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Anisotropic Violation of the Wiedemann-Franz Law at a Quantum Critical Point
Makariy A. Tanatar,1,2*Johnpierre Paglione,2,3*Cedomir Petrovic,4Louis Taillefer1,5
A quantum critical point transforms the behavior of electronsso strongly that new phases of matter can emerge. The interactionsat play are known to fall outside the scope of the standardmodel of metals, but a fundamental question remains: Is thebasic concept of a quasiparticle—a fermion with renormalizedmass—still valid in such systems? The Wiedemann-Franzlaw, which states that the ratio of heat and charge conductivitiesin a metal is a universal constant in the limit of zero temperature,is a robust consequence of Fermi-Dirac statistics. We reporta violation of this law in the heavy-fermion metal CeCoIn5 whentuned to its quantum critical point, depending on the directionof electron motion relative to the crystal lattice, which pointsto an anisotropic destruction of the Fermi surface.
1 Département de Physique et RQMP, Université de Sherbrooke, Sherbrooke, Canada. 2 Department of Physics, University of Toronto, Toronto, Canada. 3 Department of Physics, University of California, San Diego, CA 92093, USA. 4 Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, NY 11973, USA. 5 Canadian Institute for Advanced Research, Toronto, Canada.
* These authors contributed equally to this work.
Present address: Institute of Surface Chemistry, National Academyof Sciences of Ukraine, Kyiv, Ukraine.
To whom correspondence should be addressed. E-mail: louis.taillefer{at}physique.usherbrooke.ca
Discovered in 1853, the Wiedemann-Franz (WF) law (1) has stoodas a robust empirical property of metals, whereby the thermalconductivity of a sample is related to its electrical conductivity through a universal ratio. In 1927, Sommerfeld (2) used quantummechanics, applying to electrons the new Fermi-Dirac statistics,to derive the following theoretical relation
(1)
where T is the absolute temperature, kB is Boltzmann'sconstant and e is the charge of the electron. The extremelygood agreement between the theoretical constant and the empirical value played a pivotal role inestablishing the quantum theory of solids. In 1957, Landau wenton to show that, even in the presence of strong interactions,electrons in a metal can still be described as weakly interactingfermions ("quasiparticles") with renormalized mass (3). Thisis the essence of what became known as Fermi-liquid (FL) theory,the "standard model" of metals. In the limit of zero temperature,the WF law survived unchanged because it does not depend onmass. (Eq. 1 is only a law at T0, as only in that limit is energyconserved in collisions.) It has since been shown that the WFlaw remains valid as T0 for arbitrary strong scattering, disorder,and interactions (4). It is built into the fabric of matter,valid down to the quanta of conductance, respectively equalto for heat and for charge (5).
In the past decade, however, departures from FL theory havebeen observed in d- and f-electron metals when tuned to a quantumcritical point (QCP), a zero-temperature phase transition betweendistinct electronic ground states (6). These typically showup as an anomalous temperature dependence of properties at theQCP, for example, a specific heat coefficient that never saturates,growing as C/T log(1/T)(7), and an electrical resistivity thatgrows linearly with T (8). Quantum criticality also appearsto be linked to the emergence of exotic forms of superconductivity(9–11) and nematic (12) electronic states of matter.
To determine whether Landau quasiparticles survive at a QCP,we have measured the transport of heat and charge in CeCoIn5,a heavy-fermion metal with a QCP tuned by magnetic field H.In its phase diagram (Fig. 1), the QCP is located on the borderof superconductivity and marks the end of a FL regime at H =Hc = 5.0 T, where the electrical resistivity obeys the FL form = 0 + AT2 (13). A power-law fit to the A coefficient yieldsA(H – Hc)–, with 4/3 and Hc = 5.0 ± 0.1T (13). At Hc, C/T never saturates (14). The same phenomenologyis found at the field-tuned QCP of YbRh2Si2 (with 1) (15).
Fig. 1. Phase diagram of CeCoIn5. Magnetic field-temperature phase diagram for a field perpendicular to the basal plane of the tetragonal crystal lattice (shown in inset), i.e., H || c, as determined from in-plane resistivity measurements (10). The QCP is located at H = Hc = 5.0 T (vertical red line). FL behavior, = 0 + AT2, is obeyed in the blue wedge, ending at Hc. The coefficient A, proportional to the square of the electron effective mass, diverges at Hc as a power law. Below Hc, superconductivity (SC) sets in.
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In Fig. 2, we show how the thermal and electrical resistivitiesin the T = 0 limit behave in CeCoIn5 as the field is tuned towardHc. These are extrapolations to T = 0 of the low-temperaturethermal resistivity, defined as wL0T/, and electrical resistivity, for current directions parallel (J || c) and perpendicular(J || c) to the tetragonal axis of the crystal lattice. Theraw data and their extrapolation are shown in detail in (4).For H = 10 T, far away from Hc, w(T) and (T) converge as T0for both current directions. However, very close to the QCP,for H = 5.3 T, they only converge for in-plane transport. Inother words, transport along the c axis violates the WF law,with wc extrapolating to a distinctly larger value than c asT0. In the supporting material (4), we show that extrapolationsare not needed to conclude in a violation of the WF law, asthe difference data, wc(T) – c(T) versus T, shows a rigidT-independent shift from field to field. The normalized Lorenzratio, , is also seen to approach unity at 10 T but not at 5.3 T.
Fig. 2. Violation of the WF law. Residual resistivities (extrapolated to T = 0) as a function of magnetic field, for heat (solid symbols) and charge (open symbols) transport. For in-plane transport (bottom), the two resistivities track each other as a function of field, thereby obeying the WF law at all fields. For inter-plane transport (top), the electrical resistivity c is flat as HHc, where as the thermal resistivity increases, thereby causing a violation of the WF law at the QCP, with a Lorenz number L < L0.
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Our observation of a violation of the WF law at a QCP is characterizedby three distinctive features: (i) charge conduction is essentiallyunperturbed; (ii) heat conduction becomes less efficient, sothat L < L0; and (iii) the violation is qualitatively anisotropic,present in one direction and absent in the other.
The constancy of charge conduction distinguishes this from thetwo known instances of WF violation. The first occurs in thecase of superconductivity, where immediately goes to infinityas H drops below Hc2, whereas drops gradually, so that L/L0= 0. The second instance occurs in the other limit (0), realizedin the crossover from a metal to an insulator. In this limit,a violation has been observed in cuprates as the Mott insulatingstate is approached (4). The case of CeCoIn5 is neither oneof superfluid condensation nor one of charge localization, butthat of a good metal violating the WF law. The fact that itis a downward violation, L (T0) < L0, seems inconsistentwith the possibility of neutral fermionic excitations such asthose predicted to emerge at a heavy-fermion QCP (16). Instead,we will argue that the Fermi surface is destroyed, anisotropically.
In the T0 limit, the WF law holds as long as there is a stepin the Fermi distribution function, that is, as long as a sharpFermi surface exists. This step is proportional to the renormalizationparameter Z, the defining property of a Landau quasiparticle(17). In standard FL theory, Z is a measure of how stronglythe quasiparticle mass m* is enhanced by electron interactions,with Z1/m*. The anisotropic violation seen in CeCoIn5 thussuggests that a sharp Fermi surface does not exist in the cdirection but does exist in the plane. (This is consistent withthe observation of de Haas–van Alphen oscillations inCeCoIn5 for H || c (18), because these result from coherentelectron orbits in the plane.) In other words, Z must be a functionof polar angle, Z = Z(), whereby Z = 0 over a region aroundthe "poles" (c axis) and Z >0 in a region around the "equator"(basal plane), which provides evidence of anisotropic zerosin the Z parameter of a metal due to a QCP.
An anisotropic destruction of the Fermi surface is reminiscentof what occurs in the pseudogap state of underdoped high-temperaturesuperconductors, where photoemission studies have revealed aFermi surface broken into small arcs (19), shrinking to pointsalong "nodal" directions ( = /4) as T0 (20). This angle-dependentdestruction may be caused by strong antiferromagnetic (AF) correlations;it certainly is predominant at points connected by the AF orderingvector. By analogy, the uniaxial destruction of the Fermi surfacein CeCoIn5 may be caused by spin fluctuations with a uniaxialcharacter, a scenario which is consistent with both the knownfluctuation spectrum and the finite temperature properties discussedbelow.
Having focused on the T = 0 limit, we now examine how quantumcriticality unfolds as a function of T. The electrical resistivityof CeCoIn5 at the QCP is plotted up to 15 K (Fig. 3) for bothcurrent directions. c shows a purely linear T dependence, from0.4 µ cm at 25 mK all the way to 40 µ cm at 16 K.This 100-fold increase in resistivity extends by one order ofmagnitude the range over which criticality has so far been observedto persist in any material, proving beyond doubt that the powerlaw is an intrinsic property of electrons scattered by criticalfluctuations. a is qualitatively different. Its linear T dependenceis seen only above 4 K or so, crossing over to a T3/2 dependencebelow 1K (21).
Fig. 3. Anisotropic quantum criticality. Electrical resistivity at the QCP (at H = 5.3, T Hc) for inplane (a) and inter-plane (c) current directions. c (T) remains linear over a 100-fold increase in magnitude. By contrast, a is linear only above a characteristic fluctuation temperature TSF 4 K (arrow) (18). (Inset) Thermal resistivity (wcL0T/c) at the QCP, for inter-plane transport. wc is perfectly linear down to the lowest temperature.
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A comparison of heat and charge conductivities reveals informationabout the momentum dependence of inelastic scattering. Thiswas discussed in detail in the context of our study of CeRhIn5(22), the antiferromagnetic cousin of CeCoIn5, with a Néelordering temperature TN = 3.8 K. The main piece of informationthat can be extracted directly is the characteristic temperatureTSF of magnetic fluctuations, defined as the temperature abovewhich the WF law is restored, i.e., w(T) (T). (TSF is the magneticanalog of the Debye temperature, the characteristic temperaturefor the scattering of electrons by phonons, D.) In CeRhIn5,TSF 8 K (22), in good agreement with the onset of AF correlationsseen with neutron scattering. In CeCoIn5, the same approachapplied to in-plane transport yields a field-dependent TSF, 4 K at Hc and rising to match that of CeRhIn5 at high field(21). This allows us to understand the strange behavior of a.The linear-T regime above TSF arises from fluctuations withoutpreferred spatial correlations, effectively scattering electronson the whole Fermi surface and making it uniformly "hot." (AT-linear resistivity is also found in conventional metals whenT > D.) Below TSF, the emergence of AF correlations peakedat certain q vectors in the plane leads to "hot spots" and thehigher T3/2 power law at low T. The fact that c remains lineardown to the lowest temperatures suggests that TSF0 in this case,that no interplane correlations build up, and that the Fermisurface remains hot over large regions (away from the plane).[In our phonon analog, this would imply D0, a quantum meltingof the three-dimensional (3D) solid into stacks of solid sheetsseparated by nonviscous liquid or gas.]
Whereas the T = 0 intercepts are different, the linear T dependenceof the c-axis electrical resistivity is paralleled by the thermalresistivity (Fig. 3, inset): w(T) is perfectly linear down tothe lowest T. Not only is wcT but also the slope of wc isroughly equal to that of c—aconfirmation that TSF indeedvanishes in this direction. This indicates that the usual (1– cos)vertex—which makes small-angle scatteringineffective in degrading a charge current—is not workingin CeCoIn5. It may be unimportant because inelastic scatteringis dominated by large-q processes, as one would expect fromAF fluctuations, or it may be inoperative for some special reason,as in the "Kondo breakdown" model (23).
In this instance of anisotropic quantum criticality, given thatZ and TSF exhibit the same anisotropy (Z,TSF = 0 along the caxis and Z, TSF > 0 in the basal plane), it is tempting tosuggest that (i) a vanishing energy scale, Z0, and a WF lawviolation are all related, and (ii) a good indicator for theirjoint occurrence is a linear-T resistivity. Returning to ourcomparison with cuprates, a similar connection between T andZ = 0 appears to exist there as well. Indeed, a recent measurementof the (azimuthal) anisotropy of the in-plane scattering rate() in an overdoped cuprate (24) revealed that Tat = 0, wherethe Fermi surface is eventually destroyed (at lower doping),and T2 at = /4, where it survives.
It is instructive to compare our findings with the propertiesof other materials and theories of quantum criticality. A T3/2resistivity is observed in CeIn3 near the pressure-tuned QCPwhere its AF order vanishes (6). CeIn3 is the cubic parent compoundof tetragonal CeRhIn5 and, along with theincreasein c/a ratio,the ordering temperature drops from TN = 10 K in the formerto TN = 3.8 K in the latter. However, they still have comparableTSF (assuming that in CeIn3TSFTN). CeCoIn5 encounters a furtherstretch of the c/a ratio, and long-range AF order is no longerstabilized. However, it can still be viewed as a layered versionof CeIn3, with similar in-plane correlations and scattering.In this sense, the T3/2 dependence observed in CeCoIn5 can beviewed as the result of antiferromagnetic fluctuations thatare characteristic of the parent compound. Theoretically, aT3/2 resistivity is expected for AF critical fluctuations in3D from the so-called quantum spin density wave (SDW) model(17, 25, 26). In this scenario, critical scattering is peakedat "hot spots" connected by the AF wave vectors (25). As T0,one would expect the Fermi surface to remain sharp everywhereelse, and thus the WF law to prevail, as found here for in-planecurrents.
A T-linear resistivity is observed at the composition-tunedQCP of CeCu5.9Au0.1 (7) and field-tuned QCP of YbRh2Si2 (12),where AF order is thought to disappear. [In these cases, thepower law is linear in both high-symmetry directions (15, 27).]The fact that a linear power is inconsistent with the SDW modelfor AF fluctuations in 3D prompted the proposal of a 2D version(28) and of an alternate theory, where critical scattering islocal in space and therefore present at all wave vectors (29).These scenarios would lead to a more extreme breakdown of FLtheory, because the Fermi surface is "hot" not only at certainspecific spots but everywhere. It was arguedin(8) that the specificheat data on Ge-doped YbRh2Si2, which showsa C/T that exceedsthe log(1/T) dependence at low temperature, may be an indicationof such enhanced breakdown. In CeCoIn5, the fact that it isin the direction where T that the WF law is violated is certainlyconsistent with this picture. Clearly, it would be interestingto test the WF law in YbRh2Si2.
Bringing together our findings for T0 and T> 0, a pictureof qualitative anisotropy emerges, not present in either theSDW model or the local criticality model, at least in theircurrent forms. The characteristic spin fluctuation temperatureTSF vanishes at the QCP for transport along the c axis but notin the plane. As a result, the breakdown of FL theory is extremein the c direction: cT and wcT down to the lowest temperaturesand the T = 0 Fermi surface is blurred, that is, the quasiparticleZ parameter vanishes, in regions around the c-axis direction.
A possible origin for this anisotropic criticality is an anisotropicspin fluctuation spectrum. First, an AF instability is presentin all three CeMIn5 compounds (M = Co, Rh, Ir), as shown bythe fact that magnetic ordering can be induced by Cd doping(30). Second, a magnetic field does tune the magnetism. In CeRhIn5under pressure (where it becomes in many ways more similar toCeCoIn5, e.g., by developing superconductivity with the sameTc), a magnetic field stabilizes long-range magnetic order (31,32). In CeCoIn5, it is the magnetic fluctuations that are tunedby a magnetic field (21), with TSF starting at a value equivalentto that of CeRhIn5 at high fields and then lowered to a minimumat Hc. Third, the AF fluctuations in CeCoIn5 have strongly anisotropiccharacter (33), with magnetic moments well coupled in-planebut weakly coupled interplane. This is consistent with the helicalordering of moments in CeRhIn5, commensurate in-plane and incommensuratealong the c axis. Therefore, it seems natural to link this uniaxialanisotropy with the observed anisotropy in TSF, power laws,and Z(). What is not yet known is whether a scenario of AF criticalfluctuations can indeed cause a violation of the WF law at T0.
However, the AF scenario is not the only candidate for the anisotropicquantum criticality of CeCoIn5. The "Kondo breakdown" modelproposed recently (23, 34), a type of deconfined QCP where thehybridization between conduction and f electrons goes to zero,captures some of the key signatures—absence of magneticorder in the phase diagram, strong anisotropy, multiple energyscales, and a T-linear behavior of both charge and heat resistivities.Proximity to a Pomeranchuk instability of the Fermi surfacecan also cause anisotropy in electronic liquids (7). Recentcalculations show that the transport decay rate at such a QCPhas a linear T dependence everywhere on the Fermi surface exceptat "cold" points, resulting in a T3/2 dependence of the resistivity(35).
References and Notes
1. G. Wiedemann, R. Franz, Ann. Phys.89, 497 (1853).
32. G. Knebel, D. Aoki, D. Braithwaite, B. Salce, J. Flouquet, Phys. Rev. B74, 020501 (2006). [CrossRef]
33. Y. Kawasaki et al., J. Phys. Soc. Jpn.72, 2308 (2003). [CrossRef]
34. T. Senthil, M. Vojta, S. Sachdev, Phys. Rev. B69, 035111 (2004). [CrossRef]
35. L. Dell'Anna, W. Metzner, Phys. Rev. Lett.98, 136402 (2007). [CrossRef] [Medline]
36. We thank D. G. Hawthorn, R. W. Hill, F. Ronning, and M. Sutherland for experimental assistance, and P. C. Canfield, Y. B. Kim, C. Pépin, A. M. Tremblay, A. Rosch, and M. F. Smith for useful discussions. L.T. acknowledges support from the Canadian Institute for Advanced Research, a Canada Research Chair, the Natural Sciences and Engineering Research Council of Canada, the Canada Foundation for Innovation, and Le Fonds Québécois de Recherche sur la Nature et la Technologie. Part of this research was carried out at the Brookhaven National Laboratory, which is operated for the U.S. Department of Energy by Brookhaven Science Associates (DE-AC02-98CH10886).
Received for publication 2 February 2007. Accepted for publication 19 April 2007.
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and the empirical value played a pivotal role in establishing the quantum theory of solids. In 1957, Landau went on to show that, even in the presence of strong interactions, electrons in a metal can still be described as weakly interacting fermions ("quasiparticles") with renormalized mass (3). This is the essence of what became known as Fermi-liquid (FL) theory, the "standard model" of metals. In the limit of zero temperature, the WF law survived unchanged because it does not depend on mass. (Eq. 1 is only a law at T
0, as only in that limit is energy conserved in collisions.) It has since been shown that the WF law remains valid as T
for heat and 
for charge (5). 
Present address: Institute of Surface Chemistry, National Academy of Sciences of Ukraine, Kyiv, Ukraine. 
To whom correspondence should be addressed. E-mail: 
of a sample is related to its electrical conductivity 
through a universal ratio. In 1927, Sommerfeld (
log(1/T)(
= 
, with 
4/3 and Hc = 5.0 ± 0.1 T (
L0T/
, is also seen to approach unity at 10 T but not at 5.3 T. 
), whereby Z = 0 over a region around the "poles" (c axis) and Z >0 in a region around the "equator" (basal plane), which provides evidence of anisotropic zeros in the Z parameter of a metal due to a QCP. 
= 
/4) as T
cm at 25 mK all the way to 40 µ
D.) In CeRhIn5, TSF 
(
) in an overdoped cuprate (
