Colossal thermomagnetic response in the exotic superconductor URu₂Si₂

Abstract

The superconducting fluctuation effect, due to preformed Cooper pairs above the critical temperature T_c, has been generally understood by the standard Gaussian fluctuation theories in most superconductors¹. The transverse thermoelectric (Nernst) effect is particularly sensitive to the fluctuations, and the large Nernst signal found in the pseudogap regime of the underdoped cuprates^2,3 has raised much debate. Here we report on the observation of a colossal Nernst signal due to the superconducting fluctuations in the heavy-fermion superconductor URu₂Si₂. The Nernst coefficient is anomalously enhanced (by a factor of ∼10⁶) as compared with the theoretically expected value of the Gaussian fluctuations. Moreover, contrary to the conventional wisdom, the enhancement is more significant with a reduction of the impurity scattering rate. This unconventional Nernst effect intimately reflects the highly unusual superconducting state of URu₂Si₂. The results invoke possible chiral or Berry-phase fluctuations associated with the broken time-reversal symmetry^4,5,6,7 of the superconducting order parameter.

Main

Measurements of the Nernst effect provide a unique opportunity to study the superconducting fluctuations deep inside the normal state above T_c (refs 2, 3, 8, 9, 10, 11, 12). The Nernst signal N is the electric field E_y (∥y) response to a transverse temperature gradient ∇_xT (∥x) in the presence of a magnetic field H (∥z), and is given by N ≡ E_y/(−∇_xT). The Nernst coefficient, defined as ν ≡ N/μ₀H above T_c, consists of two contributions generated by different mechanisms: ν = ν^S + ν^N. The first term, ν^S, represents the contribution of superconducting fluctuations of either amplitude or phase of the order parameter, which is always positive¹³. The second term, ν^N, represents the contribution from the normal quasiparticles, which can be either positive or negative¹⁴. The second contribution is usually small in conventional metals. In almost all superconductors the superconducting fluctuation contribution to the Nernst effect can be accounted for by the Gaussian-type fluctuations^{13,15,16,17,18}. Recently, a large Nernst signal has been reported in the pseudogap state of the hole-doped underdoped high-T_c cuprates, which has been discussed in terms of possible vortex-like excitations of phase disordered superconductors^2,3,8. Although its origin is still controversial, these results imply that the fluctuation-induced Nernst signal above T_c is intimately related to the exotic superconducting state below T_c.

The heavy-fermion compound URu₂Si₂ exhibits unconventional superconductivity (T_c = 1.45 K). This compound is distinguished from the other heavy-fermion compounds by the fact that the mysterious hidden-order transition takes place at T_HO = 17.5 K, and no evidence of magnetic order has been found below T_HO (ref. 19). This system has been suggested to be a candidate as a chiral d-wave superconductor that spontaneously breaks time-reversal symmetry (TRS) in the superconducting state^4,5,6,7. Indeed, the angular dependence of the thermal conductivity and specific heat in magnetic fields indicate the presence of point nodes in the order parameter, and a chiral d-wave pairing symmetry in a complex form of k_z(k_x ± ik_y) has been proposed^4,5. Very recently, the broken TRS has also been reported as a result of polar Kerr effect measurements (A. Kapitulnik, private communications). On the basis of these results, possible Weyl-type topological superconducting states have been discussed²⁰. It is therefore highly intriguing to examine the superconducting fluctuations in URu₂Si₂.

Figure 1a shows ν(T) in the zero-field limit (see also Supplementary Fig. 2a) and Fig. 1b the in-plane resistivity ρ_xx of ultraclean URu₂Si₂ single crystals (T_c = 1.45 K) with residual resistivity ratios (RRR) of 1,080 (#1) and 620 (#2). It has been reported that for RRR ≳ 30, both T_HO and T_c are almost independent of the RRR value²¹. Above T_HO, ν(T) is negligibly small and exhibits a marked increase on entering the hidden-order state. Below T^∗ ∼ 5 K, ν(T) shows a further enhancement and increases divergently on approaching T_c (Fig. 1a, b). The inset of Fig. 1b shows the T-dependence of the ratios of Nernst coefficient and conductivity σ = 1/ρ_xx in the two crystals, r_ν ≡ ν(#1)/ν(#2) and r_σ ≡ ρ_xx(#2)/ρ_xx(#1), respectively. In contrast to the nearly T-independent r_σ, r_ν increases steeply below ∼T^∗, suggesting the appearance of an additional mechanism that generates the enhanced Nernst effect in the cleaner crystal. These results indicate that the superconducting fluctuation effect sets in below ∼T^∗. As discussed later, this is supported by the H-dependence of the Nernst effect. In magnetic fields, ν(T) vanishes just below the vortex lattice melting temperature, T_melt (Fig. 2a)⁶.

Figure 1: Transverse thermoelectric response in URu₂Si₂.

a, T-dependence of the Nernst coefficient ν in the zero-field limit (H ∥ c) for single crystals #1 (RRR = 1,080) and #2 (RRR = 620). The RRR values are determined from ρ(300 K)/ρ₀ by assuming the T-dependence of the in-plane resistivity ρ_xx as ρ_xx(T) = ρ₀ + ATⁿ, with n = 1.5 and 1.7 for #1 and #2, respectively, below 6 K. In both crystals, T_c defined by the point of zero resistivity is 1.45 K. The upper inset illustrates the crystal structure of URu₂Si₂ and the lower inset is a schematic of the measurement set-up. b, Low-temperature data of ν(T) and ρ_xx(T) for crystals #1 and #2. Below T^∗, ν rises sharply above the T-linear dependence extrapolated from higher temperatures (dashed lines). The inset shows the T-dependence of the ratios of the Nernst coefficient and conductivity of the two crystals, r_ν = ν(#1)/ν(#2) and r_σ = ρ_xx(#2)/ρ_xx(#1).

Figure 2: Anomalously large Nernst signal and thermomagnetic figure of merit.

a, The T-dependence of ν (left scale) and ρ_xx (right scale) measured at μ₀H = 1 T near the superconducting transition. Both ν and ρ_xx vanish at the vortex lattice melting transition temperature T_melt. b, Comparison of the ν(T) data at μ₀H = 1 T between samples with different scattering rates (RRR = 1,080, 620 and 30). The data for RRR ∼ 30 (expanded in the inset) is taken from ref. 23. c, Thermomagnetic figure of merit ZT_ε = N²σT/κ at 1.5 K as a function of field in crystal #1 of URu₂Si₂ (red diamonds), compared with previous data in the semimetals PrFe₄P₁₂ (blue line) and Bi (black line) at 1.2 K taken from ref. 24.

First, we discuss the Nernst signal in the range of T that is free from the superconducting fluctuations (T^∗ ≲ T ≲ T_HO). As shown in Fig. 2b, increasing RRR or the scattering time τ leads to an enhancement of ν^N. Within the Boltzmann theory, when τ is weakly dependent on energy, ν^N can be expressed as ν^N = (π²/3)(k_B²T/m^∗)(τ/ɛ_F) (refs 10, 22), where k_B is the Boltzmann constant, m^∗ is the effective mass and ɛ_F is the Fermi energy. The striking enhancement of ν below T_HO is attributed to the strong reduction of ɛ_F associated with the disappearance of carriers and concomitant enhancement of τ, both of which have been reported previously⁴. The fact that r_ν above T^∗ coincides well with r_σ (inset of Fig. 1b) provides quantitative support of ν^N ∝ τ.

At lower temperatures below T^∗, ν of clean crystals becomes huge, especially in the vicinity of T_c. Indeed, ν of the cleanest crystal #1 is comparable to that of pure semimetal Bi, with the largest Nernst coefficient reported so far²². Moreover, the combination of the large Nernst signal and high conductivity in this system leads to a sizeable thermomagnetic figure of merit ZT_ε = N²σT/κ (κ is the thermal conductivity), which exceeds by far the values of previously studied materials (Fig. 2c)^23,24.

Now we discuss the fluctuation-induced Nernst coefficient ν^S. The enhancement of r_ν(T) below T^∗ (inset of Fig. 1b) and no discernible enhancement of ν(T) near T_c for RRR ∼ 30 (inset of Fig. 2b) indicate that ν^S is greatly enhanced with τ. We stress that this τ-dependence of ν^S is opposite to that expected in the conventional Gaussian fluctuation theories, which predict ν^S ∝ ρ_xx ∝ 1/τ (refs 13, 16, 17, 18). It has also been reported that in underdoped cuprates the introduction of impurities by irradiation enhances ν^S (ref. 8), which is again opposite to the URu₂Si₂ case. It is intriguing to compare the present results with CeCoIn₅, which shares several common features with URu₂Si₂, such as heavy-fermion unconventional superconductivity with a nodal gap and similar T_c and upper critical fields. It should be stressed that in very pure CeCoIn₅, with ρ(T_c⁺) ≈ 4 μΩ cm, which is of the same order as that of our URu₂Si₂ crystals, no discernible ν^S is observed^25,26; ν^S is at least two orders of magnitude smaller in pure CeCOIn₅ than in URu₂Si₂. These results thus highlight an essential difference in the superconducting fluctuations between URu₂Si₂ and the other unconventional superconductors.

The unusual nature of the superconducting fluctuations in URu₂Si₂ is further revealed by the off-diagonal component of the thermoelectric tensor (Peltier coefficient) α_xy, which is a more fundamental quantity associated with the fluctuations than the Nernst coefficient. The relation between ν and other coefficients is given as ν = 1/μ₀H(α_xyρ_xx − S tanθ_H), where μ₀ is the vacuum permeability, S is the Seebeck coefficient and θ_H is the Hall angle. Throughout the T- and H-regions in the present study, α_xyρ_xx ≫ S tanθ_H—that is, ν ≈ α_xyρ_xx/μ₀H (Supplementary Figs 1 and 2b). Figure 3 shows the T-dependence of the fluctuation-induced Peltier coefficient α_xy^S divided by μ₀H in the zero-field limit. To determine α_xy^S, the deviation from T-linear behaviour in ν(T) (dashed lines in Fig. 1b) is attributed to ν^S. The Peltier coefficient that results from the Gaussian (Aslamazov–Larkin) fluctuations is given by

where are the fluctuation coherence lengths parallel to the ab plane and c axis, and is the magnetic length, with e being the electronic charge and ℏ Planck’s constant. The blue line in Fig. 3 shows the T-dependence of α_xy^AL/μ₀H, calculated using ξ_ab(0) = 10 nm and ξ_c(0) = 3.3 nm, which are determined by the initial slope of upper critical fields at T_c. What is most spectacular is that the observed α_xy^S/μ₀H is four to six orders of magnitude greater than α_xy^AL/μ₀H given by equation (1). Moreover, the T-dependence of α_xy^S(T) is much steeper than α_xy^AL(T).

Figure 3: Comparison with the standard theory of superconducting fluctuations.

T-dependence of the fluctuation-induced Peltier coefficient divided by the magnetic field, α_xy^S/μ₀H, for crystals #1 and #2. The blue line represents the Peltier coefficient that results from Gaussian-type (Aslamazov–Larkin) fluctuations, given by equation (1).

We stress that the observed α_xy^S(T) far exceeding α_xy^AL(T) indeed originates from the superconducting fluctuations. This is evidenced by the steep enhancement of both ν(T) and r_ν(T) below T^∗. The H-dependence of α_xy(H) provides quantitative support for this. It has been pointed out that the size of superconducting fluctuation is set by the coherence length at low fields, whereas it is set by the magnetic length at high fields. As a result, α_xy(H) peaks at a characteristic field H^∗, where ξ_ab(T) = ℓ_H(H^∗), so that μ₀H^∗ = Φ₀/(2πξ_ab²(0))ln(T/T_c), where Φ₀ is the flux quantum. The peak field H^∗ is called the ‘ghost critical field’, and has been reported both in conventional and unconventional superconductors^11,12. As shown in Fig. 4, at all temperatures of interest, α_xy(H) exhibits a peak. The inset of Fig. 4 shows the peak field plotted as a function of ln(T/T_c). The solid line, which represents H^∗(T) calculated by using ξ_ab(0) = 10 nm, is quantitatively consistent with the peak field.

Figure 4: Superconducting fluctuations and the ghost critical field.

H-dependence of the Peltier coefficient α_xy = νμ₀H/ρ_xx of crystal #2 at various temperatures. Arrows mark the peak fields. Inset: peak field plotted as a function of ln(T/T_c). Error bars are due to the uncertainties in determining the peak positions. The solid line represents μ₀H^∗ = Φ₀/(2πξ_ab²(0))ln(T/T_c), with ξ_ab(0) = 10 nm, which is the so-called ghost critical field.

We discuss several possible origins for the observed colossal Nernst signal. First, equation (1) assumes the diffusive limit, k_BT ≪ ℏ/τ, whereas the present URu₂Si₂ seems to be in the ballistic regime, k_BT ≫ ℏ/τ. Although an extension of equation (1) to the clean case (see Supplementary Information) reveals the enhancement of α_xy^S with τ, this enhancement is slower than the reduction of ρ_xx, so that ν^S ∝ α_xy^Sρ_xx is still suppressed for larger τ, which is inconsistent with Fig. 2b. Second, in the multiband system, each band with different effective coherence lengths contributes differently to the total α_xy^S. To explain the observed α_xy^S, however, small bands with extremely large effective coherence lengths, whose effective H_c2 corresponds to less than 1 mOe, are required. The multiband effect is, therefore, highly unlikely to explain the observed α_xy^S. Third, the characteristic temperature scale of phase fluctuations is given as T_Θ = A(ℏ²a/4μ₀k_Be²λ_ab²(0)), where , λ_ab is the in-plane penetration depth and A is a dimensionless number of the order of unity²⁷. Using A = 2 and λ_ab = 0.8 μm (ref. 6), we obtain T_Θ/T_c ∼ 100, suggesting that the phase fluctuations are not important.

The colossal thermomagnetic response in URu₂Si₂ seems to point to a new type of superconducting fluctuation generated by a degree of freedom which has not been hitherto taken into account. It is tempting to consider that such a degree of freedom is intimately related to the superconducting state with broken TRS, because one of the most essential differences between URu₂Si₂ and CeCoIn₅, with no discernible ν^S, is that TRS is broken in the former whereas it is not broken in the latter²⁸. In fact, a chirality-induced anomalous Nernst signal has been pointed out in ref. 29. Very recently, it has been shown that the chirality or Berry-phase associated with the superconducting state with broken TRS gives rise to a new type of fluctuation³⁰, showing that α_xy^S(T) is strikingly enhanced with τ and its T-dependence is different from that predicted by the Gaussian fluctuation theories (Supplementary Information). As shown in Supplementary Fig. 4, the experimental results are reproduced well within the framework of the model of Berry-phase fluctuations. The present results suggest that superconducting fluctuations contain a key ingredient for the topological nature of superconductors, which is a new frontier of condensed matter physics.

Methods

The ultraclean single crystals of URu₂Si₂ were grown by the Czochralski pulling method in a tetra-arc furnace²¹. The well-defined superconducting transition was confirmed by the specific heat measurements. The Nernst and Seebeck coefficients were measured by the standard d.c. method with one resistive heater, two Cernox thermometers and two lateral contacts (lower inset of Fig. 1a). For the accurate determination of these coefficients, the crystals are cut in a rectangular shape, with the long direction corresponding to the a crystalline axis. The dimensions are 1.9 × 0.56 × 0.13 mm³ (#1) and 2.0 × 0.60 × 0.15 mm³ (#2). Magnetic field is applied perpendicular to the cleavage ab-plane with a misalignment less than 1°.