Anomalous thermal transport and violation of Wiedemann-Franz law in the critical regime of a charge density wave transition

Contrary to the standard paradigm [1, 2], in more than one spatial dimension, due to the lack of perfect nesting, charge density wave (CDW) order (unlike superconducting order) only emerges for interactions greater than a critical strength. Generically, both strong electron-electron and electron-phonon interactions are involved in the “mechanism.” Among other things, the strong-coupling aspect is reflected in large values of the induced gap to $T_{c}$ ratios - in the focus material for the present study, ErTe₃, the gap associated with the primary CDW transition at $T_{CDW1}\approx 265~{}$K is $\Delta_{1}\approx 175~{}$meV, so that $2\Delta_{1}/k_{B}T_{CDW1}\approx 15$ [3, 4]. However, especially since conventional notions of weakly interacting quasi-particles and well defined phonons give a good account of the normal state at $T>T_{CDW1}$ and of the low temperature physics deep in the ordered phase, it is conventional (as is indeed common more generally in considering classical critical phenomena in metals) to adopt a phenomenological approach in which the low energy quasi-particles and the bulk of the phonons are weakly coupled to the “critical modes” associated directly with the CDW transition. This perspective is challenged by our present results.

Electrical and thermal transport measurements provide important information about the electronic structure and scattering processes in complex quantum materials. To the extent that the transport is dominated by weakly interacting (emergent) elementary excitations, the thermal conductivity can be expressed as the sum of an electronic and a phonon piece, $\kappa\approx\kappa_{el}+\kappa_{ph}$. Moreover, to the extent that the scattering processes are quasi-elastic, $\kappa_{el}$ is related to the electrical conductivity by the so-called Wiedemann-Franz (WF) law, i.e. the ratio between them, $\kappa_{el}/\sigma=L_{0}T$ is determined by the universal constant, $L_{0}=8k_{B}^{2}/\pi e^{2}\approx 2.44\times 10^{-8}$W$\Omega$K^-2. Observing this ratio gives information about how “standard” the transport is in a given electronic system, with significant violations of the WF law plausibly indicating the breakdown of the quasiparticle description.

In this paper we examine electrical and thermal transport in the layered material ErTe₃, which has two successive CDW transitions at $T_{CDW1}\approx 265~{}$K and $T_{CDW2}\approx 160~{}$K. Here $T_{CDW1}$ marks the onset of the “primary” CDW order with ordering vector $q_{CDW1}$ along the $c$-direction, while below $T_{CDW2}$ a “secondary” orthogonal CDW component with $q_{CDW2}$ along the $a$-direction appears (where $a$ and $c$ are in-plane lattice parameters in the standard space group setting for this structure type). Thus, despite the nearly tetragonal symmetry of the crystal ($a\sim c$), the phase at $T_{CDW1}>T>T_{CDW2}$ has unidirectional CDW order, while the low temperature CDW is bidirectional, but with inequivalent amplitudes in the two directions. ErTe₃ is an ideal “model system” in that it is stoichiometric and hence can be synthesized with a high degree of crystalline perfection and little disorder (in the sense both of very low residual resistivity - $\rho(T)<1~{}\mu\Omega$-cm at low $T$ - and resolution limited Bragg peaks associated with the CDW order). It boasts broad metallic bands with a plasma frequency which, depending on the global fit, is estimated in the range of $2.5~{}$eV [3] to $5.8~{}$eV [5] in the CDW state. Moreover, the effect of disorder can be explored systematically by Pd intercalation, as discussed in previous publications.[6, 7, 8]

Our primary result is that the thermal transport in a critical regime below $T_{CDW1}$ appears inconsistent with quasi-particle transport. Specifically, by assuming separate electronic and phonon contributions to $\kappa$ and that $\kappa_{el}$ satisfies the WF law, one is forced to infer an unphysically large depression of the lattice thermal transport in this regime. The regime in which we deduce the breakdown of WF law, and hence the breakdown of the quasiparticle concept, is found to be strongly asymmetric around $T_{CDW1}$, extending farther below $T_{CDW1}$ than above. In addition to this dramatic result, we also observe the following effects: i) Similar to other strongly interacting CDW systems, large anomalies are observed in the temperature derivative of the resistivity and the reflectivity (Fig. 2(b)), which, with the assumption that Fisher-Langer theory [9] applies, stands in sharp contrast to the small anomaly observed in heat capacity measurements [10]. ii) The behavior of the various linear response tensors near criticality (Figs. 1(b) and 2(a)) depends strongly on direction. As a function of decreasing $T$, the resistivity along the $a$-direction, $\rho^{a}$, has a pronounced critical singularity at $T_{CDW1}$ followed by a broad maximum before continuing to drop at lower temperatures, as has previously been discussed by Sinchenko et al.[11]. On the other hand, the critical anomaly in $\rho^{c}$ at $T_{CDW1}$ is much weaker, and neither component shows any clear non-analyticity at $T_{CDW2}$. In contrast, the thermal diffusivity has a large and sharp decrease at $T_{CDW1}$ along both the $a$- and $c$- directions, followed by a relatively faster recovery along the $c$-direction. In fact, as can be inferred from Fig. 2(a), the thermal conductivity (see also Fig. 3 below) more closely resembles the temperature derivative of the resistivity, especially the $a$-axis thermal diffusivity. iii) Thermal diffusivity in both directions show marked increases below $T_{CDW2}$, which through suppression of this effect by weak Pd-intercalation are shown to be electronic in origin (Fig. 4 below).

Samples of ErTe₃ were grown using a Te self-flux technique, which ensures purity of the melt, and produces large crystals with a high degree of structural order [12]. The material is somewhat air sensitive, and crystals must be stored in an oxygen and moisture-free environment. Crystal orientation was determined with XRD.

The heat capacity of the single-crystal samples was measured using a relaxation time technique in a Quantum Design Physical Property Measurement System (PPMS). Crystals with a mass of approximately $5~{}$mg to $10~{}$mg and flat surfaces were selected for good thermal contact with the sample platform. Data were taken in zero applied field from $1.8$ to $300~{}$K.

Measurements of resistance versus temperature were performed in a Janis Supertran-VP continuous flow cryostat. The resistivity in the $a$-$c$ plane was measured on thin rectangular crystals which had been cut with a scalpel and cleaved to expose a clean surface immediately before contacting. Crystals were cut in to bars such that current flows along the [101] axis, and contacts were attached to the surface in both longitudinal and transverse geometry, as described elsewhere [13]. In this geometry, the longitudinal and transverse contacts simultaneously measure the sum of the resistivity components along the crystal axes $\rho^{a}+\rho^{c}$ and the in-plane resistivity anisotropy $\rho^{a}-\rho^{c}$ respectively. This technique minimizes uncertainty in the resistivity anisotropy arising from geometric effects relative to using two different samples to measure $\rho^{a}$ and $\rho^{c}$ separately. Measured resistivity values were then rescaled to match existing literature [14, 13] at room temperature while preserving the same anisotropy.

Thermal diffusivity and temperature derivative reflectivity ($dR/dT$) measurements were performed using a photo-thermal microscope as described in [15]. Further details associated with the present experiment are described in the Supplemental Material section[16]. Using this apparatus, the thermal diffusivity is obtained directly, without the need to measure the thermal conductivity and specific heat separately. An advantage of this apparatus is the ability to measure the full in-plane anisotropy of the thermal diffusivity by orienting the pair of heating and probing laser spots at any arbitrary orientation with respect to the crystal axes. The mobility in the optics is further used for diagnostics of spatial uniformity of the thermal diffusivity and $dR/dT$ values. Dimensions of the measured crystal were around $400~{}\mu$m by $300~{}\mu$m by $40~{}\mu$m. To minimize surface degradation from oxygen and water, samples were cleaved using Kapton tape immediately before being placed in the photo-thermal microscope and measured.

The specific heat of two different ErTe₃ crystals over a wide temperature range is shown in Fig. 1(a) along with an ideal Debye approximation fit with a Debye temperature $T_{D}=160~{}$K. The data closely follow the Debye law both above and below each of the CDW transitions saturating at the Dulong-Petit value at high temperatures. Previous measurements of the specific heat anomaly at the CDW transition  [10] find that $\Delta c_{p}\approx 1$J/mol$\cdot$K, below the resolution of the present measurements, and surprisingly of “normal magnitude” given the large value of $2\Delta_{1}/k_{B}T_{CDW1}\approx 15$ estimated via ARPES measurements[4]. Thermal diffusivity of a different ErTe₃ crystal from the same growth batch is shown in Fig. 2(a) along with a guide to the eye curve. In contrast to the case of the specific heat, both CDW transitions produce large anomalies in the diffusivity data. In particular, at $T_{CDW1}$ the diffusivity along both axes drops by over a third from $\sim 0.21~{}$cm²/s to $\sim 0.14~{}$cm²/s.

The sharp change in diffusivity can also be compared to the much broader change in electrical resistivity. Resistivity data on the same-batch crystals is presented in Fig. 1. The resistivity was rescaled to match existing literature as described previously. This data is similar in trend to previously measured data on RTe₃ crystals [14], in particular on ErTe₃ [13, 6]. The main feature in the resistivity data of this class of materials is the strong anomaly in the direction perpendicular to the primary CDW direction ($a$-axis), as shown in Fig. 1. In contrast, the thermal diffusivity decreases along both axes. Moreover, below $T_{CDW2}$, the thermal diffusivity increases along both axes while the resistivity is little affected. However, the derivative of the resistivity $d\rho/dT$, shows effects of the CDW transitions that are more like those seen in the thermal diffusivity (Fig. 2(a)). In all cases, there is a larger change at $T_{CDW1}$ than at $T_{CDW2}$. Finally, it is illuminating to compare the critical evolution of these various DC probes with the temperature derivative of the reflectivity at $h\nu\approx 1.5~{}$eV (820 nm wavelength) as shown in Fig. 2. The derivative of the reflectivity shows a large and sharp decrease at $T_{CDW1}$ while no anomaly is visible at $T_{CDW2}$.

The effect of purposefully introduced weak disorder on the temperature dependence of the thermal diffusivity is illustrated in Fig. 4, which shows a plot of the $a$ axis diffusivity of Pd_0.003ErTe₃ in green the data on pure ErTe₃ already shown in Fig 2. $T_{CDW1}$ is slightly suppressed upon the intercalation, from $265$K to $250~{}$K, but the sharp drop in the diffusivity at the transition looks just like the one in pure ErTe₃ (as shown in [7], no disorder effects appear in the $c$-direction for this weak disorder). However, below $T_{CDW2}$ (here around $130~{}$K) there is a striking intercalation induced difference; the pronounced upturn of the diffusivity in the pure material is eliminated with Pd intercalation.

In many cases, the transport properties of metals can be successfully understood on the basis of the response of weakly interacting elementary excitations - fermionic-quasi-particles and bosonic phonons. In the last several decades, however, various transport regimes in certain “highly correlated” materials have been identified, in which the validity of such an approach has been called into question. However, it remains highly contravertial to what extent conventional quasi-particle ideas can be extended without fundamental changes in approach to strongly interacting regimes in which the quasiparticle identity is “marginally” maintained, or if entirely new paradigms (possibly involving some form of “non-Fermi liquid” or novel fractionalized quasiparticles) are needed.

One avenue of attack on this problem has been to investigate the way in which the usual quasiparticle picture can break down in the neighborhood of a quantum critical point. However, even at classical (finite $T$) critical points, the existence of non-trivial critical exponents describing the behavior in the critical regime provides clear evidence that the critical modes themselves cannot have a quasi-particle description. None-the-less, in many cases, as for instance in cases where the Fisher-Langer theory gives good account of transport anomalies, a treatment involving well-defined conduction electrons (and, presumably, phonons) weakly scattered by the critical modes, implies that the conventional mechanism of transport theory continues to apply even in the critical regime.

The dramatic failure of such an approach to give an adequate description of the thermal transport in ErTe₃, most dramatically in a $\sim 30$K range of $T$ below $T_{CDW1}$, may be a first indication of a potentially simpler context to study the breakdown of the quasi-particle paradigm. The discrepancies in the critical dependences of the thermal conductivity and the resistivity in this regime imply a complete breakdown of the WF law, the existence of independent electronic quasiparticles and phonon modes, or both. In fact, the observed behavior may more adequately described as a strongly coupled electron-phonon critical ‘soup.’

CDW formation, arising from a variety of physical mechanisms, is a relatively common phenomenon in quasi-low-dimensional materials. Although thermal transport measurements have not been performed for all such materials, they do exist for many of the well-known canonical examples, and in none of these cases has such a dramatic violation of the WF law been deduced. This raises an associated question as to why the effect should be so pronounced in this particular material system given the relative ubiquity of CDW compounds (for a recent survey of CDW systems see e.g. [24]). A wider survey of related materials might yet reveal that this effect is in fact not unique to the rare earth tritellurides, but at least for now ErTe3 occupies a unique position among known CDW compounds and presents an entirely new opportunity to explore unconventional transport properties of strongly interacting metals.

We would like to thank Alan Fang for helpful discussions. This work was supported by the Department of Energy, Office of Basic Energy Sciences, under contract no. DE-AC02-76SF00515. The photothermal apparatus was built using a grant from the Gordon and and Betty Moore Foundation through Emergent Phenomena in Quantum Systems Initiative Grant GBMF4529. JAWS was supported in part by an ABB Stanford Graduate Fellowship. AGS was supported in part by an NSF Graduate Research Fellowship (grant number DGE-1656518).