Abstract
A fundamental question for hightemperature superconductors is the nature of the pseudogap phase, which lies between the Mott insulator at zero doping and the Fermi liquid at high doping p (refs 1, 2). Here we report on the behaviour of charge carriers near the zerotemperature onset of this phase, namely at the critical doping p^{*}, where the pseudogap temperature T^{*} goes to zero, accessed by investigating a material in which superconductivity can be fully suppressed by a steady magnetic field. Just below p^{*}, the normalstate resistivity and Hall coefficient of La_{1.6−x}Nd_{0.4}Sr_{x}CuO_{4} are found to rise simultaneously as the temperature drops below T^{*}, suggesting a change in the Fermi surface with a large associated drop in conductivity. At p^{*}, the resistivity shows a linear temperature dependence as the temperature approaches zero, a typical signature of a quantum critical point^{3}. These findings impose new constraints on the mechanisms responsible for inelastic scattering and Fermisurface transformation in theories of the pseudogap phase^{1,4,5,6,7,8}.
Similar content being viewed by others
Main
At low hole doping p, hightransitiontemperature (highT_{c}) superconductors are doped Mott insulators, strongly correlated metals characterized by a low carrier density n equal to the concentration of doped holes. Indeed, Halleffect measurements on La_{2−x}Sr_{x}CuO_{4} (LSCO) at x=p<0.05 yield a Hall number n_{H}≡V/e R_{H} equal to p at low temperature^{9}, where R_{H} is the Hall coefficient, e is the electron charge and V is the volume per Cu atom. At high doping, however, these materials are Fermi liquids, metals characterized by a welldefined coherent threedimensional Fermi surface^{10} and a resistivity ρ that grows quadratically with temperature^{11}: ρ∼T^{2}. In this regime, the Fermi surface is a large cylinder containing 1+p holes^{10}, so the carrier density is high, given by n=1+p. At p≈0.25, lowtemperature measurements on Tl_{2}Ba_{2}CuO_{6+y} yield n_{H}=1+p (ref. 12). These findings naturally beg the following question: How do the electrons in copper oxide superconductors go from one state to the other?
This is intimately tied to the question of the nature of the ‘pseudogap phase’, this enigmatic region of the doping phase diagram present in all highT_{c} superconductors below a crossover temperature T^{*} (ref. 2). Here we investigate the T=0 onset of this pseudogap phase by measuring the transport properties of La_{1.6−x}Nd_{0.4}Sr_{x}CuO_{4} (NdLSCO), a material whose relatively low maximal T_{c} makes it possible to suppress superconductivity entirely with a steady magnetic field.
In Fig. 1, we show the normalstate resistivity ρ(T) of NdLSCO at a doping p=0.20. Above a temperature T^{*}=80 K, ρ(T) shows the linear temperature dependence characteristic of all holedoped copper oxides. Below this temperature, it deviates upwards and develops an upturn visible even in zero field (see Supplementary Information, Fig. S1), with a minimum at T_{min}=37 K>T_{c}=20 K, in excellent agreement with early data in zero field^{13}. By applying a magnetic field of 35 T, we were able to track the upturn in ρ(T) down to 1 K, thus revealing a pronounced rise at low temperature (Fig. 1).
The absence of magnetoresistance (see Supplementary Information, Fig. S1) implies that the magnetic field simply serves to remove superconductivity and reveal the unaltered behaviour of the underlying normal state down to T≈0. The evolution with temperature is perfectly smooth, indicating a crossover as opposed to a transition. Most significantly, ρ(T) saturates at low temperature (see Fig. 2a). This shows that the ground state is a metal and not an insulator, and that T^{*} therefore marks the onset of a crossover from one metallic state to another. Note that the loss of conductivity is substantial, by a factor of approximately ρ_{0}/ρ(T→0)=5.8, where ρ_{0}=245 μΩ cm is the resistivity measured at T=1 K and ρ(T→0)=42 μΩ cm is the value extrapolated linearly to T=0 from above T^{*}.
We identify T^{*} as the onset of the pseudogap phase, following the standard definition: the temperature below which the inplane resistivity ρ_{a b}(T) starts to deviate from its linearT behaviour at high temperature^{2,14}. (Note that the deviation can be either upwards, as in LSCO, or downwards, as in YBa_{2}Cu_{3}O_{y} (ref. 14), depending on the relative magnitude of inelastic and elastic (disorder) scattering at T^{*}; in YBa_{2}Cu_{3}O_{y}, the copper oxide material with the lowest disorder scattering, the loss of inelastic scattering below T^{*} is a much larger relative effect than in LSCO, hence the drop in ρ_{a b}(T).) In Fig. 3, we plot T^{*} as a function of doping in a p–T phase diagram. Note that the magnitude of T^{*} in NdLSCO is comparable to that found in other holedoped copper oxides, pointing to a common origin (see Supplementary Information, Fig. S2 for a comparison with LSCO).
In Fig. 4, we present the Hall coefficient R_{H}(T) measured on the same crystal (with p=0.20), and compare it directly with ρ(T). Both coefficients are seen to rise simultaneously, with T_{min} the coincident onset of their respective upturns. This is strong evidence that the cause of both upturns is a modification of the Fermi surface.
Let us now look at a slightly higher doping. Figures 1 and 4 respectively show ρ(T) and R_{H}(T) measured on a second crystal, with p=0.24. The lowtemperature behaviour has changed: ρ(T) shows no sign of an upturn and R_{H}(T) is now constant below 25 K, extrapolating to R_{H}=+0.45±0.05 mm^{3} C^{−1} as T→0. The corresponding Hall number is n_{H}=1.3±0.15, in good agreement with the carrier density n=1+p=1.24 expected for a large Fermi cylinder, and quantitatively consistent with measurements on Tl_{2}Ba_{2}CuO_{6+y} at p=0.26 and T→0, where n_{H}=1.3 (ref. 12). By comparison, at p=0.20, the magnitude of R_{H} at T→0 yields n_{H}=0.3 ±0.05. The change in the Hall number at T→0 between p=0.24 and p=0.20 is therefore Δn_{H}=1.0±0.2 hole per Cu atom. If the Hall number is interpreted as a carrier density, these values are consistent with a crossover from a metal with a large holelike Fermi surface at p^{*} (where n=1+p) to a metal with a low density of holes below p^{*} (where n≈p).
In contrast to p=0.20, the electrical resistivity at p=0.24 shown in Fig. 2b shows a monotonic temperature dependence down to 1 K, linear as T→0. The absence of any anomaly demonstrates that T^{*}=0 at that doping. Therefore, the critical doping p^{*} where the pseudogap line ends is located between p=0.20 and 0.24, inside the region where superconductivity exists in zero field. For definiteness, in Fig. 3 we set it at p^{*}=0.24, although it could be slightly lower.
As shown in Fig. 2b, not only is the inplane resistivity ρ_{a b}(T) linear as T→0 at p=0.24, but so is the outofplane resistivity ρ_{c}(T). Moreover, the fact that R_{H}(T) is flat at low temperature implies that the cotangent of the Hall angle, cotθ_{H}(T)∼ρ_{a b}(T)/R_{H}(T), is also linear at low temperature. We infer that a single anomalous scattering process dominates the electron–electron correlations at low temperature at p^{*} (or just above). This shows that the Fermiliquid behaviour observed at p=0.3 (in LSCO), where ρ_{a b}(T)∼T^{2} below T≈50 K (ref. 11), breaks down just before the onset of the pseudogap phase at p^{*}. This kind of ‘nonFermiliquid’ behaviour, whereby ρ(T)∼T as T→0, has typically been observed in heavyfermion metals at the quantum critical point where the onset temperature for antiferromagnetic order goes to zero^{3}. It is also consistent with the marginalFermiliquid description of cuprates^{15}.
In summary, our experimental findings offer compelling evidence that the pseudogap phase ends at a T=0 critical point p^{*} located below the onset of superconductivity (at p_{c}≈0.27), in agreement with previous but more indirect evidence from other holedoped copper oxides^{16}. Moreover, they impose two strong new constraints on theories of the pseudogap phase: (1) its onset below p^{*} modifies the large Fermi surface characteristic of the overdoped metallic state; (2) quasiparticle scattering at p^{*} is linear in temperature as T→0.
The existence of a quantum critical point is consistent with two kinds of theory of the pseudogap phase. The first kind invokes the onset of an order, with some associated broken symmetry^{6,7,8}. Because T^{*} marks a crossover and not a sharp transition, this order is presumably short range or fluctuating. In the electrondoped copper oxides, for example, the pseudogap phase has been interpreted as a fluctuating precursor of the longrange antiferromagnetic order that sets in at lower temperature^{17}, and the signatures of the pseudogap critical point in transport are similar to those found here: a linearT resistivity as T→0 (ref. 18) and a sharp change in R_{H}(T=0) (ref. 19). For NdLSCO and LSCO, an analogous scenario would be ‘stripe’ fluctuations, as a precursor to the static spin and charge modulations observed at lower temperature^{20}. Note that in NdLSCO at p=0.20 the onset of the upturn in ρ(T) and R_{H}(T) at T_{min}=37 K coincides with the loss of NQR intensity at T_{NQR}=40±6 K (ref. 21) (see Fig. 4). In NdLSCO at p=0.15, this socalled ‘wipeout’ anomaly in NQR at T_{NQR}=60±6 K (ref. 21) was shown to coincide with the onset of charge order measured via hardXray diffraction, at T_{ch}=62±5 K (ref. 22) (see Fig. 3). Direct evidence of a charge modulation via resonant softXray diffraction was reported recently for the closely related material EuLSCO, with T_{ch}=70±10 K at p=0.15 (ref. 23), whereas T_{NQR}=60±6 K (ref. 21) in EuLSCO at p=0.16 (see Fig. 3). Clearly, the upturn in ρ(T) is correlated with the onset of charge order in these two materials. While the correlation between T_{NQR} and T_{min} has been noted previously^{20}, the mechanism causing the upturn in ρ(T) remained unclear. Our data show that the mechanism is a change in Fermi surface, and the positive rise in R_{H}(T) imposes a strong constraint on the topology of the resulting Fermi surface. An additional constraint comes from the fact that R_{H}(T) drops to negative values near p=1/8, not only in NdLSCO (ref. 13) and other materials with ‘stripe’ order^{24,25}, but also in YBa_{2}Cu_{3}O_{y} (ref. 26).
Recent calculations of the Fermisurface reconstruction caused by stripe order are consistent with a negative R_{H} near p=1/8 in that spin stripes tend to generate an electron pocket in the Fermi surface^{27}. Interestingly, charge stripes do not^{27}, and this might explain the positive rise in R_{H} seen at higher doping, provided that stripe order involves predominantly charge order at high doping (in line with the fact that charge order sets in at a higher temperature than spin order^{20,21}).
In the other kind of theory of the pseudogap phase, the critical point reflects a T=0 transition from small hole pockets, characteristic of a doped Mott insulator, to a large hole pocket, without symmetry breaking^{4,5}. Recent work suggests that the quasiparticle scattering rate above such a critical point may indeed grow linearly with temperature^{28}. Although calculations are needed to confirm this, a change in carrier density from n≈p to n=1+p would seem natural in this kind of scenario. However, it is more difficult to see what could cause the negative values of R_{H}(T→0) near p=1/8. It seems that stripe order or fluctuations would have to be invoked as a secondary instability inside the pseudogap phase, with an onset in doping that would be essentially simultaneous with p^{*} in the case of NdLSCO.
We end by comparing our results qualitatively with those of previous highfield studies on LSCO. The resistivity shows very similar features at high temperature: linear T above T^{*} (ref. 14) and an upturn below T^{*} (ref. 29). The Hall coefficient of LSCO (ref. 30), on the other hand, has a more subtle and complex evolution with doping than that presented here for NdLSCO, which makes it harder to pinpoint p^{*} using the same criteria as we have used above. Nonetheless, it seems likely that the same fundamental mechanisms are responsible for both the linearT resistivity and the resistivity upturns, and for the onset of the pseudogap at T^{*}, in both LSCO and NdLSCO.
Methods
Single crystals of La_{2−y−x}Nd_{y}Sr_{x}CuO_{4} (NdLSCO) were grown with a Nd content y=0.4 using a travellingfloatzone technique and cut from boules with nominal Sr concentrations x=0.20 and 0.25. The actual doping p of each crystal was estimated from its T_{c} and ρ(250 K) values compared with published data, giving p=0.20±0.005 and 0.24±0.005, respectively. The resistivity ρ and Hall coefficient R_{H} were measured at the National High Magnetic Field Laboratory (NHMFL) in Tallahassee in steady magnetic fields up to 35 T and in Sherbrooke in steady fields up to 15 T. The field was always applied along the c axis. Neither ρ nor R_{H} showed any field dependence up to the highest fields. More details are available in Supplementary Information.
References
Lee, P. A., Nagaosa, N. & Wen, X.G. Doping a Mott insulator: Physics of hightemperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006).
Timusk, T. & Statt, B. The pseudogap in hightemperature superconductors: An experimental survey. Rep. Prog. Phys. 62, 61–122 (1999).
v Löhneysen, H., Rosch, A., Vojta, M. & Wölfle, P. Fermiliquid instabilities at magnetic quantum phase transitions. Rev. Mod. Phys. 79, 1015–1075 (2007).
Yang, K.Y., Rice, T. M. & Zhang, F.C. Phenomenological theory of the pseudogap state. Phys. Rev. B 73, 174501 (2006).
Haule, K. & Kotliar, G. Avoided criticality in nearoptimally doped hightemperature superconductors. Phys. Rev. B 76, 192503 (2007).
Kivelson, S. A. et al. How to detect fluctuating stripes in the hightemperature superconductors. Rev. Mod. Phys. 75, 1201–1241 (2003).
Chakravarty, S., Laughlin, R. B., Morr, D. K. & Nayak, C. Hidden order in the cuprates. Phys. Rev. B 63, 094503 (2001).
Varma, C. NonFermiliquid states and pairing instability of a general model of copper oxide metals. Phys. Rev. B 55, 14554–14580 (1997).
Ando, Y. et al. Evolution of the Hall coefficient and the peculiar electronic structure of the cuprate superconductors. Phys. Rev. Lett. 92, 197001 (2004).
Hussey, N. E. et al. Observation of a coherent threedimensional Fermi surface in a hightransition temperature superconductor. Nature 425, 814–817 (2003).
Nakamae, S. et al. Electronic ground state of heavilyoverdoped nonsuperconducting La2−xSrxCuO4 . Phys. Rev. B 68, 100502 (2003).
Mackenzie, A. P. et al. Normalstate magnetotransport in superconducting Tl2Ba2CuO6+y to millikelvin temperatures. Phys. Rev. B 53, 5848–5855 (1996).
Nakamura, Y. & Uchida, S. Anisotropic transport properties of singlecrystal La2−y−xNdySrxCuO4: Effect of the structural phase transition. Phys. Rev. B 46, 5841–5844 (1992).
Ando, Y. et al. Electronic phase diagram of highTc cuprate superconductors from a mapping of the inplane resistivity curvature. Phys. Rev. Lett. 93, 267001 (2004).
Varma, C. M. et al. Phenomenology of the normal state of Cu–O hightemperature superconductors. Phys. Rev. Lett. 63, 1996–1999 (1989).
Tallon, J. L. & Loram, J. W. The doping dependence of T^{*}—what is the real highTc phase diagram? Physica C 349, 53–68 (2001).
Motoyama, E. M. et al. Spin correlations in the electrondoped hightransitiontemperature superconductor Nd2−xCexCuO4±δ . Nature 445, 186–189 (2007).
Fournier, P. et al. Insulator–metal crossover near optimal doping in Pr2−xCexCuO4: anomalous normalstate lowtemperature resistivity. Phys. Rev. Lett. 81, 4720–4723 (1998).
Dagan, Y. et al. Evidence for a quantum phase transition in Pr2−xCexCuO4−δ from transport measurements. Phys. Rev. Lett. 92, 167001 (2004).
Ichikawa, N. et al. Local magnetic order vs superconductivity in a layered cuprate. Phys. Rev. Lett. 85, 1738–1741 (2000).
Hunt, A. W. et al. Glassy slowing of stripe modulation in (La,Eu,Nd)2−x(Sr,Ba)xCuO4: A ^{63}Cu and ^{139}La NQR study down to 350 mK. Phys. Rev. B 64, 134525 (2001).
Niemöller, T. et al. Charge stripes seen with Xrays in La1.45Nd0.4Sr0.15CuO4 . Eur. Phys. J. B 12, 509–513 (1999).
Fink, J. et al. Charge order in La1.8−xEu0.2SrxCuO4 studied by resonant soft Xray diffraction. Preprint at <http://arxiv.org/abs/0805.4352> (2008).
Adachi, T., Noji, T. & Koike, Y. Crystal growth, transport properties, and crystal structure of the singlecrystal La2−xBaxCuO4 (x=0.11). Phys. Rev. B 64, 144524 (2001).
Takeshita, N et al. Giant anisotropic pressure effect on superconductivity within the CuO2 plane of La1.64Eu0.2Sr0.16CuO4: Strain control of stripe criticality. J. Phys. Soc. Jpn. 73, 1123–1126 (2004).
LeBoeuf, D. et al. Electron pockets in the Fermi surface of holedoped highTc superconductors. Nature 450, 533–536 (2007).
Millis, A. J. & Norman, M. R. Antiphase stripe order as the origin of electron pockets observed in 1/8holedoped cuprates. Phys. Rev. B 76, 220503 (2007).
Ossadnik, M., Honerkamp, C., Rice, T. M. & Sigrist, M. Breakdown of Landau theory in overdoped cuprates near the onset of superconductivity. Preprint at <http://arxiv.org/abs/0805.3489> (2008).
Boebinger, G. S. et al. Insulatortometal crossover in the normal state of La2−xSrxCuO4 near optimum doping. Phys. Rev. Lett. 77, 5417–5420 (1996).
Balakirev, F. F. et al. Fermi surface reconstruction at optimum doping in highTc superconductors. Preprint at <http://arxiv.org/abs/0710.4612> (2007).
Nachumi, B. et al. Muon spin relaxation study of the stripe phase order in La1.6−xNd0.4SrxCuO4 and related 214 cuprates. Phys. Rev. B 58, 87608772 (1998).
Acknowledgements
We thank K. Behnia, A. Chubukov, P. Coleman, Y.B. Kim, S.A. Kivelson, G. Kotliar, K. Haule, G.G. Lonzarich, A.J. Millis, M.R. Norman, C. Proust, T.M. Rice, S. Sachdev, T. Senthil, H. Takagi and A.M.S. Tremblay for discussions, and J. Corbin for his assistance with the experiments. L.T. acknowledges support from the Canadian Institute for Advanced Research and funding from NSERC, FQRNT, and a Canada Research Chair. L.B. was supported by NHMFLUCGP and Y.J.J. by the NHMFLSchuller fellow program. J.S.Z. and J.B.G. were supported by an NSF grant. The NHMFL is supported by an NSF grant and the State of Florida.
Author information
Authors and Affiliations
Corresponding author
Supplementary information
Supplementary Information
Supplementary Informations (PDF 249 kb)
Rights and permissions
About this article
Cite this article
Daou, R., DoironLeyraud, N., LeBoeuf, D. et al. Linear temperature dependence of resistivity and change in the Fermi surface at the pseudogap critical point of a highT_{c} superconductor. Nature Phys 5, 31–34 (2009). https://doi.org/10.1038/nphys1109
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/nphys1109
This article is cited by

Hopping frustrationinduced flat band and strange metallicity in a kagome metal
Nature Physics (2024)

Absence of a BCSBEC crossover in the cuprate superconductors
npj Quantum Materials (2023)

Reconciling scaling of the optical conductivity of cuprate superconductors with Planckian resistivity and specific heat
Nature Communications (2023)

Interplay between superconductivity and the strangemetal state in FeSe
Nature Physics (2023)

Nd:YAG infrared laser as a viable alternative to excimer laser: YBCO case study
Scientific Reports (2023)