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# Universal sheet resistance and revised phase diagram of the cuprate high-temperature superconductors

Edited

^{†}by J. C. Seamus Davis, Cornell University, Ithaca, NY, and approved June 4, 2013 (received for review January 31, 2013)

## Abstract

Upon introducing charge carriers into the copper–oxygen sheets of the enigmatic lamellar cuprates, the ground state evolves from an insulator to a superconductor and eventually to a seemingly conventional metal (a Fermi liquid). Much has remained elusive about the nature of this evolution and about the peculiar metallic state at intermediate hole-carrier concentrations (*p*). The planar resistivity of this unconventional metal exhibits a linear temperature dependence (ρ ∝ *T*) that is disrupted upon cooling toward the superconducting state by the opening of a partial gap (the pseudogap) on the Fermi surface. Here, we first demonstrate for the quintessential compound HgBa_{2}CuO_{4+δ} a dramatic switch from linear to purely quadratic (Fermi liquid-like, ρ ∝ *T*^{2}) resistive behavior in the pseudogap regime. Despite the considerable variation in crystal structures and disorder among different compounds, our result together with prior work gives insight into the *p-T* phase diagram and reveals the fundamental resistance per copper–oxygen sheet in both linear (ρ_{☐} = *A*_{1☐}*T*) and quadratic (ρ_{☐} = *A*_{2☐}*T*^{2}) regimes, with *A*_{1☐} ∝ *A*_{2☐} ∝ 1/p. Theoretical models can now be benchmarked against this remarkably simple universal behavior. Deviations from this underlying behavior can be expected to lead to new insight into the nonuniversal features exhibited by certain compounds.

When exploring the properties of a material, the resistivity is the quantity that is often first measured, but last understood. It is an observable that corresponds to a weighted integration over the whole Fermi surface (see *SI Appendix* for details). Consequently, the resistivity is highly sensitive to changes in electronic behavior, and hence regularly used to detect phase transitions and determine phase diagrams. However, for the same reason, the resistivity typically exhibits complex temperature, magnetic field, etc., dependences. Nevertheless, when simple temperature dependences are observed, such as power law in a metal or exponential in a semiconductor, this usually provides important insight into the fundamental properties of a material (1). Only in rare cases, such as the quantum Hall effect in correlated 2D electron systems, has the magnitude of the resistivity been found to be the same for an entire class of materials (2).

In the case of the cuprates, the close proximity to the Mott insulator has long been argued to imply a purely electronic superconducting mechanism (3), yet the peculiar metallic state from which the superconductivity emerges upon cooling has remained an enigma. Part of the difficulty stems from the fact that the well over 100 cuprate superconductors exhibit nonuniversal temperature-doping phase diagrams due to significant differences in superconducting transition temperatures (*T*_{c}), crystal structures, disorder, competing phases, etc. (4).

Here, we first report quantitative planar dc resistivity results for the structurally simple single-CuO_{2}–layer compound HgBa_{2}CuO_{4+δ} (Hg1201). Above the pseudogap temperature *T** (5, 6), we observe the well-known yet poorly understood linear behavior ρ = *A*_{1}*T*. Upon further cooling, we find a distinct switch to a purely quadratic dependence, ρ = *A*_{2}*T*^{2}, that holds between the lower characteristic temperature *T*** and the measurable appearance of superconducting fluctuations at *T′*, just above *T*_{c} (*T** > *T*** > *T′* > *T*_{c}). We then present a systematic analysis of prior data for double-layer YBa_{2}Cu_{3}O_{6+δ} (YBCO) (7⇓⇓⇓⇓–12) as well as single-layer La_{2-x}Sr_{x}CuO_{4} (LSCO) (8, 9, 13⇓⇓–16) and Tl_{2}Ba_{2}CuO_{6+δ} (Tl2201) (13, 17). The distinct crystal structures of these four cuprates are shown in Fig. 1. Due to the quasi-2D electronic structure of the cuprates, with its underlying square-planar CuO_{2} sheet, it is natural to consider the resistance ρ_{☐} per Cu–O sheet (*SI Appendix*, section 3), although this is rarely done. Our simple and entirely model-free analysis reveals that, across a large part of the phase diagram, the sheet resistance coefficients *A*_{1☐} and *A*_{2☐} are universal. Moreover, these coefficients are simply proportional to the inverse hole concentration: *A*_{1☐} ∝ *A*_{2☐} ∝ 1/*p*. Our findings give profound insight into the phase diagram and the nature of the conducting states of the high-*T*_{c} cuprates.

## Results

### Planar Resistivity and Characteristic Temperatures.

Hg1201 crystals were prepared according to a previously reported procedure (18, 19). The in-plane dc resistivities of three underdoped samples with *T*_{c} = 80 K are displayed in Fig. 2*A*, and the magnetization characterization for one of them is shown in Fig. 2*B*. Regardless of their significantly different dimensions (e.g., *a–b* dimensions vary between about 0.01 and 1 mm), the samples exhibit identical temperature dependences (Fig. 2*C*), and hence a high degree of homogeneity. Extrapolation of the high-temperature linear behavior (ρ = ρ_{0} + *A*_{1}*T*) to zero temperature gives *T** ≈ 280 K and reveals a negligible residual resistivity. Such a small ρ_{0} is usually taken as the signature of a very clean metallic system, with minimal extrinsic and intrinsic disorder. This is noteworthy, because the underdoped cuprates typically exhibit a large ρ_{0} that increases with decreasing doping (6, 20).

The quadratic temperature dependence is demonstrated in Fig. 2*D*. For the *T*_{c} = 80 K samples, it is found below *T*** ≈ 170 K and spans about 80 K in temperature before the influence of superconducting fluctuations is noticeable at *T′*. As shown in Fig. 3 *A–C*, this behavior is also observed at lower doping (samples with *T*_{c} = 47 K and 67 K). In other cuprates, the pure underlying quadratic behavior tends to be masked: for example, we found no published evidence for Bi_{2}Sr_{2}CuO_{6+δ} (Bi2201) and Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} (Bi2212), two well-studied compounds for which disorder effects are known to be significant (4), or for other cuprates in the doping range 0.11 < p < 0.30. However, as summarized in Fig. 3, a quadratic planar resistivity has been observed in a few cases: underdoped YBCO (p = 0.03 and 0.09) (8, 10, 11), underdoped LSCO (p = 0.02 and 0.08) (8), as well as strongly overdoped LSCO (p = 0.33). The latter has been argued to be a Fermi liquid (16). In Fig. 3*E* and *SI Appendix*, section 5, we demonstrate that ρ ∝ *T*^{2} also holds at intermediate temperatures for prior data for LSCO at p = 0.01 (9). Tl2201 (13) (*T*_{c} = 15 K; Fig. 3*I*) and LSCO (14) at *p* ≈ 0.30 are in close proximity to the putative Fermi liquid regime, and the description of the planar resistivity requires only a small additional *T-*linear component. For underdoped YBCO at *p* ≈ 0.11 in a 55 T magnetic field, an approximately quadratic resistive behavior was reported down to very low temperature (12).

In Fig. 4*B* and *SI Appendix*, section 6, we demonstrate that the four characteristic temperatures and the underlying (hidden) quadratic resistive regimes of YBCO, LSCO, and Bi2201 can be identified consistently from prior contour plots of the second temperature derivative of the resistivity (20). Hg1201 and YBCO are structurally very different cuprates, with one and two CuO_{2} sheets per unit cell, respectively (Fig. 1). However, as shown in Fig. 4, the doping-dependent temperatures *T**, *T***, *T′*, and *T*_{c} determined from resistivity, which demarcate five distinct physical regions, are very similar. For both compounds, the opening of the pseudogap at *T** has been shown to be associated with a phase transition to a novel magnetically ordered state (21). The onset of the ρ ∝ *T*^{2} behavior below *T*** agrees surprisingly well with characteristic temperatures determined by two other probes: the maximum of the thermoelectric power (TEP) for both Hg1201 (22) and YBCO (23, 24) (*SI Appendix*, section 1), and the onset of a Kerr rotation signal for YBCO (25). In both compounds, superconducting fluctuations affect the dc conductivity only near *T*_{c} (below *T′*).

### Universal Sheet Resistance.

We now analyze the doping dependences of the linear and quadratic contributions to the sheet resistance for Hg1201, YBCO (7, 8, 10⇓–12), Tl2201 (13, 17), and LSCO (8, 9, 13⇓–15). As shown in Figs. 4 and 5, four primary regions need to be distinguished: the *T*-linear regime (p < *p** ≈ 0.19 and *T* > *T**), the two seemingly disconnected quadratic regimes (p < *p** and *T* < *T***; p > 0.30 and *T* < 55 K), and the intermediate “mixed” region (*p** < p < 0.30) accessed in high magnetic fields [data mostly for LSCO (14)].

For several reasons, the results in Figs. 2 and 5 are remarkable. First, for underdoped Hg1201, we observe a clear and dramatic “switch” of scattering mechanisms upon cooling: there is no discernible quadratic (linear) contribution above *T** (between *T*** and *T′*) and the residual resistivity is tiny. Second, *A*_{1☐} and *A*_{2☐} are universal, despite substantial differences in crystal structure, disorder, and optimal *T*_{c} of the four compounds (4). Consequently, the states near the Fermi level that contribute to the planar transport are essentially identical, and the underlying fundamental planar resistivity in the normal state of the cuprates is now known. Third, *A*_{1☐} (for p < *p**) and *A*_{2☐} (except near *p**) are, to a good approximation, simply proportional to the inverse hole concentration. Fourth, the scattering mechanism responsible for the linear temperature dependence of the resistivity is clearly related to fluctuations that disappear upon cooling below *T** and doping beyond *p**. This is apparent from the fact that purely *T*-linear behavior is observed only above *T**, and also from the behavior of the resistivity just above *p** (Fig. 5*B*), where *A*_{1☐} for both LSCO and Tl2201 (13, 14) decreases faster than 1/*p* and approaches zero as superconductivity disappears around p *=* 0.30.

## Discussion and Conclusions

### Discussion of Doping Dependence.

Based on the prior observation of metallic resistive behavior at low hole concentrations, a real-space picture of mesoscopic phase segregation was proposed, with a doping-dependent change of the effective volume relevant to charge transport (9). However, the evidence for such phase segregation in different cuprate families is varied, which appears difficult to reconcile with our observation of universality over a wide doping range (Fig. 5). Another viewpoint is that much of the cuprate phase diagram is controlled by an underlying quantum critical point (26, 27), which is supported by observations of novel magnetism below *T** (21, 28). In quantum critical-point theories, the effective interactions among electrons, and consequently all single-particle renormalization phenomena are assumed to be controlled by a fluctuating order parameter of some kind. Scattering off such fluctuations for *T* > *T** is proposed to cause the linear-*T* dependence of the resistivity. Interpreted in this fashion, the result in Fig. 5 indicates that the critical fluctuations either condense below *T** (p < *p**) or gradually disappear (p > *p**). We note, though, that the 1/*p* dependence of *A*_{1☐} and the quadratic resistive behavior for *T* < *T*** are not predicted by existing quantum critical-point theories.

A more specific picture is obtained by using the Drude formula ρ = *m***/*(*ne*^{2}τ), which only assumes that it is possible to separate the scattering rate (1*/τ*) from the ratio of the carrier effective mass to density (*m**/*n*) (29). The observation of distinct power-law behaviors (ρ_{☐} = *A*_{1☐}*T* and *A*_{2☐}*T*^{2}) over a wide doping range below *p** suggests that the scattering rate is proportional to *T* and *T*^{2} in the respective regions of the phase diagram. One interpretation of the result in Fig. 5 is that the doping dependences of the scattering rates and carrier densities in the Drude expression compensate exactly in such a way that *A*_{1☐} ∝ *A*_{2☐} ∝ 1/*p*. However, the simplest interpretation of this proportionality for p < *p** is to associate the doping dependence of the resistivity solely with the doped carriers: *n = p*. It follows that the respective scattering rates τ_{1} and τ_{2} as well as the effective mass *m** are doping independent and universal. This interpretation is consistent with several experimental observations (9, 30⇓–32), e.g., that the Fermi arc length in the pseudogap regime is proportional to *p* (31, 33) and that the optical effective mass enhancement is doping independent, whereas the effective optical charge density varies almost linearly with doping (32).

Furthermore, because *A*_{1☐} ∝ *A*_{2☐} ∝ 1/*p* for p < *p**, the effective number of carriers would seem to remain unaffected by the closing of the pseudogap with increasing temperature. The dominant contribution to the conductivity at all temperatures might therefore come from the relatively fast carriers in the nodal regions (34, 35). Alternatively, to explain the nodal character of the cuprates, one might invoke either the scattering off magnetic fluctuations (hot spots) associated with the antinodes or electronic scattering involving van Hove singularities near the Fermi surface (36). Finally, the antinodal states might be at the Planckian dissipation limit (14) beyond which coherent single-particle propagation is inhibited (37, 38). It has been argued that the pseudogap formation may then be viewed as the lowering of the electronic energy in response to this intense scattering, preventing the scattering from remnant quasiparticle states into the “hot” antinodal regions (14).

It is instructive to extend the above interpretation based on the Drude formula to the overdoped regime. For p > *p**, the cuprates feature a large Fermi surface volume, which in accordance with Luttinger’s theorem corresponds to 1 + *p* rather than *p* carriers (39, 40). For hole concentrations between *p** and *p* ≈ 0.30, the planar resistivity of overdoped LSCO and Tl2201 exhibits both linear and quadratic contributions below ≈200 K in a high *c*-axis magnetic field that suppresses the superconductivity (13, 14). We note that the linear term smoothly connects to the (zero-field) *A*_{1☐} ∝ 1*/p* behavior for p < *p** and *T* > *T** (Fig. 5). Above *p**, purely Fermi liquid-like quadratic resistivity has been observed for LSCO at p = *x* = 0.33, the highest doping level attained in bulk crystals (16) (*SI Appendix*, section 7). Unlike *A*_{1☐}, the determination of *A*_{2☐} near *p** is ambiguous, as it depends on the choice of polynomial or parallel resistor fit (14). However, above *p* ≈ 0.26, the contribution of the linear term is small, and the simple relation *A*_{2☐} ∝ 1/*p* appears to hold again. This suggests that the scattering mechanism in the pseudogap regime might be the same as in the putative Fermi-liquid state at high doping.

The clarification of the nodal/antinodal dichotomy, which may require the explicit theoretical treatment of oxygen degrees of freedom (38) (*SI Appendix*, section 8), appears to be essential to understand this observation.

### Pseudogap Phenomenon vs. Fermi Liquid.

The underdoped cuprates have been suggested to be “nodal metals” described by a two-component optical conductivity, with a low-energy Drude component associated with coherent quasiparticles on the Fermi arcs observed in photoemission experiments (10, 30). Photoemission experiments also indicate a nearly material and doping-independent near-nodal band structure characterized by a sizable Fermi velocity on the arcs (34, 35). Furthermore, the in-plane infrared spectral weight is found to be insensitive to the opening of the pseudogap (41). These results are consistent with our observations.

The quadratic resistive behavior in the pseudogap regime extends to rather high temperatures (Fig. 3 and *SI Appendix*, section 9) and is cut off at low temperature by either (nonuniversal) charge localization effects (9, 11) or superconducting fluctuations. We emphasize that an approximately quadratic temperature dependence was also reported for YBCO (*p* ≈ 0.11) down to low temperature (about 4 K) (Fig. 4*B*) after the suppression of superconductivity by a high magnetic field (12), and that the corresponding value of *A*_{2☐} falls on the universal plots of Fig. 5 *E* and *F*. As shown in Fig. 2*A*, for Hg1201 at temperatures below *T***, the mean-free path is considerably larger than the planar lattice constant; thus, the Ioffe–Regel criterion (42) for a good metal is satisfied (*SI Appendix*, section 10). The observation for YBCO of quantum oscillations in high magnetic fields appears to provide additional support for a Fermi-liquid state in the underdoped cuprates (43), although this phenomenon occurs in a rather narrow doping range, is suggestive of small (electron) pockets, and has not yet been shown to be universal. Additional evidence for seemingly conventional metallic behavior comes from the well-known *T*^{2} dependence of the Hall angle (*SI Appendix*, section 6) (8). Furthermore, motivated by the present work, optical conductivity measurements of Hg1201 indeed have revealed the quadratic frequency dependence and the temperature–frequency scaling expected for a Fermi liquid (44). Nevertheless, the situation is highly unconventional, as exemplified by the presence of arcs and by the evidence for a quantum critical point. Interestingly, NMR results for LSCO and Hg1201 imply a two-component local magnetic susceptibility in the pseudogap regime, one temperature-independent above *T*_{c}, i.e., Fermi-liquid-like, and the other temperature-dependent and correlated with the formation of the pseudogap (45). Further evidence for a two-component quasiparticle spectrum for p < *p** comes from a recent analysis of the electronic entropy in several cuprates (46).

### Updated Phase Diagram.

Fig. 6 shows the updated phase diagram of the cuprates. Fig. 6*A* focuses on the nonsuperconducting properties, neglecting complications (Fig. 6*B*) related to low structural symmetry, disorder and localization effects (4, 9, 11), the “stripe” instability near *p* ≈ 1/8 (47), and inhomogeneities near *p** (14). The pseudogap temperature *T** has been associated with a transition to a state with novel magnetic order in both YBCO and Hg1201 (21). The coincidence of *T*** with the onset of the polar Kerr effect in YBCO (25) raises the possibility that this temperature may be universally associated with a second phase transition above *T*_{c}. There exist at least two qualitatively different scenarios near *p** that cannot yet be distinguished. Either both *T**(*p*) and *T***(*p*) approach zero at *p** ≈ 0.2, the putative quantum critical point, or the two characteristic temperatures cross near optimal doping. A possible caveat is that, even if much of the phase diagram is controlled by a quantum critical point, this point may be inaccessible through the variation of physical parameters such as doping and magnetic field. Even though the Fermi surface undergoes a change in topology as it breaks up into arcs on the underdoped side of the phase diagram, the electronic scattering mechanism that gives rise to the quadratic resistive behavior appears to be the same at the extreme doping levels of 1% and 33%. In the quadratic resistive regime, the underdoped cuprates are rather good metals and may need to be thought of as a two-component electron fluid, as a result of correlations associated with the formation of the pseudogap state, in which the metallic component evolves from the Fermi liquid at the highest doping. A ρ ∝ *T*^{2} behavior was also found on the electron-doped side of the phase diagram. However, in contrast to our findings for the hole-doped cuprates, superconductivity appears to evolve from a state that exhibits *T-*linear resistivity (48) (*SI Appendix*, section 6).

The present work demonstrates that the quadratic planar resistivity is a universal property of the underdoped cuprates, and that the structurally simple model compound Hg1201 exhibits negligible residual resistivity and a dramatic switch from *T*-linear to quadratic behavior upon cooling. Although the resistivity is generally highly sensitive to compound-specific characteristics, we have furthermore achieved a quantitative, universal understanding of both the linear and quadratic regimes by considering the resistance per principal building block, the copper–oxygen sheet, of four structurally distinct cuprates. Analysis of compound-specific deviations (e.g., for Bi2201 and Bi2212, or for LSCO near *p* = 0.12) from the underlying behavior reported here can be expected to lead to new insights into the nonuniversal features exhibited by individual compounds. Most importantly, the present work provides a quantitative basis for the development of a comprehensive theoretical understanding of these fascinating materials.

## Acknowledgments

We gratefully acknowledge discussions with H. Alloul, A. V. Chubukov, A. Georges, S. A. Kivelson, D. van der Marel, F. Rullier-Albenque, and A. Shekhter. N.B. acknowledges, in particular, S. Barišić and L. P. Gor'kov for sharing their insights into the physics of the cuprates. The work at the University of Minnesota was supported by the Department of Energy, Office of Basic Energy Sciences. The work in Zagreb was supported by the Unity through Knowledge Fund. N.B. acknowledges support by The Alexander von Humboldt Foundation and through a Marie Curie Fellowship.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: greven{at}physics.umn.edu.

Author contributions: N.B. and M.G. designed research; N.B., M.K.C., Y.L., G.Y., X.Z., M.D., and A.S. performed research; N.B. and M.G. analyzed data; and N.B. and M.G. wrote the paper.

The authors declare no conflict of interest.

^{†}This Direct Submission article had a prearranged editor.This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1301989110/-/DCSupplemental.

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