Universal behavior in the Nernst effect of heavy fermion materials

We report the observation of a universal scaling of the Nernst coefficient over a wide intermediate temperature range in the heavy fermion superconductors CeCu$_2$Si$_2$, CeCoIn$_5$, and Ce$_2$PdIn$_8$, the ferromagnetic Kondo lattice Ce$_3$RhSi$_3$, the nonmagnetic CeRu$_2$Si$_2$, the intermediate valent YbAl$_3$, and the hidden order compound URu$_2$Si$_2$, that cover a broad spectrum of heavy fermion materials with different crystal, valence and ground state properties. The scaling formula follows exactly the magnetic susceptibility of emergent heavy quasiparticles as predicted in the two-fluid model. We give a tentative explanation of the scaling based on the skew scattering mechanism and argue that the Nernst effect is produced by the asymmetry of the quasiparticle density of states rather than that of the scattering rate. In URu$_2$Si$_2$, the giant Nernst signal in the hidden order phase is also found to follow the predicted scaling, indicating the potential involvement of hybridization physics. Our work suggests a unified picture for the Nernst effect in heavy fermion materials and provides a promising basis for the development of a better theory in the future.

The Nernst effect, a thermoelectric analog of the Hall effect, measures the transverse electric field induced by a longitudinal thermal gradient under a perpendicular magnetic field.It was first observed in elemental bismuth [1,2] but is often extremely small (nV/KT) in simple metals due to the Sondheimer cancellation [3].A large Nernst signal of the order of µV/KT may be realized in anisotropic metals with nonspherical Fermi surfaces [4] or multiband systems where the presence of both electron and hole carriers can give rise to a large ambipolar contribution [5].But unlike those in semiconductors and semimetals [6], the Nernst effect in correlated systems has not been paid much attention until an anomalously enhanced signal was discovered and attributed to the movement of vortices or vortexlike excitations in the pseudogap phase of the underdoped La 2−x Sr x CuO 4 [7].This resembles that of conventional or high-T c superconductors [8][9][10] and immediately stimulated intensive interest in the community of correlated electrons [11][12][13][14][15][16][17][18][19].
In heavy fermion materials, large Nernst signals have since been reported and explained by various scenarios besides vortex motion [20].Magnetic fluctuations, the proximity of a quantum critical point, or some collective modes have all been considered as possible origins, but a generic understanding has not been achieved, thus preventing the development of a unified theory that could potentially cover the rich variety of experimental observations.In this work, we report an unexpected discovery of universality in the Nernst effect via a systematic examination of existing data in a number of prototypical heavy fermion compounds.Independent of material details, we find that the Nernst coefficients in all compounds exhibit a universal scaling over a wide intermediate temperature range, which follows exactly the predicted magnetic susceptibility of emergent heavy quasiparticles in the two-fluid model [21].This motivates us to develop a tentative explanation based on the skew scattering mechanism and relate the Nernst coefficient with the asymmetry of the quasiparticle density of states at the Fermi energy.Interestingly, the same scaling is also observed in the hidden order phase of URu 2 Si 2 , indicating potential involvement of hybridization physics in this mysterious state.
Figure 1 collects and compares the Nernst data reproduced from the literature for the heavy fermion superconductors CeCu 2 Si 2 [22], CeCoIn 5 [19,23], and Ce 2 PdIn 8 [24], the ferromagnetic Kondo lattice Ce 3 RhSi 3 [25], the nonmagnetic CeRu 2 Si 2 [20], and the intermediate valence compound YbAl 3 [26].These cover a broad spectrum of heavy fermion systems with different crystal, valence and ground state properties.The magnitude of their Nernst coefficients also varies over two orders of magnitude and has been given different explanations.In CeCu 2 Si 2 , an enhanced Nernst coefficient was observed to correlate with the thermopower and ascribed to the asymmetric (or skew) Kondo scattering [22].In YbAl 3 , the correlation is absent, possibly due to its intermediate valence.A finite Nernst signal was observed at high temperatures and attributed to other scattering processes such as acoustic phonons [26].In the ferromagnetic Kondo lattice Ce 3 RhSi 3 [25], the Nernst coefficient shows a similar temperature dependence as the Hall coefficient, most likely originating from the skew scattering [27], but the thermopower behaves very differently and cannot be explained by available theories.In CeCoIn 5 , the maximal Nernst coefficient can reach 1 µV/kT at low field, much larger than the residual value in cuprates above T c [19].Various origins have been assigned such as antiferromagnetic fluctuations [28], long-range phase coherence [29], or unconventional density wave [30].Moreover, ν/T was found to be dramatically enhanced at 6 T, most probably FIG.1: Collection and comparison of the Nernst coefficients (ν) in a number of prototypical heavy fermion compounds including CeCu2Si2 [22], CeRu2Si2 [20], Ce2PdIn8 [24], Ce3RhSi3 [25], CeCoIn5 [19], and YbAl3 [26].The data were reproduced from the literature, possibly with different sign conventions.
due to a suppressed Fermi energy induced by the proximity of a field-indued quantum critical point [28].In Ce 2 PdIn 8 , the Nernst coefficient diverges logarithmically below 7 K, implying the presence of an underlying quantum critical point [24].At intermediate temperatures, it exhibits strong field and temperature dependence, possibly associated with an anisotropic scattering time due to antiferromagnetic fluctuations.No ambipolar enhancement was detected at high temperatures despite of its multiband electronic structure.
Given such complexity, it seems unlikely to develop a unified picture.However, one may notice that the Nernst coefficients in all above compounds share certain kind of common features, namely a weak temperature dependency at high temperatures and a strong enhancement at intermediate temperatures, followed by a rapid suppression as the temperature approaches zero.With increasing magnetic field, as shown for CeCoIn 5 , Ce 2 PdIn 8 , and Ce 3 RhSi 3 , the Nernst coefficient is systematically suppressed and the peak shifts towards higher temperature, possibly following the increase of the Zeeman energy.This common trend suggests the possibility of a generic mechanism and motivates us to make a systematic comparison of existing experiments, which, quite unexpectedly, immediately reveals the presence of a true universality.As plotted in Fig. 2, all data can be scaled FIG.2: Universal scaling of the normalized Nernst coefficients as a function of the dimensionless temperature for all data presented in Fig. 1.The high temperature constants are subtracted.The solid line is the predicted scaling of emergent heavy quasiparticles in the two-fluid model [31].
to fall on a single curve over a wide intermediate temperature range.In most cases, the scaling works pretty well from T * down to the peak temperature.Moreover, it follows exactly the magnetic susceptibility (or density of states) of the emergent heavy quasiparticles predicted in the two-fluid model [31], where T * is the onset temperature, ν * is a constant from the high temperature background, and ν 0 is the normalization factor for the scaling.What is then the underlying mechanism?The above connection suggests a tentative explanation based on the skew scattering mechanism.Quite generally, we have in the Boltzmann picture, where k B is the Boltzmann constant, e is the electron charge, H is the magnetic field, and ǫ F is the Fermi energy.The Hall angle is defined as tan Θ H = ρ xy /ρ, in which ρ xy and ρ are the transverse and longitudinal resistivity, respectively.We have assumed energy dependency in all quantities.For dominant skew scattering, the Hall coefficient is given by R H = R 0 + rρχ, where R 0 is the normal contribution from background conduction electrons, r is a constant prefactor, and R s = rρχ is the skew scattering contribution in proportion to the magnetic susceptibility χ [27].Accordingly, the Nernst coefficient also has two contributions: a background term, which is typically small as in metals, and a skew scattering term given by ν s ∝ ∂χ(ǫ)/∂ǫ following the Boltzmann equation.Now the two-fluid model states, χ = χ l + χ h , where χ h is the susceptibility of emergent heavy quasiparticles and χ l is that of residual unhybridized moments.
If the compound is in the Kondo limit, the local moments stay deep below the Fermi level so that χ l must depend weakly on ǫ.We have then ∂χ l /∂ǫ| ǫF ≈ 0 and consequently a weak Nernst signal at high temperatures in the fully localized regime.This is indeed the case for all compounds in Fig. 1.In YbAl 3 , the somewhat larger Nernst coefficient at high temperatures is most probably due to its intermediate valence [26].By contrast, the heavy electron spectral weight is always located near the Fermi energy and provides the major contribution, ν s ∝ ∂χ h (ǫ)/∂ǫ| ǫF , which is, unfortunately, still difficult to model in theory.To proceed, we consider the possibility of an E/T scaling in the quasiparticle spectra, which has been observed in the scanning tunneling and angle-resolved photoemission spectroscopies of CeCoIn 5 [32,33].It is likely supported by the universal nature of the emergent heavy quasiparticles as manifested in many experiments [21].One may then assume a generic form for the susceptibility (or density of states), χ h (ǫ) = χ h (T )g(x), where χ h (T ) gives the temperature dependence of the quasiparticle spectrum and g(x) is a regular function of x = (ǫ−ǫ F )/T describing the shape.We have ∂χ h (ǫ)/∂ǫ| ǫF = −χ h (T )T −1 ∂g(x)/∂x| x=0 , which immediately yields the observed scaling ν s ∝ χ h (T ).Two remarks are in order concerning this probably oversimplified derivation.First, the enhanced Nernst coefficient follows the emergence of itinerant heavy quasiparticles.Our fit yields T * ≈ 110 K for YbAl 3 , 59 K for CeRu 2 Si 2 , 78 K for CeCu 2 Si 2 , 98-107 K for Ce 3 RhSi 3 , 25 K for CeCoIn 5 and 16-18 K for Ce 2 PdIn 8 .These values agree well with those estimated from many other measurements for YbAl 3 (120 ± 10 K), CeRu 2 Si 2 (60 ± 10 K), and CeCu 2 Si 2 (75 ± 20 K) [34].However, for CeCoIn 5 , previous estimates give T * ≈ 40 K along planar direction, which are twice larger than that from the Nernst fit.The same is also seen in Ce 2 PdIn 8 .This is very puzzling.It might be that the formula of R s fails below the coherence temperature.In fact, similar χ h scaling has been observed in the Hall coefficient of CeCoIn 5 and Ce 2 PdIn 8 [35].It is not clear if the discrepancy in these two compounds is accidental or demands an alternate explanation.At the moment, we cannot exclude other possibilities, but it seems hard to derive the scaling formula on a different basis.On the other hand, our Nernst data for CeCoIn 5 is only below 25 K.It will be interesting to see if higher temperature data might follow the same scaling with a higher T * .In recent pump probe experiment on CeCoIn 5 [36], two collective modes have been detected to emerge below these two temperatures, respectively.It could be possible that there exist multiple hybridization processes and the Nernst effect is more sensitive to the lower one.In any case, the Nernst coefficient seems very different from the Hall coefficient.While the latter is mostly dominated by the local moment contribution, our observed scaling here indicates that the local moment contribution is largely eliminated in the Nernst data, presumably by the energy derivative in Eq. ( 2).In this respect, the enhanced Nernst signal is primarily associated with the asymmetry of the quasiparticle density of states rather than that of the scattering rate.Otherwise, a different scaling would be expected in the intermediate valence compound YbAl 3 .
Second, there should be a tentative correlation between the magnitude of the Nernst signal and the value of T * .The latter sets roughly the effective bandwidth of the heavy quasiparticle spectra [37] and is inversely proportional to the density of states and its slope at the Fermi energy [38].In Fig. 1, a large ν 0 as in CeCoIn 5 or Ce 2 PdIn 8 seems quite typically to have a small T * , while all other compounds have relatively larger T * and smaller ν 0 .Indeed, it has been shown previously that the Nernst coefficient in CeCoIn 5 is correlated with the quasiparticle effective mass or the specific heat [20], both of which are inversely proportional to T * [38].It is known that T * is typically the order of 10-100 K, only a few hundredths of the bandwidth of conduction electrons in a normal metal.
Hence the large Nernst signal in heavy fermion materials is primarily associated with their small quasiparticle bandwidth.Clearly, such a correlation is not exact and the Nernst signal may vary a lot owing to the complicated variation of ground state properties.Figure 3 compares the derived ν 0 as a function of the magnetic field for CeCoIn 5 , Ce 2 PdIn 8 and Ce 3 RhSi 3 .We see no common trend of ν 0 versus H in three compounds.For Ce 3 RhSi 3 , ν 0 is almost field independent, possibly associated with its ferromagnetic property.For CeCoIn 5 and Ce 2 PdIn 8 , ν 0 decreases monotonically with increasing field, reflecting the suppression of heavy quasiparticles at high field.On the other hand, if we look closer into the data of CeCoIn 5 , there appears to be a change of slope around the quantum critical point at 4.1 T [39,40], but more field data will be needed to clarify this possibility.
Putting together, the Nernst effect in heavy fermion materials may be typically categorized into three regimes, a high temperature regime with an almost constant background, a wide intermediate temperature regime with enhanced signal and universal temperature dependence due to emergent heavy quasiparticles with a narrow bandwidth, and a low temperature regime where the Nernst coefficient tends to be suppressed.Similar suppression is also present in the Hall data [41].While the high temperature constant comes from conduction electrons or f valence bands (as in YbAl 3 ), the low temperature limit varies a lot depending on the Fermi liquid or other ordered states.Ideally, one may expect a continuing increase near a quantum critical point as observed in Ce 3 RhSi 3 at low field.In the Fermi liquid, the scaling will break down and the Nernst effect will be governed by well-defined Landau quasiparticles.Then a different formula should be expected as for a normal metal.
With the above picture in mind, we turn to the mysterious heavy fermion compound URu 2 Si 2 .Intriguingly, a giant Nernst signal has been observed in the hidden order phase below 17.5 K [42] rather than the coherence temperature of about 55 K [34].Thermopower and Hall measurements point up the possibility of a small Fermi energy, a low carrier density, and a long scattering time [42].Other more exotic scenarios involve the presence of chiral orders or chiral fluctuations [43][44][45].With increasing magnetic field, the Nernst signal shows successive anomalies, suggesting a series of Fermi surface reconstructions [46].But still, as shown in Fig. 4, all data can be scaled to collapse on the same universal curve with a derived T * of 17-22 K, just around the hidden order temperature and depending weakly on the magnetic field.Again, our data is limited below 25 K.It will be interesting to see if the normal state data follow the above scaling with a higher T * .On the other hand, the validity of the same scaling in the hidden order phase seems to suggest that the hidden order also possesses certain aspect of hybridization physics [47], among many other dazzling theories [48].This is in agreement with the point contact measurement  [31].The inset shows the derived ν0 decreasing linearly with field.The dashed line is a guide to the eye.[49], in which a significant Fano resonance has been observed in the hidden order phase, suggesting the presence of hybridization [50].Previously, it has also been shown in experiment that a band of heavy quasiparticles drops below the Fermi level across the transition [51], which probably causes the change in the Nernst effect.With increasing field, the derived value of ν 0 decreases linearly as shown in the inset of Fig. 4(b), indicating a diminishing hybridization in accordance with the suppression of the hidden order [52].
To summarize, we discover a universal scaling of the Nernst coefficient over a wide intermediate temperature range in heavy fermion materials, regardless of their crystal, valence and ground state properties.The scaling follows the predicted magnetic susceptibility of the emergent heavy quasiparticles in the two-fluid model.This reveals a close connection between the enlarged Nernst signal and the asymmetry of the quasiparticle density of states and allows for a tentative but generic explanation of the experimental data.The scaling is further confirmed in the hidden order phase of URu 2 Si 2 , suggesting potential involvement of hybridization physics.Our work provides the basis for a unified understanding and urges on the development of a more elaborate theory of the Nernst effect in heavy fermion systems.

3 FIG. 3 :
FIG.3:The derived normalization factor ν0 as a function of the magnetic field for CeCoIn5, Ce2PdIn8, and Ce3RhSi3.The data for CeCoIn5 is multiplied by -0.3.The dashed line is a guide to the eye.

FIG. 4 :
FIG. 4: (a)The Nernst coefficient of URu2Si2 under different magnetic fields reproduced from the literature[42].(b) The universal scaling of the normalized Nernst data as a function of the dimensionless temperature.The value of T * varies slightly from about 22 to 17 K with increasing field.The solid line is the predicted scaling formula of emergent heavy quasiparticles in the two-fluid model[31].The inset shows the derived ν0 decreasing linearly with field.The dashed line is a guide to the eye.