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Phonon Thermal Transport of URu2Si2: Broken Translational Symmetry and Strong-Coupling of the Hidden Order to the Lattice

A dramatic increase in the total thermal conductivity (k) is observed in the Hidden Order (HO) state of single crystal URu2Si2. Through measurements of the thermal Hall conductivity, we explicitly show that the electronic contribution to k is extremely small, so that this large increase in k is dominated by phonon conduction. An itinerant BCS/mean-field model describes this behavior well: the increase in kappa is associated with the opening of a large energy gap at the Fermi Surface, thereby decreasing electron-phonon scattering. Our analysis implies that the Hidden Order parameter is strongly coupled to the lattice, suggestive of a broken symmetry involving charge degrees of freedom.

At symmetry breaking (2 nd order) phase transitions, an "order parameter" must characterize the non-symmetric, low temperature state [1]. The heavy-fermion URu 2 Si 2 undergoes a 2 nd order phase transition at ~ 17.5 K, yet the order parameter has remained "hidden" for over 20 years. This is an example of one of the most fundamental problems in solid state physics: the symmetries and associated order parameters are far from obvious in many interesting systems. In fact, the order parameters of antiferromagnets (the staggered magnetization) and superconductors (the phase of the condensate wavefunction) were similarly "hidden" for many years. New discoveries of such nontrivial hidden-order systems, such as spin glasses, or 2D XY magnets (where the "order" is of a purely topological character) thus continually expand our understanding of symmetry in solids. It for this reason that uncovering the "Hidden Order" (HO) phase in URu 2 Si 2 is so interesting.
Almost all conceivable experimental probes have been utilized in an attempt to uncover the HO phase. Yet still, this diverse body of experimental data does not seem to convincingly converge to any existing model [2][3][4]. For example, this 2 nd order transition seems to have some characteristics of antiferromagnetic (AF) ordering: a large peak appears in the specific heat. However, the observed small staggered-moment is inconsistent with the amount of entropy associated with the transition [4], and is instead likely due to a small AF impurity-phase [5]. In addition, due to strong hybridization between the conduction electrons and nearly-localized U-5f 2 moments, neither a purely local-nor itinerant-electron model can be applied unambiguously. In fact, there is strong evidence for a Fermi-liquid comprised of Γ 5 non-Kramers doublet degrees of freedom [6,7].
We present another route towards uncovering the HO in URu 2 Si 2 via thermal conductivity (κ) measurements. Surprisingly, κ(T) undergoes a very large increase at the HO transition. Although this feature has been observed some time ago [8], and then reported again recently [9], previous authors have not been able to model this effect in a way that elucidates the HO. By studying both pure and Rh-doped compounds, we have succeeded in explicitly showing that this feature is dominated by phonons. We apply a mean-field/BCS framework to describe κ phonon whereby electron-phonon (e-ph) scattering decreases as a gap (∆) in the electronic density of states (DOS) opens at the HO transition. An unusually large ∆(0)/k B T C ratio of ~4 is needed in order for the model to best describe the data, in agreement with STM estimates of ∆(T), that is suggestive of a coupling of the order parameter to the lattice. Our analysis suggests that an itinerant model associated with translational symmetry breaking of the charge degrees of freedom is needed in order to uncover the Hidden order parameter. κ(T) and the thermal Hall conductivity, κ XY , are measured using the steady state method with two calibrated Cernox thermometers up to 15 T and down to 2 K with an absolute error of no more than 10 %. Single crystal U(Ru 1-x Rh x ) 2 Si 2 samples are grown using the Czochralski method with two separate batches [10] yielding identical results.
Rh is doped onto the Ru site at a concentration of up to 4% (x=0.04). Rh concentration is increased, this feature becomes significantly reduced until it is entirely lost at 3% Rh doping, for which AF sets in at T N . Unlike in the HO state, the AF state induced via Rh doping displays a tiny feature upon ordering. Increasing the doping to 4% destroys both HO and AF so that no feature is seen at all in κ(T). For each sample, a transition is observed where it is expected according to the known phase diagram [10].
The absolute magnitude of κ at T>T HO for each sample is very similar and within the error bars. This trend is unlike that of the electrical conductivity (σ), which noticeably increases with Rh doping. Judging from the behavior of the non-ordered 4% sample, it appears that κ(T) sharply increases at the HO transition in URu 2 Si 2 . Fig. 1b shows κ(T) for Q//H//c and Q//a in zero field for pure URu 2 Si 2 . Clearly, κ a (T)>κ c (T), in contrast to σ(T) for which σ c >σ a [11]. Interestingly, as H is applied, the peak in κ(T) is reduced in a more gradual way than the effects of doping. T HO is also gradually reduced with H//c in accordance with the H vs. T phase diagram [12].
Before investigating any implications of the observed effect for the HO, one must determine the dominant contribution(s) to κ. We argue here that the observed increase in κ is dominated by phonons rather than electrons [13], suggested by the data shown in Fig.   1. As already mentioned, the κ anisotropy and systematic change with Rh doping is opposite to that of σ. If electronic thermal conduction were important, the behavior of κ should be more strongly correlated with that of σ. In addition, κ obeys a T 2 power law behavior at low T (inset Fig. 1a), which is consistent with dominant phonon conduction in the presence of e-ph scattering [15].
The thermal Hall (Righi-Leduc) conductivity (κ XY ) is the best way of ascertaining κ E . Though the Weidemann-Franz law (WFL), κ E =L 0 σT, is often used to estimate the electronic contribution, the results of Fig. 1 imply that κ phonon is dominant and so the WFL is highly uncertain. Assuming that e --ph scattering dominates, the WFL fails because of the preponderance of small angle scattering events (phonon wavevector q<<k F ) at low, yet intermediate T, which disproportionately affect the electronic heat (as opposed to charge) current [14]. κ XY provides a direct measurement of electronic heat conduction since electrons moving perpendicular to H will be deflected in a transverse direction given by the Lorentz force, whilst phonons will not [15,16].
Our determination of κ E is shown in Fig. 2. In Fig. 2b, we have shown the raw ∆T Hall signal, measured isothermally as a function of H for T=2, 5, 10 and 20 K [17]. The thermal hall conductivity is calculated using the expression κ XY = (∇ Hall T /∇ a T)κ c [16].
∇ a T is the longitudinal T gradient produced by a Q(//a) identical to that which produces Fig. 2 (c), approximately increasing linearly from zero at low H.
The longitudinal κ E can be derived from the κ XY (H) plot using the assumption: This relationship is more robust than the WF law [16]. The quantity σ XY /σ c =ρ XY /ρ a =tanΘ H (T) is then measured (Fig. 2b, inset) to calculate κ E (H) at each T, which is extrapolated at low H to 0 T to determine κ E (0 T) (Fig. 2a). The error bars shown for the κ E (T) plot indicate the cumulative errors from each successive step of the analysis. At 2 and 5 K, the error bars are approximately the size of the symbols. Due to errors in geometry, the error in the absolute magnitude of κ E may be up to 50 %.
Clearly, κ E makes up a quite small proportion of the total κ, shown more clearly in the inset of Fig. 2a, along with κ E expected from the WFL. At high T, the WFL clearly overstates the magnitude given by our estimate of κ E . However, the two estimates approach each other at low T as expected. The WF T-dependence near T HO is also incorrect based on our results. κ E itself appears to also undergo an increase, though not precisely at T HO and of a much smaller magnitude than κ phonon . This is consistent with the claim that the electronic mean free path increases at the transition [18]. The primary assumption in previous reports has been to attribute some significance to the Lorenz ratio L=κ/σT using the raw κ data. Here we show that this contains a large phonon component that should not be meaningfully included in this quantity. In particular, we present the low-H transverse Lorenz number L XY (T, µ 0 H=2T)=κ XY /σ XY . The magnitude and Tdependence of L XY (and by implication, L) are very different from previous reports [8,9].
Since κ E plays no important role in the large κ increase, we analyze our data assuming only phonon conduction. It is clear that some scattering process drastically changes at T HO . There is independent evidence [19,20] that a gap, ∆(T), opens over parts of the Fermi surface (FS) at the transition. This will naturally lead to a decrease in e-ph scattering that can be modeled using the BCS formalism [21]. In particular, the effect of ∆(T) on κ(T) has been explored (BRT), and applied to superconductors (SC) [22].
In order to perform the analysis, we first fit κ(T) [23] of the 4% Rh doped sample, for which no transition is observed. This sample serves as a "background" that yields an independent way to extract the magnitudes of the various scattering rates for the undoped sample. The modification to e-ph scattering via the BRT formalism is then applied to describe URu 2 Si 2 . Near T HO , e-ph scattering appears to be the most important. The fit, along with the experimental data points are shown in Fig. 3a for the 4% sample. dependence is still incorrect. However, as α is increased, the calculation comes closer to the experimental κ(T) for the 2% sample shown in Fig. 1a. Namely, the peak in κ becomes rapidly reduced with even small amounts of doping. This simulation implies that adding small amounts of Rh (naively adding electrons) strongly increases the DOS at certain places in the FS, which also must reduce ∆ because T HO is sharply reduced.
Note also that adding Rh increases σ [10], implying a weakened coupling of the f-and conduction electrons. The minute amount of Rh needed to produce such a strong change in κ may imply an unusual instability in the FS, which leads to the HO ground state.
We further explore this model by inserting the BCS formula for ∆(T) into the BRT scattering rate. As shown in Fig. 3b, the use of a BCS ∆(T) comes closer to the observed κ increase (as well as the BRT/STM model) as the value of ∆(0)/k B T C is progressively increased from the "weak-coupling" limit (1.764) to a "strong-coupling" limit (e.g. 4). Furthermore, it is likely that ∆(0)/k B T C >4 because the residual DOS for E<∆ is ignored in the model. The value ∆(0)/k B T C~4 is considerably larger than typical "strong-coupled" superconductors (~2 in Pb and ~2.5 in K 3 C 60 [25]), and in particular, is beyond the scope of the Migdal-Eliahsberg theory [26]. However, typical CDW systems show ∆(0)/k B T C~3 -12 [21,27]. These very large values are usually interpreted as being a consequence of the enormous elastic energy costs associated with a coupling between the CDW and the lattice. Unlike the CDW ground state, URu 2 Si 2 does not undergo a large lattice distortion at T HO . However, the large value of ∆(0)/k B T>4 (intermediate between a strong coupled SC and CDW) implies a similarly significant coupling to the lattice.
The fact that this mean field (BCS) model for κ phonon reproduces the general form of the experimental results is an important clue to the mechanism for the HO transition.
In particular, this model is essentially of an itinerant nature, implying instabilities in the FS that open a gap. For example, both the SC and CDW ground states can be described using the BCS formalism [21]. In fact, a κ phonon increase at SC/CDW/SDW transitions has been observed to occur [14,28] in a similar way as observed here. One must therefore conclude that our results support an itinerant model of the HO. In combination with this conclusion, the large value of ∆(0)/k B T C (~ 4) that is needed to describe our data strongly suggest that the Hidden order parameter must be strongly coupled to the lattice. This claim is consistent with anomalies detected in the elastic constants at T HO , in particular the observed softening of a shear mode implies a lattice instability [29]. If the HO is associated with the modulation of charge degrees of freedom (e.g. U electrical quadrupolar moments implicit to Γ 5 doublets [6,7]) this would naturally lead to a strong coupling to the lattice due to Coulomb forces. Note that exotic d-wave models [2,3] should not have such a strong coupling to the lattice because the associated charge distributions possess no net electrical moment. There may well be similarities between the HO and localized charge-modulated states, such as the Charge-Ordered (CO) state.
Very similar increases in κ(T) have been observed upon entering into various CO states in, e.g. the manganites, nickelates, and cuprates [30].
We  Lines are guides for the eye.