Seebeck and Nernst coe ffi cients of the heavy-electron metals - - PowerPoint PPT Presentation

Seebeck and Nernst coefficients of the heavy-electron metals

Romain Bel, Alexandre Pourret & Hao Jin (Paris) In collaboration with: Pascal Lejay, & Jacques Flouquet (Grenoble) Koichi Izawa & Yuji Matsuda (Tokyo) Daisuke Kikuchi, Yuji Aoki & Hideyuki Sato (Tokyo) Didier Jaccard(Genève)

Kamran Behnia

Ecole Supérieure de Physique et de Chimie Industrielles - Paris

Contents

• 1. Introduction to q, the thermopower-to-

specific heat ratio and its utility

• 3. The case of CeCoIn5: thermoelectricity

near a QCP

• 5. Giant Nernst effect in the ordered states of

URu2Si2 and PrFe4P12

Nernst and Seebeck coefficients in the Boltzmann picture JQ

hot cold

Remarkably relevant even in presence of strong correlation!

The Seebeck coefficient

This yields:

transport thermodynamic

If we forget the first term…

Thermopower and specific heat

Thermopower is a measure of specific heat per carrier The dimensionless ratio: is equal to –1 (+1) for free electrons (holes) Is there a correlation between S/T and γ in real metals in the zero-temperature limit?

Heavy electrons in the T=0 limit

Replotting data two decades old!

Behnia, Jaccard & Flouquet,

• J. Phys. : condens. Matter 16, 5187 (2004)

The data cluster around the two q=±1 lines!

A rigorous treatment (Miyake & Kohno, JPSJ, 74, 254 (2005) ) confirms this naïve approach

In both unitary and Born limits, q ~1 is expected!

Even in the T=0 limit, the transport term is NOT negligible!

Putting the dimensionless q under scrutiny

• Should scale inversely with the number of carriers per f.u.
• Experimentally: ~+1 for Ce-based and ~-1 for Yb-based HFs
• What about U-based HFs?

Our recent studies : the case of UPt3

Pourret et al., unpublished S/T = 1.6 ± 0.3 µV/ K2 γ = 420 mJ/K2 mol q =0.35 ± 0.07 3 f electrons per formula unit, q=0.33 expected!

Our recent studies : the case of PuCoGa5

JC Griveau et al., unpublished S/T = -0.18 ± 0.03 µV/ K2 γ = 77 mJ/ (K2 mol) q =-0.22 ± 0.03 5 f electrons per formula unit [if all f electrons are itinerant], then q=0.2 expected!

In heavy fermions with a low carrier density: q becomes very large!

• Example: the HF semi-metal CeNiSn

S/T ~ 50 µV/ K2 γ ~ 45 mJ/ (K2 mol) q= 107 (Hiess et al., ’94)

Only 10-2 of [very heavy] carriers per f.u. !

This is also the case of the ordered states of URu2Si2 and PrFe4P12!

Summary of the first part

• It is instructive to look at the

thermopower-to-specific heat ratio!

• In appropriate units, this ratio is close to

unity for a wide range of compounds!

• When this correlation breaks down,

interesting non-trivial physics may emerge!

• II. Thermoelectricity in the vicinity of a

Quantum Critical Point

• Does thermopower and specific heat

scale in the vicinity of a QCP?

Theoretical answers:

• Yes, according to Paul & Kotliar; Phys. Rev. B 64, 184414 (2001)

[S and C are both expected to diverge logarithmically!]

• Yes for a FM QCP but No for an AFM-QCP, according to Miyake

& Kohno, JPSJ, 74, 254 (2005) [S/C should become very small!]

The case of CeCoIn5

Proximity of a QCP leads to a …

Sidorov et al., PRL(2002)

The case of CeCoIn5

…a logarithmic divergence of γ …

Petrovic et al., JPCM (2001)

The case of CeCoIn5

Kim et al., PRB (2001)

…even at zero magnetic field …

The case of CeCoIn5

Nakajiama et al., JPSJ (2003)

…and a linear resistivity!

Thermoelectricity is anomalous too!

Anomalously … low!

At zero field and T ~Tc, q~0.06 !!!

The anomaly disappears in a magnetic field of 5T!

At 5T, q becomes close to unity!

Field-induced restoration of the Fermi-liquid state detected by resistivity and specific heat measurements (See Paglione et al. & Bianchi et al.; PRL 2003)

Another thermoelectric anomaly…

Giant Nernst effect in the zero-field limit!

Superconducting vortices produce a Nernst signal!

(Ri et al. 1994)

The Nernst coefficient is finite in the vortex liquid state!

Nernst effect in metals

e-

N ~ S ΘH

Ey JQ

Nernst effect in metals

Absence of charge current leads to a counterflow of hot and cold electrons: e- e-

Ey JQ JQ ≠ 0 ; Je= 0

Nernst effect in metals

Absence of charge current leads to a counterflow of hot and cold electrons: e- e-

Ey JQ JQ ≠ 0 ; Je= 0

In an ideally simple metal, the Nernst effect vanishes!(~0.1nV/KT in gold)

A case of vortex/quasi-particle duality!

In response to a thermal gradient: Vortices generate a transverse electric field! Quasi-particles generate a longitudinal electric field!

But, beware of oversimplification!

A word of caution: Ambipolar Nernst effect in NbSe2!

ü In a multi-band metal Sondheimer cancelletion is absent!

e- h+

JQ ≠ 0 ; Je =0 Jq

Bel et al. , ‘03

N ~ S ΘH

End of digression: Back to CeCoIn5!

Giant Nernst effect in the zero field limit!

Bel et al., 04

The vortex Nernst signal is owerwhelmed!

Nernst signal remains negative in the vicinity of the superconducting transition!

Vortex contribution leads to a faster collapse of the Nernst signal!

By plotting N(T) – N(Tc) S(T)/S(Tc), the vortex Nernst signal can be extracted.

The large Nernst effect fades away with increasing field!

• AT B=0T, Electric field tends to

become orthogonal to the heat current!

• The magnetic field reduces the

misalignment !

Origin of anomalous thermoelectricity in CeCoIn5 at zero field

• Where does the missing thermopower go?
• Cancellation of hole-like and electron-like contributions?
• Localization of the f-electron?
• …
• Where does the large Nernst effect come?
• Exotic excitations coupling flux to entropy?
• …

Is there simple scenario providing a common answer to these two questions? Yes!

Since transport and thermodynamics diverge…

… the scattering rate does not track the density of states!

Back to the origins:

• There is an independent way to estimate the first

term! and

Therefore:

To check this, one should compare…

Linking signs and magnitudes of four experimental quantities with NO fitting parameter!

Summary of the second part

• Both the large Nernst and the small

Seebeck coefficients in CeCoIn5 can be explained by assuming a strong energy- dependence of the elastic scattering time at zero field.

• The ratio of the Nernst coefficient to the

Hall angle scales inversely with the Fermi

• energy. The three quantities are linked by

the Mott formula.

Part III – Nernst effect and exotic electronic orders

An order of magnitude larger than in high-Tc superconductors!

How large can the Nernst coefficient of a metal become?

How large can the Nernst coefficient of a metal become?

How large can the Nernst coefficient of a metal become?

How large can the Nernst coefficient of a metal become?

How large can the Nernst coefficient of a metal become?

Can quasi-particles produce a Nernst coefficient of this size?

A crude estimation : N= 285 µV/K X ΘH X kBT/ εF

Yes!

Recall: A dilute liquid of heavy electrons in a clean metal can produce a giant Nernst signal!

The enigmatic order of URu2Si2!

Palstra et al., ‘85 Wiebe et al., ‘04 A lot of entropy is lost, but only a tiny magnetic moment appears!

Theoretical models for a « hidden order » in URu2Si2

• Barzykin & Gorkov, ’93 (three-spin correlation)
• Santini & Amoretti, ’94 (Quadrupole order)
• Kasuya, JPSJ ‘97, (U dimerization)
• Ikeda & Ohashi,’98 (d-density-wave)
• Onuki & Miyake, ’98 (CEF and Quantum fluctuations)
• Chandra, Coleman et al., ’02, (Magnetic orbital order)
• Dora & Maki, ’03 (unconventional SDW)
• Mineev & Zhitomirsky, ’04 (SDW)
• Varma & Zhu, ’05 (Helicity order)
• Kiss & Fazekas’04, (Octupolar order )

Mysterious phase transition in PrFe4P12!

Aoki et al., ‘01

Suspected to be an antiferro-qudrupolar

• rdering!

Hao et al., ‘03

The order parameters are yet to be identified, but the consequences of ordering on transport look similar!

Palstra ‘86 Sato ‘03 A drop in carrier density and an increase in carrier mean-free-path

The ordered states of URu2Si2 and PrFe4P12 share common features:

a) The carrier density is low (a gap destroys much of the

FS)

b) The mean-free-path is long (the phase space

becomes restricted in the ordered state)

c) Electrons are heavy (much more than suspected ) These features conspire to create a large Nernst effect!

• dilute
• heavy
• have a long mean-free-path

A survey of experimental evidence suggesting that carriers in the

• rdered states are :

I–Thermopower and specific heat (q is large!)

The entropy per carrier increases in the hidden-order state! URu2Si2 , q ~10 PrFe4P12, q ~20 (and ~1 when the order is destroyed!) Zero-field γ ~ 0.1 J/ (K2 mol) γ ~ 0.065 J/ (K2 mol)

II- The Hall coefficient becomes very large

Ordering leads to a multi-fold increase of the Hall

• coefficient. In the T=0 limit, it is orders of magnitude

larger than other ordinary HFs!

III- Thermal transport

Thermal conductivity is enhanced in the

• rdered state!

The Lorenz number (L= κ /σΤ)

Phonon heat transport suddenly increases! Which means: a) strong e-ph coupling b) sudden drop in electron density

L0 = (π2/3) (kB/e)2

Electronic and lattice heat conductivities!

A naive separation assuming the validity of the Wiedemann-Franz law at finite temperatures!

IV-The Hall angle

(confirms the rise in the carrier lifetime!)

ΘH ∝ l / kF

Reminiscent of the case of high- Tc cuprates

The opening of a d-wave gap, restricts the phase space and leads to an increase in the mean- free-path of the nodal quasi-particles! Seen by microwave and thermal transport!

Thus the ingredients for a large Nernst signal,

a large effective mass, a long mean-free-path and a small Ferm vactor are all present in the ordered state.

Nernst coefficient of PrFe4P12!

(Larger than any other metal!) ?

A new line on the phase diagrame

The final word

• Giant Nernst signals arise in ordered states
• f URu2Si2 and PrFe4P12.
• They may simply reflect the long lifetime

and the low density of the electrons in the

• rdered state.
• Thermoectricity as a probe of exotic states
• f correleted electrons is still largely

underexplored.