## Abstract

The presence or absence of a quantum critical point and its location in the phase diagram of high-temperature superconductors have been subjects of intense scrutiny. Clear evidence for quantum criticality, particularly in the transport properties, has proved elusive because the important low-temperature region is masked by the onset of superconductivity. We present measurements of the low-temperature in-plane resistivity of several highly doped La_{2–x}Sr_{x}CuO_{4} single crystals in which the superconductivity had been stripped away by using high magnetic fields. In contrast to other quantum critical systems, the resistivity varies linearly with temperature over a wide doping range with a gradient that scales monotonically with the superconducting transition temperature. It is maximal at a critical doping level (*p*_{c}) ∼ 0.19 at which superconductivity is most robust. Moreover, its value at *p*_{c} corresponds to the onset of quasi-particle incoherence along specific momentum directions, implying that the interaction that first promotes high-temperature superconductivity may ultimately destroy the very quasi-particle states involved in the superconducting pairing.

An important theme in strongly correlated electron systems is quantum criticality and the associated quantum phase transitions that occur at zero temperature upon tuning a nonthermal control parameter, *g* (e.g., pressure, magnetic field *H* or composition), through a critical value, *g*_{c}. One feature of such a system is the influence that critical fluctuations have on the physical properties over a wide region in the (*T, g*) phase diagram above the quantum critical point (QCP), inside which the system shows marked deviations from conventional Landau Fermi-liquid behavior. A number of candidate non–Fermi-liquid systems have emerged, particularly in the heavy fermion family (*1*), although there are others, for example, certain transition metal oxides (*2*), that display similar characteristics.

The physics of copper-oxide high-temperature superconductors may also be governed by proximity to a QCP. The generic temperature-doping (*T, p*) phase diagram resembles that seen in the heavy fermions, with an apparent funnel-shaped region that either pierces or skirts the superconducting dome (*3*). Above this region, cuprates display an in-plane resistivity, ρ_{ab}, that varies linearly with temperature over a wide temperature (*4*) yet narrow doping (*5*) range. This *T*-linear resistivity has been widely interpreted, in tandem with other anomalous transport properties (*6*), as a manifestation of scale-invariant physics borne out of proximity to the QCP. This viewpoint has remained untested, largely because of the high upper critical field *H*_{c2} values in high-*T*_{c} cuprates that restrict access to the important limiting low-temperature region below *T*_{c}(*p*). We used a combination of persistent and pulsed high magnetic fields to expose the normal state of La_{2–x}Sr_{x}CuO_{4} (LSCO) over a wide doping and temperature range and studied the evolution of ρ_{ab}(*T*) with carrier density, from the slightly underdoped (*p* = 0.15) to the heavily overdoped (*p* = 0.33) region of the phase diagram. Our analysis reveals the presence of a singular doping concentration in LSCO at which the electronic response changes, although in a manner distinct from that observed in other candidate quantum critical systems.

**In-plane resistivity of La _{2–x}Sr_{x}CuO_{4}**. A series of high-field ρ

_{ab}(

*T, H*) measurements were carried out on overdoped LSCO single crystals with doping levels of

*p*= 0.18, 0.21, and 0.23 (labeled hereafter LSCO18, LSCO21, and LSCO23, respectively) with the field aligned perpendicular to the CuO

_{2}planes in order to suppress the superconductivity. Figure 1A shows the ρ

_{ab}(

*T, H*) data obtained on LSCO23. In order to track the temperature dependence of the zero-field resistivity ρ(

*T*, 0) below

*T*

_{c}, we used a simple, transparent technique to extrapolate the high-field ρ

_{ab}(

*T, H*) data to the zero-field axis (Fig. 1B). The resultant ρ(

*T*, 0) values, plotted in Fig. 1C together with the zero-field ρ

_{ab}(

*T*) curve below 70 K, are found to exhibit a

*T-*linear dependence down to 1.5 K. For comparison, we also plotted the absolute values of ρ(

*T*, 48) at a fixed high field of 48 T obtained directly from the vertical dashed line in Fig. 1A. The temperature dependence of the latter (analysis-free) values is identical to that of ρ(

*T*, 0) and is consistent with earlier 60-T data taken on LSCO22 (

*7*), showing that the analysis itself has not introduced any additional, artificial temperature dependence in ρ(

*T*, 0). Similar pulsed-field measurements and analysis were carried out for the two other doping levels as summarized in fig. S1.

Figure 2 shows the resultant ρ(*T*, 0) values plus zero-field ρ_{ab}(*T*) data for seven different concentrations ranging from optimal doping (*p* = 0.17) to the heavily overdoped, nonsuperconducting region (*p* = 0.33). The gradual crossover in the temperature dependence of ρ_{ab}(*T*), from quasi-linear for LSCO17 to approximately quadratic for LSCO33, is evident in the raw data and is consistent with previous studies carried out above *T*_{c} (*5*, *8*, *9*). At low temperatures, however, ρ_{ab}(*T*) develops predominantly *T*-linear behavior for the entire doping range 0.18 ≤ *p* ≤ 0.29 [for *p* = 0.17, data exists only above *T*_{c}(*H* = 0)]. Although evidence for a low-*TT*-linear resistivity has emerged for single doping concentrations in both electron- (*10*) and hole-doped (*11*, *12*) cuprates, our measurements show that the low-*T* linearity in fact persists over a broad range of doping.

**Single-component analysis.** In heavy fermion systems, Δρ(*T*), the *T*-dependent part of ρ(*T*), is often described by a single term α_{n}*T ^{n}* whose exponent

*n*(

*T, H*) evolves from the Fermi-liquid value

*n*= 2 to some anomalous value less than 2 over a narrow temperature and magnetic field window (

*13*–

*15*). The anomalous exponent in Δρ(

*T*) persists to low temperatures only at the critical field,

*H*

_{c}. In Fig. 3, we plotted a comparative

*n*(

*T, p*) = d(lnΔρ)/d(ln

*T*) for LSCO by using the resistivity curves shown in Fig. 2.

For *T* > 50 K, the resultant phase diagram resembles that seen in prototypical quantum critical systems, with a narrow region in which ρ_{ab}(*T*) is approximately (although not strictly) *T*-linear separated from a region where ρ_{ab}(*T*) varies approximately as *T*^{2}. As the temperature is lowered, however, the situation becomes markedly different. Rather than collapsing to a single (critical) point, the *T*-linear region in LSCO fans out and dominates the low-*T* response. Intriguingly, this *T*-linear regime (or more precisely, the region where *n* < 1.1) is coincident with both the *T*_{c} parabola (long-dashed white line) and the superconducting fluctuation regime (short-dashed white line) and has thus been obscured until now by the veil of superconductivity.

**Dual-component analysis.** Previously, Δρ_{ab}(*T*) in overdoped, hole-doped cuprates has been expressed either as above, that is, as α_{n}*T ^{n}* (1 ≤

*n*≤ 2) (

*16*), or as the sum of two components, α

_{1}

*T*+ α

_{2}

*T*

^{2}(

*11*,

*17*,

*18*). In fig. S2, we describe in detail why the latter is in fact the more appropriate expression for LSCO. In Fig. 4, A and B, we show the doping dependences of α

_{1}and α

_{2}, respectively, for two different fitting protocols. The solid squares are coefficients obtained from least-square fits of the ρ

_{ab}(

*T*) curves for

*T*≤ 200 K to the expression ρ

_{ab}(

*T*) = α

_{0}+ α

_{1}

*T*+ α

_{2}

*T*

^{2}, whereas the solid circles are obtained from fits over the same temperature range to a parallel-resistor formalism 1/ρ

_{ab}(

*T*) = 1/(α

_{0}+ α

_{1}

*T*+ α

_{2}

*T*

^{2}) + 1/ρ

_{max}, where ρ

_{max}(=900 ± 100 μohm·cm) represents a maximum (saturation) resistivity value (

*19*,

*20*). The inclusion of ρ

_{max}helps to account smoothly for the escalation of ρ

_{ab}(

*T*) to higher temperatures and to make the values of α

_{1}and α

_{2}insensitive to the temperature range of fitting. [For a discussion of the physical meaning of ρ

_{max}and its relevance to the data, please refer to the supporting online material (SOM).] For all samples, the parallel-resistor fits (blue dashed curves in Fig. 2) are nearly indistinguishable from the ρ

_{ab}(

*T*) curves over a fitting range that extends (with the exception of LSCO17) over 2 decades in temperature and a factor of 2 in doping. The open symbols in Fig. 4 relate to coefficients obtained from corresponding fits to earlier zero-field ρ

_{ab}(

*T*) data (

*5*) for comparison.

For both sets of data and analysis, α_{1} is seen to grow rapidly with decreasing *p*, attaining a maximum value of around 1 μohm cm/K at *p*_{c} = 0.185 ± 0.005. In the parallel-resistor fit, α_{2} remains finite at all doping concentrations and essentially *p*-independent down to the same doping level, at which point it rises sharply. Without ρ_{max} in the fitting procedure, α_{2} becomes negligible around *p* = 0.20. It is important to note that none of the main findings of this study—the ubiquity of the *T*-linear term, the nondivergence of α_{2}, the position of the kink, nor the value of α_{1} at *p* = *p*_{c}—are dependent on the type of dual-component analysis used.

The coexistence of two distinct components to ρ_{ab}(*T*) is suggestive of some form of two-fluid model, although the fact that the two components add in the resistivity (i.e., in series) rather than in the conductivity (in parallel) implies that the two subsystems would have to coexist on a microscopic scale. Indeed, it is well documented that cuprates possess substantial microscopic inhomogeneity (*21*), and overdoped LSCO in particular is prone to microscopic phase separation into superconducting and non-superconducting regions with a physical extent on the order of the superconducting coherence length (*22*, *23*). One might then be tempted to attribute the *T*-linear component to the superconducting region (because its value scales roughly with *T*_{c}) and the quadratic term to that portion of the sample with an effective carrier density *p* > 0.27.

An alternative approach, however, is to assign the two additive coefficients to two distinct, independent quasi-particle scattering processes that coexist on the cuprate Fermi surface. Such a picture was recently proposed on the basis of angle-dependent magnetoresistance (ADMR) measurements on overdoped Tl_{2}Ba_{2}CuO_{6+δ} (Tl2201) that revealed that the scattering rate was composed of two distinct terms, a *T* ^{2} scattering term that is isotropic within the basal plane and a *T*-linear scattering rate that is strongly anisotropic, exhibiting a maximum near the Brillouin zone boundary, where the pseudogap is maximal and vanishing along the zone diagonal (*18*). In a subsequent doping-dependent study, it was intimated that this anisotropic *T*-linear term correlates with *T*_{c}, whereas the quadratic term remains constant as a function of *p* (*24*). Both of these trends in Tl2201 now appear to be confirmed in LSCO but over a much wider range of doping.

**Anomalous criticality in LSCO.** Critical fluctuations are a definitive signature of a QCP. In nearly ferromagnetic (FM) or antiferromagnetic (AFM) metals, for example, critical spin fluctuations give rise to physical properties with anomalous temperature dependences whose critical exponents depend on the nature of the spin fluctuations (FM or AFM) and on the dimensionality. For two-dimensional (2D) AFM spin fluctuations, a *T*-linear resistivity is expected that extends to very low temperatures at or near the QCP (*25*). Away from the critical point, the resistivity shows a crossover to a Fermi-liquid–like *T*^{2} resistivity as *T* decreases. As the critical point is approached, the crossover temperature *T*_{F} decreases, and the coefficient of the *T*^{2} resistivity diverges. Such behavior is typified by the heavy-fermion compound YbRh_{2}Si_{2}, where Δρ is found to be strictly *T*-linear over 2 decades in temperature at a critical magnetic field, *H*_{c} (*13*). Either side of *H*_{c}, Δρ is proportional to *T*^{2} with a coefficient α_{2} (*T*_{F}) that diverges (falls linearly) as *H* approaches *H*_{c}. Similar behavior is exhibited in the quasi-2D heavy-fermion compound CeCoIn_{5}, where the QCP coincides with the upper critical field *H*_{c2} (*14*), and in the bilayer ruthenate Sr_{3}Ru_{2}O_{7} (*2*), where the QCP coincides with a metamagnetic transition.

There are three aspects of the resistivity behavior in LSCO that conflict with this conventional quantum critical picture: (i) the two *T*-dependent coefficients appear to coexist for all *T* and *p*, rather than simply merge into one another; (ii) the coefficient of the *T*^{2} term does not diverge—it either remains constant or diminishes as one approaches *p*_{c} from the overdoped side; and (iii) the *T*-linear scattering term in LSCO persists at low *T* over an extended doping range. This is evident not only in the evolution of α_{1} (Fig. 4A) but also in the single-exponent plot (Fig. 3). This extended regime of *T*-linear resistivity is a possible manifestation of a second tuning parameter other than doping, that is, a third axis in the phase diagram (e.g., magnetic field or disorder) along which the QCP may be located. In YbRh_{2}Si_{2}, for example, alloying with Ge broadens the field range over which the *T*-linear resistivity extends to low *T* (*13*). Alternatively, the breadth of the critical region in LSCO suggests that the strange metal physics in cuprates is associated not with a QCP but with a novel, extended quantum phase, as seen, for example, in the itinerant ferromagnet MnSi beyond a critical pressure (*26*).

**Discussion.** The marked kinks in α_{1} and α_{2} at *p*_{c} = 0.185 ± 0.005 are the most revealing findings of this study. The value of *p*_{c} coincides with that inferred from a wealth of other experimental probes (*3*) and is associated with the opening of the pseudogap, an anisotropic (nodal) gap in the normal state excitation spectrum. Below *p*_{c}, a new energy (*E**) or temperature (*T**) scale emerges, which grows sharply with decreasing doping and appears to extrapolate to a value about equal to the AFM exchange energy, *J*. There are two competing scenarios describing the effect of the pseudogap on the Fermi surface in cuprates: one involving Fermi surface reconstruction into hole and electron pockets (*27*) and the other involving Fermi surface degradation into a series of disconnected Fermi patches or arcs, centered about the zone diagonals, with an arc length that is progressively reduced upon underdoping (*28*). Although the actual doping dependence of the pseudogap has been hotly debated (*3*), the consistent and systematic analysis reported here appears to confirm that there is indeed a fundamental change in the transport properties of cuprates at this specific, well-defined doping level sandwiched between optimal doping and the upper edge of the superconducting dome (*16*, *29*).

A particular feature of Fig. 4 is the anticorrelation of α_{1} and α_{2} for *p* < *p*_{c}; whereas α_{2} rises or recovers, α_{1} falls away from its doping trajectory inferred from *p* > *p*_{c} (thin dashed line in Fig. 4A). Recent experiments on less-disordered cuprates with *p* ≤ 0.1 have shown that this trend continues well into the underdoped regime with ρ_{ab}(*T*), becoming purely *T*^{2} at low to intermediate temperatures (*30*). The distinct momentum dependences of the two scattering processes, suggested by the ADMR experiments on Tl2201 (*18*), provides a means to interpret qualitatively the anticorrelation of α_{1} and α_{2} within a scenario of progressive destruction of the underlying Fermi surface (*31*). As *p* falls below *p*_{c}, sections of Fermi surface near the Brillouin zone boundary begin to disappear. Although this may not necessarily have a detrimental effect on the isotropic term α_{2}, for α_{1} the reduction in the total number of coherent states is more than offset by the removal of the strong scattering sinks near (π, 0), leading to an overall decrease, or at best saturation, in α_{1} with further reduction in carrier density.

Recent angle-resolved photoemission spectroscopy (ARPES) studies of underdoped cuprates have hinted at a gradual erosion of the Fermi arc length with decreasing temperatures below *T** (*32*). For 0.15 ≤ *p* ≤ 0.19, *T** is about equal to *E**/2 and is of the order of 100 K (*33*). Because our fitting range extends up to 200 K, the changes observed in α_{1}(*p*) and α_{2}(*p*) for *p* < *p*_{c} imply that actually quasi-particle states are lost at temperatures *T* > *T**. Moreover, given that our fitting parameters are relatively insensitive to the temperature range, we conclude that the antinodal states must play little or no role in the conductivity at any given temperature. One hint toward understanding this lies in the absolute value of α_{1} (=1 μohm·cm K^{–1}) at *p* = *p*_{c}. For a 2D metal, dρ_{ab}/d*T* = (2π*ħd*/*e*^{2}*v*_{F}*k*_{F})d(1/τ)/d*T*, where *d* is the interlayer spacing (=0.64 nm in LSCO), *v*_{F} is the Fermi velocity, and *k*_{F} the Fermi wave vector. Taking typical (*p*-independent) values of *v*_{F} (=8.0 × 10^{4} ms^{–1}) and *k*_{F} (=7 nm^{–1}) for the antinodal states in overdoped LSCO (*34*), we find that at *p* = *p*_{c} the momentum-averaged scattering rate *ħ*/τ ∼ π*k*_{B}*T*. Given that the anisotropic scattering rate varies as cos^{2}2φ within the plane (φ being the angle between the *k* vector and the Cu-O-Cu bond direction) (*18*), we find that *ħ*/τ ∼ 2π*k*_{B}*T* for states near (π, 0). This level of scattering intensity is consistent with the so-called Planckian dissipation limit (*35*) beyond which Bloch-wave propagation becomes inhibited, that is, the quasiparticle states themselves become incoherent.

The picture that emerges then is the following. As the carrier number falls and the system approaches the Mott insulating state at *p* = 0, the effective interaction responsible for the (anisotropic) *T*-linear scattering term becomes progressively stronger. Because the strength of this term is closely correlated with *T*_{c}, the same interaction also drives up both the superconducting transition temperature and the condensation energy, the latter reaching a maximum at *p* = *p*_{c} (*33*, *36*). At this point, however, scattering intensity is so strong that those states near (π, 0) begin to de-cohere, and, given the proportionality with temperature, incoherence occurs at all finite temperatures, not just below *T**. With further reduction in doping, the strength of the interaction continues to rise, causing yet more states to lose coherence and leading to an overall reduction or saturation in α_{1}(*p*) and a corresponding suppression in the condensation energy (*36*) and the superfluid density (*37*).

One is then left with the intriguing possibility that the pseudogap itself forms in response to this intense scattering, the electronic ground state lowering its energy through gapping out the incoherent, highly energetic antinodal states and, in so doing, preventing scattering of the remnant quasi-particle states into those same regions. The temperature-invariant violation of the Planckian dissipation limit then provides a possible explanation as to why the pseudogap itself is a non–states-conserving gap (*3*, *33*): In contrast to a conventional (Bardeen-Cooper-Schrieffer) superconducting gap where the full density of states is recovered above the (two-particle) gap energy, pseudogapped states near (π, 0) are effectively removed at all energies up to the bandwidth (*38*) or the Coulomb energy (*39*). Along the nodal directions, where this anisotropic scattering term is absent, however (*18*), the nodal quasi-particle states remain intact (protected) at all *T* and *p* across the entire superconducting dome.

**Concluding remarks.** It has been a long-standing mystery why the pseudogap in cuprates vanishes at a singular doping level, *p*_{c}, beyond optimal doping. By tracking the evolution of the in-plane resistivity from the overdoped side, we have uncovered a change in the electrical transport at *p* = *p*_{c} that appears to coincide with the onset of incoherence (at the antinodal regions), thus offering a possible explanation for its precise location in the cuprate phase diagram and the subsequent reduction in the strength of the superconductivity with lowering carrier concentration. And although Planckian dissipation is quantum criticality of sorts, it does not appear to be consistent with any simple one-parameter scaling hypothesis (*40*) in just the same way as the doping evolution of the transport coefficients is difficult to reconcile with conventional quantum criticality.

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/1165015/DC1

Materials and Methods

Figs. S1 and S2

References