Seebeck and Nernst coe ffi cients of the heavy-electron metals Kamran Behnia Ecole Supérieure de Physique et de Chimie Industrielles - Paris 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)
Contents 1. Introduction to q, the thermopower-to- specific heat ratio and its utility 3. The case of CeCoIn 5 : thermoelectricity near a QCP 5. Giant Nernst e ff ect in the ordered states of URu 2 Si 2 and PrFe 4 P 12
Nernst and Seebeck coe ffi cients in the Boltzmann picture hot cold J Q Remarkably relevant even in presence of strong correlation!
The Seebeck coe ffi cient 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!
The data cluster around the two q= ± 1 lines! Behnia, Jaccard & Flouquet, J. Phys. : condens. Matter 16, 5187 (2004)
A rigorous treatment (Miyake & Kohno, JPSJ, 74, 254 (2005) ) confirms this naïve approach Even in the T=0 limit, the transport term is NOT negligible! In both unitary and Born limits, q ~1 is expected!
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Our recent studies : the case of UPt 3 Pourret et al., unpublished 3 f electrons per formula unit, q=0.33 expected! S/T = 1.6 ± 0.3 µ V/ q =0.35 ± 0.07 K 2 γ = 420 mJ/K 2 mol
Our recent studies : the case of PuCoGa 5 5 f electrons JC Griveau et al., per unpublished formula unit [if all f electrons are itinerant], then q=0.2 expected! S/T = -0.18 ± 0.03 µ V/ q =-0.22 ± 0.03 K 2 γ = 77 mJ/ (K 2 mol)
In heavy fermions with a low carrier density: q becomes very large! • Example: the HF semi-metal CeNiSn S/T ~ 50 µ V/ K 2 � γ ~ 45 mJ/ (K 2 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 URu 2 Si 2 and PrFe 4 P 12 !
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 CeCoIn 5 Sidorov et al., PRL(2002) Proximity of a QCP leads to a …
The case of CeCoIn 5 Petrovic et al., JPCM (2001) …a logarithmic divergence of γ …
The case of CeCoIn 5 Kim et al., PRB (2001) …even at zero magnetic field …
The case of CeCoIn 5 Nakajiama et al., JPSJ (2003) …and a linear resistivity!
Thermoelectricity is anomalous too!
Anomalously … low! At zero field and T ~T c , q~0.06 !!!
The anomaly disappears in a magnetic field of 5T! Field-induced restoration of the Fermi-liquid state detected by resistivity and specific heat measurements (See Paglione et al. & Bianchi et al.; PRL 2003) At 5T, q becomes close to unity!
Another thermoelectric anomaly… Giant Nernst e ff ect in the zero-field limit!
Superconducting vortices produce a Nernst signal! (Ri et al. 1994) The Nernst coe ffi cient is finite in the vortex liquid state!
Nernst e ff ect in metals e - J Q E y N ~ S Θ H
Nernst e ff ect in metals Absence of charge current leads to a counterflow of hot and cold electrons: J Q ≠ 0 ; J e = 0 e - e - J Q E y
Nernst e ff ect in metals Absence of charge current leads to a counterflow of hot and cold electrons: J Q ≠ 0 ; J e = 0 e - e - J Q E y In an ideally simple metal, the Nernst e ff ect 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 e ff ect in NbSe 2 ! ü In a multi-band metal Sondheimer cancelletion is absent! Bel et al. , ‘03 J Q ≠ 0 ; J e = 0 e - h + J q N ~ S Θ H
End of digression: Back to CeCoIn 5 ! Bel et al., 04 Giant Nernst e ff ect in the zero field limit!
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(T c ) S(T)/S(T c ), the vortex Nernst signal can be extracted.
The large Nernst e ff ect 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 CeCoIn 5 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 e ff ect 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 coe ffi cients in CeCoIn 5 can be explained by assuming a strong energy- dependence of the elastic scattering time at zero field. • The ratio of the Nernst coe ffi cient to the Hall angle scales inversely with the Fermi energy. The three quantities are linked by the Mott formula.
Part III – Nernst e ff ect and exotic electronic orders
An order of magnitude larger than in high-T c superconductors!
How large can the Nernst coe ffi cient of a metal become?
How large can the Nernst coe ffi cient of a metal become?
How large can the Nernst coe ffi cient of a metal become?
How large can the Nernst coe ffi cient of a metal become?
How large can the Nernst coe ffi cient of a metal become?
Can quasi-particles produce a Nernst coe ffi cient of this size? Yes! A crude estimation : N= 285 µ V/K X Θ H X k B T/ ε F Recall: A dilute liquid of heavy electrons in a clean metal can produce a giant Nernst signal!
The enigmatic order of URu 2 Si 2 ! Wiebe et al., ‘04 Palstra et al., ‘85 A lot of entropy is lost, but only a tiny magnetic moment appears!
Theoretical models for a « hidden order » in URu 2 Si 2 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 PrFe 4 P 12 ! Aoki et al., ‘01
Suspected to be an antiferro-qudrupolar ordering! 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 URu 2 Si 2 and PrFe 4 P 12 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 e ff ect!
A survey of experimental evidence suggesting that carriers in the ordered states are : � � � dilute � � � heavy � � � have a long mean-free-path
I–Thermopower and specific heat (q is large!) URu 2 Si 2 , q ~10 PrFe 4 P 12 , q ~20 (and ~1 when the order is destroyed!) γ ~ 0.065 J/ (K 2 mol) Zero-field γ ~ 0.1 J/ (K 2 mol) The entropy per carrier increases in the hidden-order state!
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