Abstract
Longrange order is destroyed in a superconductor warmed above its critical temperature (T_{c}). However, amplitude fluctuations of the superconducting order parameter survive^{1} and lead to a number of wellestablished phenomena, such as paraconductivity^{2}: an excess of charge conductivity due to the presence of shortlived Cooper pairs in the normal state. According to theory^{3}, these pairs generate a transverse thermoelectric (Nernst) signal. In two dimensions, the magnitude of the expected signal depends only on universal constants and the superconducting coherence length, so the theory can be rigorously tested. Here, we present measurements of amorphous superconducting films of Nb_{0.15}Si_{0.85}. In this dirty superconductor, the lifetime of Cooper pairs exceeds the elastic scattering lifetime of quasiparticles in a wide temperature range above T_{c} and, consequently, their Nernst response dominates that generated by the normal electrons. We resolved a Nernst signal, which persists deep inside the normal state. Its amplitude is in excellent agreement with the theoretical prediction. This result provides an unambiguous case for a Nernst effect produced by shortlived Cooper pairs.
Main
The Nernst effect, the generation of a transverse electric field by a longitudinal thermal gradient, has attracted considerable attention since the observation of a puzzling Nernst signal in the normal state of highT_{c} cuprates^{4,5,6,7,8,9,10}. In the context of the debate on the origin of this signal, Ussishkin, Sondhi and Huse (USH) calculated the contribution of amplitude fluctuations of the superconducting order parameter on thermoelectric transport^{3} and concluded that these fluctuations, described in the gaussian approximation and responsible for the wellestablished phenomenon of paraconductivity^{2} in the normal state of superconductors, should also generate a Nernst signal. In their model, the main contribution to the Nernst signal comes from the Aslamazov–Larkin term, which represents Cooper pairs with a finite lifetime above T_{c} (ref. 1). This lifetime decreases with increasing temperature. Therefore, in the presence of a thermal gradient, the pairs diffusing towards low temperature live longer than those diffusing towards high temperature, and thus the temperature gradient induces a net drift of pairs towards low temperature. The deflection of this current by a magnetic field produces a transverse voltage and hence a Nernst effect.
According to the USH calculations, the contribution of gaussian superconducting fluctuations to thermoelectricity leads to a finite offdiagonal component in the Peltier conductivity tensor, α_{x y}, which is the ratio of the longitudinal charge current to the transverse thermal gradient (). In particular, in two dimensions, and for low magnetic fields B≪φ_{0}/2πξ^{2}, where ξ is the coherence length and φ_{0} is the flux quantum, α_{x y} is expected to follow this simple expression:
Here, ℓ_{B}=(ħ/e B)^{1/2} is the magnetic length scale. Note that in equation (1), the three universal constants (Planck, Boltzmann and the charge of electron) combine to generate the quantum of thermoelectric conductance (k_{B}e/h=3.3 nA K^{−1}), a less celebrated concept compared with the quanta of electric^{1} (e^{2}/h) or thermal^{11} (π^{2}k_{B}^{2}T/3h) conductance. However, in the notation used by USH (as is often the case for theoretical papers), k_{B} is taken as equal to unity and the quantum of thermoelectric conductance does not appear explicitly. As the magnitude of the coherence length, ξ, is the only parameter in equation (1), this theory is particularly apt for an unambiguous confrontation with experiment.
We have tested this theory by measuring the Nernst coefficient in amorphous films of Nb_{x}Si_{1−x} (refs 12–14), which exhibit the widelydocumented features of superconductor–insulator transition in dirty twodimensional superconductors^{15}. The competition between superconducting and insulating ground states is controlled by the Nb concentration, x, the thickness, d, or the magnetic field^{16}.
The data for the two samples shown in Figs 1 and 2 show that a finite measurable Nernst signal persists well above T_{c}. Before comparing this observation with the theoretical prediction by USH, let us argue that neither the excitations of the normal state, nor the superconducting phase fluctuations could be a plausible source for the observed Nernst signal. A rough scale for the normalstate Nernst signal is the product of the Seebeck coefficient (S) and the Hall angle (tanθ=R_{H}/ρ_{x x}, where R_{H} is the Hall coefficient and ρ_{x x} is the longitudinal resistivity). As seen in Fig. 2, in the entire range of our measurements, the Nernst coefficient, ν, is three orders of magnitude larger than Stanθ. In a multiband metal, the contribution of carriers with different signs to Stanθ cancel out and its overall value could become smaller than ν (ref. 17), but such a possibility can be easily ruled out here. The hypothetical existence of two very small Fermi surface pockets hosting carriers of opposite sign with long meanfreepath seems implausible. The small value of tanθ≈2×10^{−5} simply reflects an extremely short electronic meanfreepath (of the order of interatomic distance ∼0.25 nm) and a conventional carrier density (the magnitude of R_{H}=4.9×10^{−11} m^{3} C^{−1} is comparable to that reported^{18} for bulk Nb).
Could this signal be caused by phase fluctuations of the superconducting order parameter? This is also unlikely. In contrast to the underdoped cuprates, the carrier density in Nb_{0.15}Si_{0.85} is comparable to any conventional metal. As the ‘phase stiffness’ of a superconductor is determined by its superfluid density^{19}, there is no reason to speculate on the presence of preformed Cooper pairs without phase coherence in a wide temperature window above T_{c} as has been the case in the pseudogap state of the cuprates. In contrast to granular superconductors^{20}, decreasing the thickness leads to a shift of the sharp superconducting transition and does not reveal a temperature scale other than the meanfield BCS (Bardeen–Cooper–Schrieffer) critical temperature. The variation of T_{c} with thickness has been attributed to the enhancement of the Coulomb interactions with the increase in the sheet resistance, R_{square} (ref. 21).
On the other hand, there is no reason to doubt the presence of amplitude fluctuations of the superconducting order parameter invoked by the USH theory. Now, the theory makes a precise prediction on the magnitude of α_{x y}, but what is directly measured by the experiment is the Nernst coefficient, ν, which is intimately related to it. When the Hall angle is small and the contribution of superconducting fluctuations to charge conductivity is also small, there is a simple relationship between α_{x y}, ν and the sheet resistance R_{square}:
where σ is the electric conductivity.
The validity of both conditions was checked in our experiment: tanθ≈2×10^{−5} and σ^{SC}=(e^{2}/16ħ)(T_{c}/(T−T_{c})) (ref. 1) is a few per cent of σ_{x x} when T>1.1T_{c}.
Because α_{x y}^{SC}∝B, it follows from relation (2) that ν should be independent of the magnetic field, in the low magnetic field region considered by this model. The data show that this is indeed the case. Therefore, we can directly determine α_{x y}/B for each temperature in the zerofield limit using equation (2) and the data for ν and R_{square}. Now, equation (1) can be rewritten as:
The value of ξ obtained in this way for the two samples is shown in Fig. 3 as a function of the reduced temperature ɛ=(T−T_{c})/T_{c} and allows a direct verification of the theory. Theoretically, the coherence length, ξ, should vary as ɛ^{−1/2} (ref. 1). Moreover, its absolute magnitude is expected to scale inversely with , which is in conformity with the ξ value found here: the ratio of the coherence lengths ξ_{1} and ξ_{2} for samples 1 and 2 is ξ_{1}(ɛ=1)/ξ_{2}(ɛ=1)=1.48, and the ratio . More quantitatively, the coherence length of a twodimensional dirty superconductor is^{1}:
where v_{F} and ℓ are the Fermi velocity and the electronic meanfreepath. The most direct way to estimate v_{F}ℓ is to use the known values of the electric conductivity, σ≈6.4×10^{4} Ω^{−1} cm^{−1} and the electronic specific heat, γ_{e}≈108 J K^{−1} m^{−3} (see the Methods section). The generic relationship between the specific heat and the thermal conductivity (κ), combined with the Wiedemann–Franz law yields:
This enables us to directly estimate v_{F}ℓ=4.35×10^{−5} m^{2} s^{−1} and, using equation (3), plot ξ_{d}(ɛ). As seen in Fig. 3, for both samples, for small values of ɛ, there is an excellent agreement between these two estimations of the coherence length. As the temperature rises, decreases faster than ξ_{d}. This discrepancy is not surprising because the ɛ^{−1/2} dependence of the coherence length ξ_{d}(ɛ) and the USH theory are valid only for small (<1) values of ɛ. Moreover, ξ becomes much smaller than the film thickness and the 2D limit is no longer valid. It is remarkable, however, that even for ɛ=10, the two values obtained for ξ differ by a mere factor of two.
In retrospect, it is not surprising that this effect is unambiguously observed for the first time in a dirty superconductor. According to the theory, what is universal is the magnitude of α_{x y}^{SC}, the expected Nernst signal is therefore larger when the normalstate conductivity is lower. Moreover, owing to the short meanfreepath of electrons, the normalstate Nernst effect becomes negligible, making the detection of the signal produced by superconducting fluctuations easier. On a more fundamental level, in a dirty superconductor, the lifetime of a fluctuating Cooper pair (τ_{GL}≈ξ^{2}/v_{F}ℓ (ref. 1)) exceeds the elastic lifetime of a normal electron (τ_{el}≈ℓ/v_{F}) in a wide temperature window above T_{c}, paving the way for a dominant contribution of the gaussian fluctuations to the Nernst signal.
Charge conductivity, even in the presence of gaussian fluctuations, is dominated by the contribution of normal electrons. As we saw above, this is not the case for the Nernst effect, which (owing to the smallness of the normalstate Nernst effect) can be totally dominated by these fluctuations. This makes the Nernst effect a powerful probe of superconducting fluctuations.
We conclude by considering the field dependence of the Nernst coefficient. The USH calculation has been carried out for weak fields (ξ≪ℓ_{B}) and was only tested here in the zerofield limit. For T>T_{c}, as seen in Fig. 1, the Nernst signal reveals a field scale in its field dependence that increases with increasing temperature. This seems to simply reflect the decrease in the field scale associated with ξ(B^{*}=ħ/(e ξ^{2})). For both samples, the Nernst signal does not vanish even with the application of a field as large as 4 T. This field is larger than all three field scales that can be associated with the destruction of superconductivity. These are (1) the critical field for the superconductor–insulator transition defined as the crossing field of R(B) curves at low temperatures, B^{SI} (ref. 16); (2) the Pauli limit B^{P}=1.84T_{c} (ref. 22); and (3) the orbital limit B_{orb}=φ_{0}/2πξ^{2}. The values for both samples are shown in Table 1, where it seems that the upper critical field B_{c2} is set by the Pauli limit, that is, B_{c2}=B^{SI}≈B^{P}<B^{orb}. It is natural to assume that the superconducting longrange order is indeed destroyed at B_{c2}, but the superconducting fluctuations persist and gradually fade away above B_{c2}, as they do above T_{c}. The contribution of the gaussian fluctuations to the Nernst effect in high magnetic fields remains a challenging question for both theory and experiment.
Methods
The two amorphous thin films of Nb_{0.15}Si_{0.85} used in this study were prepared as described elsewhere^{12,13}. The nominal concentration of Nb in the two samples used in this study was the same (x=0.15) and the difference in T_{c}s, (0.165 K in sample 1 with d=12.5 nm and 0.38 K in sample 2 with d=35 nm), is mainly due to the difference in thickness of the two samples. The critical temperature was defined as the midheight of the resistive transition at zero field. A setup with one resistive heater, two RuO_{2} thermometers and two lateral contacts^{23} was used to measure the thermoelectric and the electric coefficients of each sample in a dilution cryostat. At T∼0.19 K, we could resolve a d.c. voltage of 1 nV and a temperature difference of 0.1 mK. The magnitude of the electronic specific heat used for the estimation of the coherence length is based on the magnitude of γ_{e} in bulk Nb (ref. 24) and the concentration of itinerant electrons provided by the Nb fraction, as confirmed by direct measurements of specific heat in Nb_{x}Si_{1−x} thin films of a lower concentration^{13}.
References
Larkin, A. & Varlamov, A. Theory of Fluctuations in Superconductors (Clarendon, Oxford, 2005).
Glover, R. E. Ideal resistive transition of a superconductor. Phys. Lett. A 25, 542–544 (1967).
Ussishkin, I., Sondhi, S. L. & Huse, D. A. Gaussian superconducting fluctuations, thermal transport, and the Nernst effect. Phys. Rev. Lett. 89, 287001 (2002).
Xu, Z. A. et al. Vortexlike excitations and the onset of superconducting phase fluctuation in underdoped La2−xSrxCuO4 . Nature 406, 486–488 (2000).
Wang, Y. et al. Onset of the vortexlike Nernst signal above Tc in La2−xSrxCuO4 and Bi2Sr2−yLayCuO6 . Phys. Rev. B 64, 224519 (2001).
Wang, Y. et al. Dependence of upper critical field and pairing strength on doping in cuprates. Science 299, 86–89 (2003).
Capan, C. et al. Entropy of vortex cores near the superconductorinsulator transition in an underdoped cuprate. Phys. Rev. Lett. 88, 056601 (2002).
Wen, H. H. et al. Twodimensional feature of the Nernst effect in normal state of underdoped La2−xSrxCuO4 single crystals. Europhys. Lett. 63, 583–589 (2003).
RullierAlbenque, F. et al. Nernst effect and disorder in the normal state of highTc cuprates. Phys. Rev. Lett. 96, 067002 (2006).
Wang, Y., Li, L. & Ong, N. P. Nernst effect in highTc superconductors. Phys. Rev. B 73, 024510 (2006).
Schwab, K., Henriksen, E. A., Worlock, J. M. & Roukes, M. L. Measurement of the quantum of thermal conductance. Nature 404, 974–977 (2000).
Dumoulin, L., Bergé, L., Lesueur, J., Bernas, H. & Chapellier, M. NbSi thin films as thermometers for low temperature bolometers. J. Low Temp. Phys. 93, 301–306 (1993).
Marnieros, S., Bergé, L., Juillard, A. & Dumoulin, L. Dynamical properties near the metalinsulator transition: evidence for electronassisted variable range hopping. Phys. Rev. Lett. 84, 2469–2472 (2000).
Lee, H.L., Carini, J. P., Baxter, D. V., Henderson, W. & Grüner, G. Quantumcritical conductivity scaling for a metalinsulator transition. Science 287, 633–636 (2000).
Goldman, A. M. & Markovic, N. Superconductor–insulator transitions in the twodimensional limit. Phys. Today 51, 39–44 (1998).
Aubin, H. et al. Magneticfieldinduced quantum superconductorinsulator transition in Nb0.15Si0.85 . Phys. Rev. B 73, 094521 (2006).
Bel, R., Behnia, K. & Burger, H. Ambipolar Nernst effect in NbSe2 . Phys. Rev. Lett. 91, 066602 (2003).
Gilchrist, J. le G. & Vallier, J.C. Hall effect in superconducting niobium and alloys. Phys. Rev. B 3, 3878–3886 (1971).
Emery, V. J. & Kivelson, S. A. Importance of phase fluctuations in superconductors with small superfluid density. Nature 374, 434–437 (1995).
Jaeger, H. M., Haviland, D. B., Orr, B. G. & Goldman, A. M. Onset of superconductivity in ultrathin granular metalfilms. Phys. Rev. B 40, 182–196 (1989).
Maekawa, S., Ebisawa, H. & Fukuyama, H. Upper critical field in twodimensional superconductors. J. Phys. Soc. Jpn 52, 1352–1360 (1983).
Clogston, A. M. Upper limit for the critical field in hard superconductors. Phys. Rev. Lett. 9, 266–267 (1962).
Bel, R. et al. Giant Nernst effect in CeCoIn5 . Phys. Rev. Lett. 92, 217002 (2004).
Leupold, H. A. & Boorse, H. A. Superconducting and normal specific heats of a single crystal of niobium. Phys. Rev. 134, A1322–A1328 (1964).
Acknowledgements
This work is partially supported by Agence Nationale de la Recherche. We are grateful to C. Capan, A. Kapitulnik, M. Grilli, D. Huse, S. Kivelson, S. Sondhi and I. Ussishkin for useful discussions.
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C.A.M.K., L.B. and L.D. prepared the samples. A.P. (assisted by H.A. and K.B.) carried out the measurements. H.A. and K.B. (discussing with A.P. and J.L.) interpreted and analysed the data. K.B. wrote the text.
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Pourret, A., Aubin, H., Lesueur, J. et al. Observation of the Nernst signal generated by fluctuating Cooper pairs. Nature Phys 2, 683–686 (2006). https://doi.org/10.1038/nphys413
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DOI: https://doi.org/10.1038/nphys413
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