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# Field-induced quantum critical route to a Fermi liquid in high-temperature superconductors

Edited by Sudip Chakravarty, University of California, Los Angeles, CA, and accepted by the Editorial Board March 4, 2008 (received for review December 28, 2007)

## Abstract

In high-transition-temperature (*T*_{c}) superconductivity, charge doping is a natural tuning parameter that takes copper oxides from the antiferromagnet to the superconducting region. In the metallic state above *T*_{c}, the standard Landau's Fermi-liquid theory of metals as typified by the temperature squared (*T*^{2}) dependence of resistivity appears to break down. Whether the origin of the non-Fermi-liquid behavior is related to physics specific to the cuprates is a fundamental question still under debate. We uncover a transformation from the non-Fermi-liquid state to a standard Fermi-liquid state driven not by doping but by magnetic field in the overdoped high-*T*_{c} superconductor Tl_{2}Ba_{2}CuO_{6+x}. From the *c*-axis resistivity measured up to 45 T, we show that the Fermi-liquid features appear above a sufficiently high field that decreases linearly with temperature and lands at a quantum critical point near the superconductivity's upper critical field—with the Fermi-liquid coefficient of the *T*^{2} dependence showing a power-law diverging behavior on the approach to the critical point. This field-induced quantum criticality bears a striking resemblance to that in quasi-two-dimensional heavy-Fermion superconductors, suggesting a common underlying spin-related physics in these superconductors with strong electron correlations.

Quantum criticality refers to a phase transition process between competing states of matter governed not by thermal but by quantum fluctuations demanded by Heisenberg's uncertainty principle (1). It has emerged at the front and center of the physics of strongly correlated electron systems known to host competing quantum orders, and is witnessed by a proliferation of reports on heavy Fermions (2⇓⇓–5), itinerant (quantum) magnets (6), and high-transition-temperature (high-*T*_{c}) superconductors (7), with quantum matter tuned (at times arguably) through a transition by pressure, magnetic field, or doping—arguably because one has to rely on long shadows cast by quantum criticality far above zero temperature (8), for, obviously, *T* = 0 K cannot ever be attained.

The often-invoked hallmark of quantum criticality is an unconventional behavior of resistivity. For resistivity contribution, the standard Fermi liquid (FL) theory of metals predicts a quadratic temperature dependence ρ(*T*) = ρ(0) + *AT*^{2} at low temperatures. In high-*T*_{c} cuprates, however, the baffling *T*-linear resistivity over a huge temperature range near optimal (hole) doping has been observed (9), flagging, in this sense, a non-Fermi liquid (n-FL) behavior in the metallic state above *T*_{c}. This has led to new theoretical concepts, some related [e.g., phenomenology of “marginal Fermi liquid” (10)] and some unrelated [e.g., “strange metal” state (11)] to quantum criticality. In most considerations of cuprates near quantum critical points (QCPs) the tuning parameter is charge doping (1, 10); and although there is some experimental support (7, 12) for a doping-driven QCP, it is still to be broadly confirmed. Thus, it is of primary import to probe experimentally how the n-FL state transforms into the conventional FL state, and whether and how charge or spin degrees of freedom are involved.

Here, we report on the transformation from such “strange” n-FL state to the conventional FL metallic state in high-*T*_{c} superconductors in high magnetic fields. Our experiments measuring charge transport in overdoped Tl_{2}Ba_{2}CuO_{6+x} reveal an unanticipated quantum criticality in a cuprate that is not doping-induced but field-induced. The results are in close correspondence with the quantum criticality in the quasi-two-dimensional heavy-Fermion superconductors having strong antiferromagnetic fluctuations, suggesting common fundamental physics of magnetic origin responsible for the observed QCP.

To have access to large regions of the metallic regime at low temperatures, we use magnetic fields to destroy superconductivity in heavily doped Tl_{2}Ba_{2}CuO_{6+x} (*T*_{c} ≈ 15 K). This material has a single CuO_{2} layer per unit cell and is relatively clean among cuprates as evidenced by the high *T*_{c} (up to 93 K) that can be controlled with oxygen content. We focus here on the *c*-axis longitudinal magnetotransport (*H* ‖ *c*), because it should be less affected by orbital contributions than the transverse geometry, and because in our overdoped system it is expected that Fermi surface is three-dimensional-like and coherent (13), as revealed by the fact that the temperature dependence of *c*-axis resistivity, ρ_{c}(*T*), can be well scaled by that of *ab*-plane resistivity, ρ_{ab}(*T*) (see below).

## Results and Discussion

Fig. 1 shows the temperature dependence of *c*-axis resistivity at zero and 45-T fields. At zero field, ρ_{c}(*T*) is metallic all of the way down to *T*_{c}. This represents a clear contrast with the semiconducting-like upturn in ρ_{c} observed at lower dopings of Bi_{2}Sr_{2}CaCu_{2}O_{8+y} (14, 15) in the pseudogap state (16). We can examine our data within the overall temperature dependence ρ_{c}(*T*) = ρ_{c0} + *A*_{0}*T*^{2} + *CT*, which reproduces the temperature dependence of ρ_{ab}(*T*) (17). In addition, it can be as easily fitted by a power law with the exponent 1.3 (<2) (Fig. 1 *Inset*). Regardless of the choice, the temperature dependence is not *T*-quadratic as in a conventional FL; it marks an n-FL state even in the heavily overdoped region.

When we apply 45 T along the *c* axis, the superconductivity is destroyed, and the entire temperature dependence up to 100 K can now be fitted with the simple FL form ρ_{c}(0) + *AT*^{2}. This clearly demonstrates that sufficiently high magnetic fields destroy all remnants of the n-FL behavior, recovering the all familiar Fermi-liquid metal; i.e., in this overdoped cuprate there exists a field-induced transformation from the n-FL to FL state.

To follow the temperature dependence of ρ_{c} at different fields, we plot it against *T*^{2} in Fig. 2. It is evident that the *AT*^{2} dependence is observed below a field-dependent temperature *T*_{FL} indicated by the arrows. At higher temperatures, the ρ_{c}(*T*) data deviate from the *T*^{2} behavior as can be seen more clearly by subtracting ρ_{c}(0) + *A*(*H*)*T*^{2} in Fig. 2 *Upper*. We note that although the change is gradual, the power in the temperature dependence unmistakably changes from 2 at low temperatures (*T* < *T*_{FL}) to <2 at high temperatures (*T* > *T*_{FL}). The field dependence of the *T*_{FL} is depicted in the *T–H* diagram in Fig. 3. At 45 T, FL state extends up to 100 K, and at lower fields, the Fermi liquid breaks down crossing to an n-FL behavior above *T*_{FL}. With decreasing field, *T*_{FL}(*H*) decreases linearly and extrapolates to zero in the vicinity of the upper critical field *H*_{c2}(0) (see below), terminating at a putative QCP. We conclude then that in zero temperature limit, the normal state above *H*_{c2} in Tl_{2}Ba_{2}CuO_{6+x} is a Fermi liquid, in agreement with the recent observation in this system of the Wiedemann–Franz law (18).

Next, we examine the field dependence of ρ_{c} at constant temperatures, plotted in Fig. 4*a*. At low temperatures, the resistivity is zero below the so-called irreversibility field *H*_{irr} (14) in the vortex solid state. We note that above *H*_{irr}, the magnetoresistance is always positive. We recall that in less doped pseudogapped Bi_{2}Sr_{2}CaCu_{2}O_{8+y}, the observed magnetoresistance is negative over a large field range (14), consistent with filling of the low-energy states within the pseudogap in the applied magnetic field. We surmise then that at this doping, the pseudogap is either way below the superconducting energy scale or perhaps entirely absent.

The superconducting coherence can survive up to a characteristic field *H*_{sc}, above which the quasi-particle conductivity overcomes the vortex contribution (19⇓–21). This often underestimates the upper critical field *H*_{c2} near *T*_{c} in high-*T*_{c} cuprates; it is notoriously difficult to obtain from transport owing to large thermal fluctuations. However, previous studies of *c*-axis magnetotransport (20) revealed that in the overdoped regime in the low-*T* limit, *H*_{c2} is very near *H*_{sc}(0). In our sample, we evaluate μ_{0}*H*_{c2}(0) ∼ 8 T. Above this limiting field, ρ_{c}(*H*) at low *T* is strictly *H*-linear in the normal state over the entire field range.

To take a closer look at higher *T*, we subtract the high-field linear term from ρ_{c}(*H*) and obtain δρ_{c}, which quantifies the deviation from the *H*-linear dependence. This analysis highlights a noticeable deviation from the field-linearity below a temperature-dependent characteristic field *H*_{FL} (see Fig. 4*b*). The obtained *H*_{FL}(*T*) is also plotted in Fig. 3. Remarkably and consistently, it follows the *T*_{FL}(*H*) line within the experimental error bars. We surmise then that although the *H*-linear and large magnetoresistance is a nontrivial finding in its own right that needs to be further understood, here it clearly is a phenomenon of the Fermi liquid. Indeed, several theoretical accounts within the Fermi-liquid picture derive large *H*-linear ρ_{c}(*H*) (22, 23).

We remark that at low temperatures below 5 K, the standard FL state is confirmed by the classical Kohler's rule for magnetoresistance (see Fig. 4*c*). At higher temperatures, where the low field data below *H*_{FL} [including ρ_{c}* ^{n}*(0)] no longer follow what is expected in the simple FL state, the scaling is clearly violated; and while the violation of Kohler's rule at high temperatures can be caused by other mechanisms, the low-temperature data are consistently in correspondence with the field-induced FL state. The temperature-dependent violation further indicates that here the magnetoresistance is not simply governed by ω

_{c}τ (a product of the cyclotron frequency and scattering time). From this, we conclude that the observed field-induced

*AT*

^{2}behavior is an intrinsic effect and not an artifact due to ω

_{c}τ.

At finite temperatures, the observed field-induced transformation appears to be crossover-like. So now we will ask whether the *T* → 0 K terminus of *H*_{FL}(*T*) indicates a true phase transition at QCP. We note the conspicuously strong field dependence of the FL coefficient *A*: it increases with decreasing field and decreasing *T*_{FL} (see Fig. 3 *Inset*). Indeed, we find that the field dependence can be fitted to
where *A _{0}* and

*D*are constants and α and

*H*

_{QCP}are the relevant parameters of the fit. As we discussed earlier, in zero field, ρ

_{c}(

*T*) can be analyzed either by a power law or by the ρ

_{c0}+

*A*

_{0}

*T*

^{2}+

*CT*dependence. In the analysis of the field dependence, the two different forms would require different values of

*A*

_{0}in Eq.

**1**. In the former case, we take

*A*

_{0}= 0 μΩcm/K

^{2}, and we can fit the

*A*(

*H*) by α = 0.62 and

*H*

_{QCP}= 7.4 T. In the latter case, we use the finite coefficient

*A*

_{0}= 0.86 μΩcm/K

^{2}(see Fig. 1), and the fit gives α = 1.04 and

*H*

_{QCP}= 5.8 T.

Thus, experimental *A*(*H*) algebraically diverges at *H*_{QCP}. Within Fermi-liquid theory, as *T* → 0 K, the energy dependence of the total scattering rate near the Fermi surface takes the form 1/τ = 1/τ_{0} + *a*(*E* − *E*_{F})^{2} (1/τ_{0} comes from the impurity scattering, *a* is a constant in energy *E*, and *E*_{F} is the Fermi energy) (see, e.g., ref. 24). At the QCP, the singularity of *a*(*H*) will mirror that of *A*(*H*)—the two coefficients are related through the quasi-particle–quasi-particle scattering cross-section. The found divergence of *A* thus gives us confidence in assigning *H*_{QCP} as the QCP field, and α as the exponent characterizing quantum criticality. We remark that strongly correlated electron systems commonly obey the Kadowaki–Woods relation *A* ∝ γ^{2} (25), where γ is the electronic coefficient of specific heat and a measure of the effective mass *m** of a Landau quasi-particle. Although this relation is complex [and sometimes violated (5)], we note that with large [≈10^{3} (17)] resistivity anisotropy in Tl_{2}Ba_{2}CuO_{6+x}, the obtained *A* values near *H*_{QCP} imply enhanced γ ∼ 30 mJ/mol-K^{2}, comparable with that, e.g., in superconducting Sr_{2}RuO_{4} (26), where similarly anisotropic *A* values between the *c*-axis and in-plane resistivities have been observed. This enhancement of *A* and a lack of saturation may also be related to the enhanced susceptibility χ_{0} in the overdoped Tl_{2}Ba_{2}CuO_{6+x} (9). We surmise then that at finite temperatures the system is governed by the quantum fluctuations, generating the n-FL state that crosses over to the conventional FL above *H*_{FL}.

The n-FL state with non-*T*^{2} dependence of resistivity (2) and a violation of Kohler's rule (3) has also been observed in heavy-Fermion superconductors having strong antiferromagnetic fluctuations. Notably, in CeCoIn_{5} with quasi-two-dimensional electronic structure, a quite similar field-induced QCP has been identified by the transport and specific heat measurements (27⇓–29). In high magnetic fields, the resistivity recovers the *AT*^{2} dependence at low temperatures in a similar manner near the upper critical field *H*_{c2}(0) (≈5 T). It has been pointed out (28) that the underlying antiferromagnetic fluctuations (30) become critical in the immediate vicinity of the superconductivity, preventing development of magnetic order. We note that anisotropic violation of the Wiedemann–Franz law in CeCoIn_{5} was recently found near the QCP (31), where the FL renormalization parameter *Z* (≈1/*m**) tends to zero in the *c* direction but remains finite in the *ab* plane. This suggests that the *c* direction is more susceptible to instabilities related to QCP.

An intriguing question to ask is whether field-induced *H*_{QCP} ≈ *H*_{c2}(0) in a highly overdoped cuprate is a shear coincidence or whether they are inherently linked. In particular, one may ask whether an extended regime of superconducting fluctuations can promote the observed n-FL state. In the heavy-Fermion superconductor CeCoIn_{5}, FL coefficient *A* also diverges at the QCP located very near *H*_{c2}(0), with α close to unity (27). By applying pressure, *H*_{QCP} is strongly suppressed and is no longer coincident with *H*_{c2}(0) (29). This rather compellingly points to a QCP controlled by a competing order, most likely related to antiferromagnetism (32). In cuprates, neutron-scattering experiments (33, 34) show that magnetic field can induce a distinct static magnetic order, and a surprisingly much enhanced spin fluctuations at low *T* within the vortex cores, also detected by a spatially resolved NMR (35). Thus, spin correlations in cuprates seem to experience a field-induced boost.

Our work, in a departure from previous studies, probes the high-field regime at very high hole doping—much distanced from the antiferromagnetic “mother order.” That the antiferromagnetic fluctuations (36) could have such long reach (37) and play a role in the uncovered field-induced QCP is quite extraordinary. We expect that the true nature of the quantum critical fluctuations that produce the n-FL state in the highly overdoped Tl_{2}Ba_{2}CuO_{6+x} is complex because here we are not far from the superconductivity's charge-doping end point (38). From our experiments, with salient similarities found between a cuprate and a heavy-Fermion compound, all evidence here points to a spin-controlled QCP universal to these strongly correlated electron systems.

## Materials and Methods

Single crystals of Tl_{2}Ba_{2}CuO_{6+x} were grown by a flux method (39). In this system, the doping can be tuned by oxygen content covering a range from somewhat overdoped (*T*_{c} ≈ 93 K) up to heavily overdoped (*T*_{c} ≈ 0 K) (9). In our study, we used a homogeneous highly overdoped crystal with a sharp transition at *T*_{c} ≈ 15 K (see Fig. 1). The *c*-axis resistivity ρ_{c}(*T*, *H*) was measured in the 45-T hybrid magnet at NHMFL (comprising an 11.5-T superconducting outsert and 33.5-T resistive insert magnets) by the standard four-probe method using an ac resistance bridge (40). The temperature at high fields was controlled to ≈50 mK by using a capacitance censor at low temperatures, where the magnetoresistance of Cernox resistive sensors is not negligible.

## Acknowledgments

We thank A. I. Buzdin, S. Chakravarty, S. Fujimoto, N. E. Hussey, H. Kontani, and C. M. Varma for discussions, and B. Brandt for technical assistance at NHMFL. This work was supported in part by Grants-in-Aid for Scientific Research from the Japan Society for the Promotion of Science, and for the 21st Century COE “Center for Diversity and Universality in Physics” from the Ministry of Education, Culture, Sports, Science, and Technology, Japan.

## Footnotes

- ↵
^{‡}To whom correspondence should be addressed. E-mail: shibauchi{at}scphys.kyoto-u.ac.jp

Author contributions: T.S. and L.K.-E. designed research; T.S., L.K.-E., M.H., Y.K., and R.O. performed research; T.S. and L.K.-E. analyzed data; and T.S., L.K.-E., and Y.M. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. S.C. is a guest editor invited by the Editorial Board.

- Received December 28, 2007.

- © 2008 by The National Academy of Sciences of the USA

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