TK TS TN Tc TL TK α A

• 0.5

CeRu2Ge2

T (K) p (GPa)

Kondo

FL afm

SC II

VF

Phase diagram derived from transport data How to explain these features? AFM

Modeling unstable 4f ions 4f0 4f1 V 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. Two possible local configurations. W Ef

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

U → ∞

Hd = Σij,σ(tij − µδij)d†

iσdjσ

Hf = Σl,η(fη − µ)f †

lηflη − UΣl,σ>ηf † lσflσf † lηflη

Hfd = 1 √ N Σk,l,σ(Vkc†

kσflσ + h. c.)

Infinite correlation 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?

A(ω) = − 1 π Im Gf(ω + i0+)

Spectral function Green’s function

Gf(z) = 1 z − (f − µ) − Γ(z) − Σ(z)

Transport relaxition time

1 τ(ω) = cNπV 2A(ω)

Lij = σ0 ∞

−∞

dω

• −d

f(ω) dω

• τ(ω)ωi+j−2

Transport integrals

Self-consistent NCA solution: Hybridization parameter Bosonic Green’s function Fermionic Green’s function Fermionic self energy Bosonic self energy

G0(ω) = 1 ω − 0 − Π(ω) Π(ω) =

• ∆

n∆

f

• d G∆

f (ω + )Γ()f()

Σ(ω) =

• d G0(ω + )Γ(−)f()

G∆

f (ω) =

1 ω − ∆

f − Σ(ω)

Γ(ω) =

• V 2()ρc( − ω)

Additional self-consistent loop for spectral functions: B-spectral function F-spectral function Self-consistency eqns. Partition function

b(ω) = |G0|2

• d a∆(ω + )Γ(−)f()

a∆(ω) = |G∆|2

• d b(ω + )Γ()f()

Z = e−βω0

• dω[b(ω) +
• ∆

a∆(ω)] a∆() = e−β(−ω0) πZ ImG∆() b() = e−β(−ω0) πZ ImG0()

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

P>0 E0 < 0 Ef - E0 P=0 4f1 E0 > 0 W Ce summary of calculations: Pressure increases Γ and reduces nf. E0 and Ef are measured with respect to μ 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. Ef Ef - E0 Ef

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

TS TS

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

TS TS

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

TS TS

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

TS TS

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

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

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

Eu ion is found in the 4f7 or 4f6 Hund’s rule state. Configurational fluctuations give rise to Kondo effect 4f7 4f6

Ef

Eu 3+ Eu 2+ Configurational splitting is Ef conduction electron Sz=7/2 J=0 Jz =7/2 Jz=7/2 Eu2Cu2(SixGe1-x)2 example

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

Summary of Yb calculations: Pressure shifts Ef and reduces nf. Γ is unchanged. 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. Ef - E0 P>0 W

4f13

E0= 0 Ef - E0 P=0

4f13.5

E0> 0

NCA results for Yb ions: Experiment Chemical pressure effects

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

Δ>Γ

Transport is defined by the spectrum of elementary excitation

T≈Δ T≈Δ/2 T≈T0 T>T0 T=2 K T=200 K T=41 K T=700 K

(small hybridization)

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

T=2 K: T=200 K: T=700 K:

T0

zoom x10 zoom x 100 Δ>Γ

T=2 K: T=200 K: T=700 K:

T<T0 T>T0

Δ < Γ < 2Δ

Spectrum of elementary excitation T0 T0

zoom x10 zoom x 100 Δ>Γ T=41 K T=2 K T=200 K T=700 K zoom x10

T≈Δ T≈Δ T=200 K

Γ=2Δ Γ>2Δ

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

T0 T0

Calculated phase diagram

Kondo Calculated phase diagram

Kondo

FL

Calculated phase diagram

Kondo

FL VF Calculated phase diagram

Kondo

FL VF

AFM

Calculated phase diagram

• Thermopower in Ce, Eu, and Yb intermetallics can be understood

from the fixed point analysis of the effective single impurity Anderson model.

• S(T) depends on the number of electrons and the relative magnitude
• f Γ/Ef and Γ/∆.
• Shape of S(T) follows from the redistribution of the spectral weight

within the Fermi window.

• Pressure changes Ef in Yb and Γ in Ce.
• Combining the NCA and the Fermi liquid approximations we find

solution for any T. Summary of NCA thermopower calculations

Conclusions

• Above the coherence temperature ( Tc~T0 ), we do not see

significant effects due to the QCP.

• Single ion Kondo effect seems to be doing all the work. Effective f-

degeneracy changes with T and P. Local environment is most important (CF splitting, ligands).

• High-concentration data and low-concentration data are not related

by a simple scaling law. Shape of S(T) changes with concentration (chemical pressure).

Thermopower (α) versus entropy (sN)

j=<e-βH j >/<e-βH> current q=<e-βH q >/<e-βH> heat current H contains inter-particle interactions and couplings to electrical and gravitational fields. Gradient expansion leads to transport equations (Luttinger) j = -σ∇φ-σα∇T q = (φ+Π)j-κ∇T eNA(α/SN)=NA/N=qN

} ⇒

Seebeck effect: current generation by heat flow Seebeck coefficient: transport eq. for j=0 α=∆V/∆T

T+ΔT q T ΔV≠0 ∇T ∇T j To

Peltier effect: heat transfer by current flow Peltier coefficient: transport eq. for ∇T=0

Π(T) = q j

T+ΔT q T ΔV≠0 j To ∇T ∇T

Onsager: α = Π/T

T q T j Q,e Q,e Stationary state in isothermal condition (Peltier experiment): dQ/dt = -div q = 0 div q= Tj∇α Integrating over the interface: qs-ql=T(αs- αl)j q has a discontinuity at the interface. j is always continuous. Stationary flow: j = nev q = Qv v - drift velocity n - particle density

α = q jT

T q T j Q,e Q,e Stationary state in isothermal condition (Peltier experiment): dQ/dt = -div q = 0 div q= Tj∇α Integrating over the interface: qs-ql=T(αs- αl)j q has a discontinuity at the interface. j is always continuous. Stationary flow: j = nev q = Qv v - drift velocity n - particle density

α = q jT = Q/T en

dQ/dt = -div q = 0 div q= Tj∇α Integrating over the interface: qs-ql=T(αs- αl)j q has a discontinuity at the interface. j is always continuous. Stationary flow: j = nev q = Qv v - drift velocity n - particle density T q T j Q,e Q,e Stationary state in isothermal condition (Peltier experiment):

eNA α γT = NA N = qN

Analysis of transport equation

• Free electrons: qN=1
• Anderson model: qN~1
• Falicov-Kimball model: qN=1
• Periodic Anderson model (NFL) α ~ sN

(N/NA) is proportional to the Fermi volume of charge carriers

Conclusions

• The ratio qN=(NAe)(α/ϒT) calculated from transport equation is

almost constant and qN ~ N/NA, where NA/N is the Fermi volume

• f the interacting system.
• But qN is not a universal number, as heat and charge currents are

not always proportional to the particle density.

• qN refers to α and ϒ of the charge carriers. Experimental data

should be corrected for the phonon, magnon, etc. contributions.

• Reliable model calculations of the thermopower for interacting

electrons on the lattice are badly needed.