Anomalous density fluctuations in a strange metal
- aDepartment of Physics, University of Illinois at Urbana–Champaign, Urbana, IL 61801;
- bMaterials Research Laboratory, University of Illinois at Urbana–Champaign, Urbana, IL 61801;
- cInstitute for the Theory of Condensed Matter, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany;
- dDepartment of Physics and Astronomy, University of Oklahoma, Norman, OK 73069;
- eCondensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, NY 11973
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Edited by Steven A. Kivelson, Stanford University, Stanford, CA, and approved April 10, 2018 (received for review December 10, 2017)

Significance
The strange metal is a poorly understood state of matter found in a variety of quantum materials, notably both Cu- and Fe-based high-temperature superconductors. Strange metals exhibit a nonsaturating, T-linear electrical resistivity, seemingly indicating the absence of electron quasiparticles. Using inelastic electron scattering, we report a momentum-resolved measurement of the dynamic charge susceptibility of a strange metal, optimally doped Bi2.1Sr1.9CaCu2O8+x. We find that it does not exhibit propagating collective modes, such as the plasmon excitation of normal metals, but instead exhibits a featureless continuum lacking either temperature or momentum dependence. Our study suggests the defining characteristic of the strange metal is a singular type of charge dynamics of a new kind for which there is no generally accepted theory.
Abstract
A central mystery in high-temperature superconductivity is the origin of the so-called strange metal (i.e., the anomalous conductor from which superconductivity emerges at low temperature). Measuring the dynamic charge response of the copper oxides,
The nonsuperconducting normal state of the high-temperature superconductors, usually referred to as the “strange metal,” has many properties that cannot be explained by the conventional Landau–Fermi liquid theory of metals (1, 2). These include a resistivity that is linear in temperature and exceeds the Mott–Ioffe–Regel limit (2⇓⇓–5), an in-plane conductivity exhibiting an anomalous power-law dependence on frequency (6, 7), a magnetoresistance that violates Kohler’s rule (8), a quasiparticle damping,
The main spectroscopic signature of the strange metal is a featureless continuum observed to the highest measurable energy in Raman scattering experiments (16, 17). Its origin is still unknown, exemplifying the need for a new experimental probe of the collective excitations of strange metals, particularly one that could determine how this continuum evolves at finite momentum, q. Generically, the most direct measure of the collective excitations of any material is its dynamic charge susceptibility,
Here we report a millielectronvolt-resolved,
Probing anomalous density fluctuations in the normal state of cuprates. (A) Scattering geometry of the M-EELS experiment.
M-EELS measures the surface density–density correlation function,
Fig. 2A shows
Continuum collapse in OP BSCCO. (A) Dynamic charge susceptibility,
As the momentum is increased to beyond
The momentum dependence of
The broad plasmon linewidth at small momentum is evidence that the continuum is present even for q < 0.15 r.l.u., which would lead to decay of the plasmon via Landau damping (18). To evaluate this possibility, we determined the polarizability of the system,
Determining
For this reason, we modeled the particle-hole continuum using the empirical expression (26)
Having established a plausible form for
Stated more succinctly, the polarizability
The observed power laws might be interpreted as evidence for a quantum critical point near optimal doping claimed by many authors (1). To evaluate this possibility, we repeated our experiment on OD BSCCO with Tc = 50 K, which is widely believed to exhibit a cross-over to a more Fermi liquid-like phase at low temperature (1, 5, 15). One would expect to observe deviation from simple power law behavior at low temperature. Fig. 3 shows the temperature dependence of the M-EELS spectra from OD BSCCO compared with that of the OP material. At T = 295 K, the spectra are similar, indicating that the power-law region persists over a finite range of doping at high temperature. As the temperature is lowered, however, a gap-like feature appears in the OD spectra below 0.5 eV, indicating the emergence of an energy scale not present at optimal doping. This behavior is, at first glance, consistent with the emergence of a more Fermi liquid-like phase at low temperature, and the presence of a fan-shaped quantum critical region centered on optimal doping (1, 26, 28, 29).
The data do not, however, exhibit the generic properties of a quantum critical point (29). For one, the ∼0.5-eV gap-like feature is more than an order of magnitude larger than the temperature scale on which it emerges,
Furthermore, the M-EELS spectra are momentum-independent even in OD samples in which the gap-like feature is observed. Fig. 4 shows the momentum dependence of the OD data at T = 115 K (SI Appendix, Fig. S4 shows the data at 295 K). The spectra show very little q dependence, just as in the OP case. We fit the data using Eqs. 1 and 2, though only for
Continuum collapse in OD BSCCO. (A) Dynamic charge susceptibility,
We close by speculating about the underlying cause of the density fluctuations we observe. Our results bear a striking similarity to the so-called marginal Fermi liquid (MFL) hypothesis, which asserts that the strange metal is a consequence of a featureless continuum of fluctuations that pervades all time and length scales (30). This continuum is conjectured to arise from quantum fluctuations of some hidden order parameter that exhibits “local criticality,” meaning that the spatial correlation length
Another possibility is that the response functions of the strange metal are dominated by disorder. BSCCO is known to be electronically inhomogeneous (32) and also exhibits an incommensurate supermodulation due to structural misfit between the CuO2 and BiO layers (33). Disorder breaks translational symmetry and can explicitly broaden features in a momentum-resolved measurement such as M-EELS. Furthermore, random disorder has been shown, in simple spin models, to give rise to singular, frequency-independent correlation functions of the sort we observe here (34, 35). Further studies of the response of strongly correlated systems to disorder are needed to clarify this issue.
Recently, theoretical approaches have been developed to address the strange metal problem from a completely new perspective. The anti-de Sitter/conformal field theory correspondence, which relates a gravity theory in a curved spacetime to a strongly interacting quantum field theory on its boundary (36), is one such approach that has already been used to reproduce some properties of the strange metal holographically (37). Rapid developments in this area may shed new light on this problem.
In summary, we present a q-resolved measurement of the dynamic charge susceptibility of a strange metal at the millielectronvolt scale. We have uncovered a type of charge dynamics in which the fluctuations are local to such a degree that space and time axes are effectively decoupled. Explaining this observation may require a new kind of theory of interacting matter.
Methods
Sample Growth and Characterization.
OP single crystals of Bi2.1Sr1.9Ca1.0Cu2.0O8+x with Tc = 91 K were grown by the floating-zone method (38). Overdoping was achieved by annealing in a hot, isostatic press with gas pressure 6.8 kbar at temperature 500 °C for 100 h. The gas mixture was 20% O–80% Ar with oxygen partial pressures up to 1.35 kbar. The Tc values were determined using superconducting quantum interference device magnetometry.
M-EELS Measurements.
M-EELS measurements were performed using an Ibach-type HR-EELS spectrometer (39) that was motorized and mated to a custom, multiaxis sample goniometer. Centering of the rotation axes was done using remote cameras and reference scatterers, as described previously (19). Experiments were done in a magnetically shielded ultrahigh vacuum (UHV) chamber at 5 × 10−11 torr vacuum and residual field of 3 mG using a 50-eV beam energy at 170 pA current and overall resolutions of 4 meV in energy and 0.02 Å−1 in momentum. The BSCCO surfaces were prepared by cleaving along a (001) surface normal in a UHV prep chamber. Measurements were performed on three different OP and four OD crystals and tested on several cleaves of the same sample. Each figure of this paper reports data collected on a single, independent cleave. The crystals were oriented by locating elastic scattering from the (1,0) and (0,0) (specular) Bragg reflections and building an orientation matrix relating goniometer angles to momentum space (19). In this paper, Miller indices
Determining the Susceptibility from M-EELS Data.
The M-EELS cross-section is given by (19, 21)
where
where q is the in-plane component of the momentum transfer and
The overall scale was determined by applying the f-sum rule [4],
where m is the bare electron mass and
Random Phase Approximation Calculations.
The charge susceptibility,
Acknowledgments
We thank C. M. Varma for discussions and assistance analyzing the data and J. Zaanen, N. D. Goldenfeld, and P. W. Phillips for helpful discussions. This work was supported by the Center for Emergent Superconductivity, an Energy Frontier Research Center funded by the US Department of Energy (DOE), Office of Basic Energy Sciences under Award DE-AC02-98CH10886. Crystal growth was supported by DOE Grant DE-SC0012704. P.A. acknowledges support from the EPiQS program of the Gordon and Betty Moore Foundation, Grant GBMF4542. B.U. acknowledges NSF CAREER Grant DMR-1352604. M.M. acknowledges support by the Alexander von Humboldt Foundation through the Feodor Lynen Fellowship program.
Footnotes
- ↵1To whom correspondence may be addressed. Email: mmitrano{at}illinois.edu or abbamont{at}illinois.edu.
↵2Present address: Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139.
Author contributions: P.A. designed research; M.M., A.A.H., S.V., A.K., M.S.R., S.I.R., J. Schmalian, B.U., J. Schneeloch, R.Z., G.D.G., and P.A. performed research; M.M. and A.A.H. analyzed data; J. Schneeloch, R.Z., and G.D.G. grew and characterized samples; and M.M. and P.A. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
Data deposition: The data reported in this paper have been deposited on Zenodo (available at https://zenodo.org/record/1229614#.WuDdum4vxpg).
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1721495115/-/DCSupplemental.
Published under the PNAS license.
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