Crossover of Charge Fluctuations across the Strange Metal Phase Diagram

TL;DR

This study investigates charge fluctuations in the strange metal phase of Bi-2212, revealing a persistent featureless continuum across the phase diagram and highlighting a crossover in spectral weight behavior at optimal doping.

Contribution

It demonstrates that plasmons are absent in Bi-2212 and identifies a universal continuum feature in the strange metal phase, with a crossover in spectral weight behavior at optimal doping.

Findings

Continuum persists throughout the strange metal region.

Spectral weight increases in underdoped samples at low temperature.

Spectral weight decreases in overdoped samples at low temperature.

Abstract

A normal metal exhibits a valence plasmon, which is a sound wave in its conduction electron density. The mysterious strange metal is characterized by non-Boltzmann transport and violates most fundamental Fermi liquid scaling laws. A fundamental question is: Do strange metals have plasmons? Using momentum-resolved inelastic electron scattering (M-EELS) we recently showed that, rather than a plasmon, optimally-doped Bi$_{2.1}$Sr$_{1.9}$Ca$_{1.0}$Cu$_{2.0}$O$_{8+x}$ (Bi-2212) exhibits a featureless, temperature-independent continuum with a power-law form over most energy and momentum scales [M. Mitrano, PNAS 115, 5392-5396 (2018)]. Here, we show that this continuum is present throughout the fan-shaped, strange metal region of the phase diagram. Outside this region, dramatic changes in spectral weight are observed: In underdoped samples, spectral weight up to 0.5 eV is enhanced at low temperature, biasing the system towards a charge order instability. The situation is reversed in the overdoped case, where spectral weight is strongly suppressed at low temperature, increasing quasiparticle coherence in this regime. Optimal doping corresponds to the boundary between these two opposite behaviors at which the response is temperature-independent. Our study suggests that plasmons do not exist as well-defined excitations in Bi-2212, and that a featureless continuum is a defining property of the strange metal, which is connected to a peculiar crossover where the spectral weight change undergoes a sign reversal.

Figures

Equations

$${\int }_0^{2\text{eV}}\omega {\chi }^{\prime \prime }(\mathbf{q},\omega )d\omega =-\pi \frac{q^2}{2m}{N}_{\text{eff}},$$

$$\chi (\mathbf{q},\omega )=\frac{\Pi (\mathbf{q},\omega )}{{ϵ}_{\infty }-V(\mathbf{q})\Pi (\mathbf{q},\omega )},$$

$$V(\mathbf{q})\propto \frac{e^{-qd}}{q}.$$

$$\xi =-{\int }_{0.1\text{eV}}^{0.5\text{eV}}\left[{\chi }^{\prime \prime }(q^{\ast },\omega )-{\chi }_{\text{ref}}^{\prime \prime }(q^{\ast },\omega )\right]d\omega ,$$

$$\frac{1}{{ϵ}_{\infty }}|\Pi (\mathbf{q},\omega )-\Pi (0.5,\omega )|=|\frac{\chi (\mathbf{q},\omega )}{1+V_{\text{fit}}(q)\chi (\mathbf{q},\omega )}-\chi (0.5,\omega )|.$$

$$\underset{q\to 0}{\lim }\Pi (q,0)=-n^2\kappa ,$$

Full text

I Introduction

The enigmatic and poorly understood strange metal has been found within the phase diagrams of many strongly-correlated systems, including transition metal oxides, heavy fermion materials, organic molecular solids, and iron-pnictide superconductors Norman (2011); Hussey et al. (2004, 2006); Stewart (2001, 2006); Dressel (2011); Kasahara et al. (2010). This phase is characterized by violation of fundamental Fermi liquid scaling laws, and by its close proximity to other exotic phases such as unconventional superconductivity, charge or spin density waves, and nematicity Stewart (2001); Dressel (2011); Norman (2011); Ginsberg (1998). For example, the prototypical copper-oxide strange metals, which are also high temperature superconductors, exhibit a resistivity that is linear in temperature and exceeds the Mott-Ioffe-Regel limit Keimer et al. (2015); Hussey et al. (2004), an optical conductivity exhibiting an anomalous power law dependence on frequency Basov and Timusk (2005); van der Marel et al. (2003), a magnetoresistance that is linear in field, violating Kohler’s rule Giraldo-Gallo et al. (2018), a quasiparticle decay rate that scales linearly with energy Valla et al. (2000), and an NMR spin relaxation rate that violates the Korringa law Berthier et al. (1993). No generally accepted theory of matter can explain these properties, which appear to be incompatible with fundamental assumptions of Boltzmann transport theory. The strange metal has thus become one of the great unsolved problems in condensed matter physics.

The quasiparticle dynamics of strange metals have been studied extensively with angle-resolved photoemission (ARPES) and scanning tunneling microscopy (STM) techniques, which directly measure the one-electron spectral function Damascelli et al. (2003); Reber et al. (2015); Bok et al. (2016); Vishik et al. (2010); Chatterjee et al. (2011); Fujita et al. (2012). However, little is known about the two-particle charge response, which directly reveals the strongly correlated nature of this phase Zaanen et al. (2006).

The fundamental charge collective mode of an ordinary metal is its plasmon, which is essentially a sound wave in its valence electron density Nozières and Pines (1999). On the other hand, we recently showed that in optimally doped Bi2.1Sr1.9Ca1.0Cu2.0O8+x (Bi-2212), a known strange metal, the plasmon is overdamped and rapidly decays into a momentum-, energy-, and temperature-independent continuum extending up to an energy of $\sim 1$ eV Mitrano et al. (2018). Initial measurements away from optimal doping, on heavily overdoped samples, showed a surprising depletion of spectral weight below 0.5 eV at low temperature Mitrano et al. (2018). This energy scale is more than $20\times$ larger than the temperature scale on which these changes take place, suggesting strong interactions are at play Basov et al. (2011). It is therefore crucial to map out the exact region of the phase diagram where a featureless continuum is present, so its connection to the strange metal and other neighboring phases can be established.

Here, we present a study of the density fluctuations across the doping-temperature phase diagram of the strange metal Bi-2212 using momentum-resolved electron energy-loss spectroscopy (M-EELS) Vig et al. (2017). This technique measures the surface dynamic charge response of a material, $\chi(\mathbf{q},\omega)$, and directly reveals the charged bosonic collective modes of the system Vig et al. (2017). Note that, while generally regarded as a surface technique, the probe depth of M-EELS is given by the inverse of the in-plane momentum transfer, $q^{-1}$, making it somewhat more bulk sensitive than single-particle spectroscopies like ARPES and STM Vig et al. (2017). In this study we focus on the energy regime 0.1 eV $<\omega<2$ eV, which is relevant to the high-temperature normal state out of which superconductivity and other instabilities form.

II Experiment

Single crystals of Bi-2212 were grown using floating zone methods described previously Wen et al. (2007). The current study was done on underdoped crystals with $T_{c}$=50 K (UD50K) and 70 K (UD70K), optimally doped crystals with $T_{c}$=91 K (OP91K), and overdoped crystals with $T_{c}$=50 K (OD50K).

Measurements of the charge fluctuation spectra were performed using meV-resolution, momentum-resolved electron energy-loss spectroscopy (M-EELS) Vig et al. (2017). M-EELS is a variant of surface HR-EELS Ibach and Mills (1982) in which the momentum transfer of the probe electron is measured with both high resolution and accuracy Vig et al. (2017). Measurements were performed on cleaved single crystals of Bi-2212 at 50 eV incoming electron beam energy and 4 meV energy resolution at a fixed out-of-plane momentum transfer, $q_{z}=4.10\textrm{ \AA}^{-1}$. We use Miller indices, $(H,K)$, to designate an in-plane momentum transfer $q=(2\pi H/a,2\pi K/a)$, where $a=3.81\textrm{ \AA}$ is the tetragonal Cu-Cu lattice parameter Damascelli et al. (2003). Unless otherwise specified, all momenta are along the (1,-1) crystallographic direction, i.e., perpendicular to the structural supermodulation in this material Damascelli et al. (2003). The sample orientation matrix was determined in situ using the (0,0) specular and (1,0) Bragg reflections, and verified by observing the (1,-1) reflection, establishing a quantitative relationship between the momentum, $q$, and the diffractometer angles. M-EELS spectra were taken from 0 to 2 eV energy loss and binned into 30 meV intervals for improved statistics.

M-EELS measures the dynamic structure factor of a surface, $S(\mathbf{q},\omega)$, which is proportional to the surface dynamic charge susceptibility, $\chi^{\prime\prime}(\mathbf{q},\omega)$, by the fluctuation-dissipation theorem Ibach and Mills (1982); Kogar et al. (2014); Vig et al. (2017). $\chi^{\prime\prime}$ was determined from the raw data by dividing the M-EELS matrix elements, which depend on the momentum transfer Vig et al. (2017), and antisymmetrizing to remove the Bose factor Vig et al. (2017); Mitrano et al. (2018). The data were placed on an absolute scale by performing the partial $f$-sum rule integral,

[TABLE]

where $N_{\textrm{eff}}$ was determined by integrating the $q=0$ ellipsometry data from Ref. Levallois et al. (2016) over the same energy interval at the corresponding values of temperature and doping.

III M-EELS Data

Figure 1 shows the M-EELS spectra for a selection of momenta at room temperature (300 K) and at low temperature (100 K or 115 K, depending upon the doping). The intensity rise below 0.1 eV in all spectra is due to the well-known Bi-2212 optical phonons Vig et al. (2017); Mills et al. (1994); Qin et al. (2010). Looking at the 300 K data, the lowest momentum (Fig. 1(a)) shows a highly damped plasmon reported previously Mitrano et al. (2018),which appears in the spectrum as a local maximum at $\omega\sim 1$ eV energy loss. This peak has very similar shape in all four dopings studied, and has the same energy and width as the plasmon observed in transmission EELS measurements at the same in-plane momentum Vig et al. (2017); Mitrano et al. (2018); Nücker et al. (1991). Previous studies indicate that the optical properties of the cuprates are dominated below $\sim 1$ eV by excitations in the CuO2 planes, but that interband transitions, potentially involving the BiO layers, contribute above 1.5 eV and may create features around 2.3 eV and 3.8 eV Quijada et al. (1999); Bozovic (1990); Giannetti et al. (2011). We therefore expect the M-EELS continuum principally arises from the CuO2 planes but may have other contributions at high energy.

At larger momenta (Fig. 1 (b)-(d)) the plasmon is no longer present, by which we mean that a local maximum is no longer observed in the spectra. Instead, the plasmon decays into an energy-independent continuum, as reported previously Mitrano et al. (2018). As shown in Appendix B, this continuum appears also in transmission EELS experiments, and is thus a bulk property of Bi-2212. This continuum is not a subtle effect; it saturates the $f$-sum rule and is the primary feature of the charge response of Bi-2212. As reported in Ref. Mitrano et al. (2018), the M-EELS spectra exhibit very little $q$-dependence at momenta greater than 0.16 r.l.u., which implies that charge excitations barely propagate in this material. Additionally, Fig. 1 (a)-(d) shows that, at $T=300$ K, the continuum also has little doping dependence across the composition range.

Surprisingly, the spectra become dramatically doping-dependent at low temperature (Fig. 1(e)-(h)), which is the main finding of this work. In overdoped samples, the spectral weight below 0.5 eV is greatly suppressed at low temperature Mitrano et al. (2018). By contrast, in underdoped materials, the weight in this energy range at low temperature is enhanced. The 0.5 eV energy scale of this spectral weight rearrangement at all compositions is more than an order of magnitude larger than the temperature scale on which it occurs, indicating that the effect arises from strong electron-electron interactions. The enhancement in the low-energy susceptibility at low temperature suggests that underdoped Bi-2212 should have a tendency to form charge density waves (CDW), which may be connected to recent observations from resonant x-ray scattering da Silva Neto et al. (2014); Comin and Damascelli (2016) (note that we have not seen evidence for a true CDW in Bi-2212 with M-EELS, though further study may yet uncover such effects). All spectra are doping- and temperature-independent above 1 eV where the system exhibits a universal $\sim 1/\omega^{2}$ form.

IV Properties of the polarizability

The continuum is essentially momentum-independent at all doping values for $q>0.16$ r.l.u., where there is hardly any discernible difference between M-EELS spectra at different $q$ values (Fig. 1). This implies that the susceptibility, $\chi^{\prime\prime}(\mathbf{q},\omega)$, is constant over $90\%$ of the Brillouin zone, apart from an overall $q^{2}$ dependence required by the $f$-sum rule (Eq. 1). Generally speaking, the momentum dependence of the susceptibility $\chi(\mathbf{q},\omega)$ is determined by two separate effects: the intrinsic polarizability of the system, $\Pi(\mathbf{q},\omega)$, and the momentum dependence of the Coulomb interaction itself, $V(\mathbf{q})$. It is therefore important to determine whether the deviation from a constant in the remaining portion of the Brillouin zone, $q<0.16$ r.l.u., is due to the Coulomb interaction, the polarizability, or the combination of the two.

Physically, the polarizability $\Pi(\mathbf{q},\omega)$ can be understood as the density response of the system with a completely screened Coulomb interaction (i.e. the system with neutral rather than charged density fluctuations) Mahan (2000); Nozières and Pines (1999). The polarizability $\Pi(\mathbf{q},\omega)$ and susceptibility $\chi(\mathbf{q},\omega)$ are related by Mahan (2000); Nozières and Pines (1999),

[TABLE]

where $V(\mathbf{q})$ is the Coulomb interaction and $\epsilon_{\infty}$ is the background dielectric constant, which for Bi-2212 is $\epsilon_{\infty}=4.5$ Levallois et al. (2016).

In practice, the most important difference between the polarizability and the susceptibility is that the former does not feature plasmon excitations, whose existence is a consequence of the direct Coulomb interaction between particle and hole excitations. Instead, the polarizability $\Pi(\mathbf{q},\omega)$ reveals the particle-hole excitation spectrum itself.

We argued in Ref. Mitrano et al. (2018) that $\Pi(\mathbf{q},\omega)$ could be extracted from the M-EELS data by assuming a two-dimensional form for the Coulomb interaction,

[TABLE]

where $d$ is on the order of the interlayer spacing in Bi-2212. Here, we apply this procedure to all four doping values by extrapolating $\chi^{\prime\prime}(\mathbf{q},\omega)$ with a $1/\omega^{2}$ tail and Kramers-Kronig transforming to acquire the real part, $\chi^{\prime}(\mathbf{q},\omega)$. We then evaluate $\Pi(\mathbf{q},\omega)$ for each doping from Eq. 2 using $d=15.62\,\textrm{\AA}$, which is close to the bilayer spacing for Bi-2212, and a proportionality constant in Eq. 3 for $V(\mathbf{q})$ of $7.9\cdot 10^{3}\,\textrm{eV}\cdot\textrm{\AA}^{2}$ (see Appendix A). The result is displayed in Fig. 2, which shows the scaled quantity, $\Pi^{\prime\prime}(\mathbf{q},\omega)/q^{2}$, at low temperature for each of the four doping values. The spectra for all momenta collapse onto a single curve that is different for each doping value. This collapse implies that $\Pi(\mathbf{q},\omega)$ is constant for all momenta (again, apart from an overall $q^{2}$ factor required by the sum rule), even down to the lowest momentum measured, $q=0.1$ r.l.u.

The overarching point of Fig. 2 is that particle-hole excitations in Bi-2212 cannot propagate, at least for energies $\omega>0.1$ eV. The modest $q$-dependence in the M-EELS data below 0.16 r.l.u., where the plasmon-like maximum is visible in the data, is purely an effect of $V(\mathbf{q})$ in Eq. 2. The particle-hole spectrum itself, $\Pi^{\prime\prime}(\mathbf{q},\omega)$, appears to be constant even down to the lowest momentum studied, $q=0.1$ r.l.u. This is extremely surprising, since the individual quasiparticles in this energy range exhibit significant dispersion Damascelli et al. (2003); Kaminski et al. (2003); Vishik et al. (2010); Chatterjee et al. (2011). We conclude that the textbook Lindhard description of the polarizability, in terms of particle-hole excitations across a Fermi surface and RPA treatment of the Coulomb interaction Mahan (2000); Nozières and Pines (1999), completely fails in Bi-2212. This failure occurs at all compositions we examined—even overdoped materials that are usually considered to be good Fermi liquids Keimer et al. (2015).

A second implication of Fig. 2 is that the peak in M-EELS data at low temperature in the overdoped regime (Fig. 1(e)-(h)) is not a plasmon. While the overdoped susceptibility in Fig. 1(e)-(h) exhibits a peak in a similar energy range as Fig. 1(a) with a local maximum near $\sim 0.8-1$ eV, the peak in the overdoped susceptiblity in Fig. 1(e)-(h) is also present in the polarizability (Fig. 2(d)), which as discussed above does not show plasmon effects. The peak in $\chi^{\prime\prime}(\mathbf{q},\omega)$ in the low-temperature overdoped regime is thus inherited from $\Pi^{\prime\prime}(\mathbf{q},\omega)$ and it is therefore more appropriate to think of it as a feature of the particle-hole excitation spectrum rather than as a collective, plasmon mode.

A final, striking implication of Fig. 2 is that, at energy $\omega>0.1$ eV, the polarizability factors, i.e., $\Pi^{\prime\prime}(\mathbf{q},\omega)\sim f(\mathbf{q})\cdot g(\omega)$, where $f(\mathbf{q})\sim q^{2}$. Since the original literature on the marginal Fermi liquid (MFL) phenomenology of the cuprates Zhu et al. (2015); Varma (2017), this factoring has been considered a a key signature of “local quantum criticality,” an exotic phase of matter in which the spatial correlation length $\xi_{x}\sim\ln\xi_{t}$, where $\xi_{t}$ is the temporal correlation length Varma et al. (1989); Zhu et al. (2015); Varma (2017); Mitrano et al. (2018); Krikun et al. (2019). One can think of such a phase as being characterized by a dynamical critical exponent $z=\infty$. The new lesson we have learned here is that this factoring takes place at all measured dopings and temperatures in the phase diagram. It is therefore a general materials property of Bi-2212, and not a feature of a critical point or a particular doping value.

V Spectral weight trend

The changes in spectral weight follow a distinct trend across the phase diagram. The fine temperature dependence is displayed in Fig. 3, which shows spectra at a fixed momentum $q^{*}=0.24$ r.l.u. at which $\chi^{\prime\prime}(\mathbf{q},\omega)\approx\Pi^{\prime\prime}(\mathbf{q},\omega)$. At optimal doping the spectra are temperature-independent, but the overdoped material shows a suppression of spectral weight below 0.5 eV as the system is cooled, indicating the emergence of an energy scale Mitrano et al. (2018). In the underdoped case, this trend reverses: the weight below 0.5 eV is enhanced as the system is cooled, exhibiting a power-law form at low temperature (Fig. 3(a)-(b)). This enhancement may be a consequence of slowing CDW fluctuations in underdoped materials da Silva Neto et al. (2014); Comin and Damascelli (2016). The optimally doped case corresponds to a turning point between regions with opposite behavior, where the resulting response is temperature-independent (Fig. 3(c)). The normal state at this doping corresponds to the center of the strange metal regime in which resistivity is linear over the widest temperature range Chatterjee et al. (2011); Keimer et al. (2015).

A distinct trend is now clear: The M-EELS response is featureless and doping-independent at room temperature, but not at low temperature (Fig. 1). At the same time, the response is temperature-independent at optimal doping, but not at other dopings (Fig. 3(a)-(d)). The overall behavior may be summarized using a parameter that quantifies the deviation of the response from its featureless form at high temperature,

[TABLE]

where the reference spectrum, $\chi^{\prime\prime}_{\textrm{ref}}(q^{*},\omega)$, is taken at optimal doping at $T=150$ K and $q^{*}=0.24$ r.l.u. The quantity $\xi$ measures, as a function of temperature and doping, the degree to which the spectra deviate from the integrated spectral weight of $\chi^{\prime\prime}_{\textrm{ref}}(q^{*},\omega)$ which has a constant value of $12.94\cdot 10^{-5}\,\textrm{\AA}^{-3}$. Note that this parameter is also a measure of the change in the Coulomb energy of the system Levallois et al. (2016).

The behavior of $\xi$ is summarized in Fig. 3(e). Its value is small at 300 K for all four dopings. As the system is cooled, $\xi$ becomes negative in the overdoped case, where spectral weight is suppressed at low temperature, and positive for underdoped materials, where the weight is enhanced. At optimal doping, where the response is always featureless, $\xi$ remains small at all temperatures. Note that the value of $\xi$ as a function of doping at low temperature shows the same change in sign as the Coulomb energy determined from ellipsometry experiments Levallois et al. (2016), though the magnitude of the effect observed here is significantly larger (Appendix B).

The trend is illustrated in another way in Fig. 4, which shows the spectral weight change $\xi$ as a function of doping and temperature superposed on the known phase diagram of Bi-2212 constructed from the phase boundaries in Refs. Vishik et al. (2010); Chatterjee et al. (2011); Barisic et al. (2013); Proust et al. (2016). The region over which $\xi$ is small, in which the response remains featureless, closely coincides with the fan-shaped, strange metal region. Fig. 4 suggests a connection between a featureless form for the density response and the existence of a linear-in-$T$ normal state resistivity.

VI Discussion

The highly unconventional behavior of the density response across the Bi-2212 phase diagram can be summarized as consisting of a flat continuum of density fluctuations within the fan-like, strange metal region, and dramatic changes in spectral weight up to 0.5 eV outside this region with a sign-reversal at optimal doping. Here, we discuss the implications of the presence of this continuum for some other widely known properties of the cuprates.

First, the continuum provides a natural, qualitative explanation for the normal state quasiparticle lifetimes in Bi-2212 Damascelli et al. (2003); Kaminski et al. (2003); Vishik et al. (2010); Chatterjee et al. (2011). Broadly speaking, the poorest quasiparticle coherence is observed in underdoped materials at low temperature. The coherence increases with increasing doping, with the longest lifetimes observed at low temperature on the overdoped side Vishik et al. (2010); Chatterjee et al. (2011). This behavior can be understood by recognizing that the continuum we observe should provide a decay path for quasiparticle damping, amplifying the imaginary part of the self-energy, $\Sigma^{\prime\prime}(\omega)$ Damascelli et al. (2003); Vishik et al. (2010); Chatterjee et al. (2011). The strong damping of quasiparticles in underdoped materials can then be understood as a consequence of the enhancement of the M-EELS continuum at low temperature in this regime (Fig. 3(a)-(b)). Similarly, the increased quasiparticle coherence in overdoped materials arises because the continuum in this regime is suppressed (Fig. 3(d)).

At optimal doping, in the strange metal phase, the imaginary part of the self-energy, $\Sigma^{\prime\prime}(\omega)$, is linear in $\omega$, which has been shown to be consistent with an Eliashberg function that is frequency-independent Bok et al. (2016). This behavior is highly consistent with the observation that the continuum is frequency-independent in this regime (Fig. 3(c)), and suggests that the M-EELS data may have a direct relationship with the Eliashberg function itself.

It is critical to keep in mind, however, that while the continuum provides a clear decay channel for quasiparticles, the mechanism by which the quasiparticles in turn generate the continuum is less clear. Because the quasiparticles are highly dispersive in Bi-2212 Chatterjee et al. (2011); Vishik et al. (2010), a standard Lindhard calculation of the density response using RPA Nozières and Pines (1999); Mahan (2000) would yield a continuum that is also highly momentum-dependent, which is inconsistent with the M-EELS data at all compositions studied. It may be the case that beyond-RPA effects, such as excitonic, local field, or vertex correction effects dominate the response properties of strange metals. New theoretical approaches using nonperturbative techniques going beyond RPA, such as those based on the AdS-CFT correspondence, may provide progress in understanding the density response of the strange metal Krikun et al. (2019); Zaanen (2019).

It is worth considering whether the momentum-independence of the density fluctuations could be a consequence of strong disorder, which might eliminate momentum conservation by explicitly breaking translational symmetry, resulting in $q$-integrated response functions in all measurements. Even in materials in which the degree of disorder is low by traditional crystallographic standards, translational symmetry could be broken by emergent electronic heterogeneity, for which there is ample evidence in the cuprates Fujita et al. (2012). Such heterogeneity has even been proposed as the origin of the linear-in-$T$ resistivity in strange metals Pelc et al. (2019). Unfortunately, such a view of the M-EELS data is inconsistent with ARPES and STM studies, which report clearly dispersing quasiparticle excitations in Bi-2212 where the charge response is $q$-independent Damascelli et al. (2003); Reber et al. (2015); Bok et al. (2016); Vishik et al. (2010); Chatterjee et al. (2011); Fujita et al. (2012). Moreover, recent RIXS experiments on electron-doped cuprates, which are not strange metals but have a similar degree of disorder Armitage et al. (2010), have clearly shown conventional dispersing plasmon excitations, in agreement with Hubbard model-based RPA calculations Hepting et al. (2018). Also, dispersing collective modes have been observed with M-EELS in other materials with similar degree of disorder Kogar et al. (2017). So it seems unlikely that disorder is the sole cause of the $q$-independence we see. On the other hand, it remains possible that the strong coupling physics of Bi-2212 conspires with disorder in such as way as to make momentum irrelevant in the density response, though not other observables.

Finally, we consider the question of whether the continuum could be a sign of some kind of quantum critical behavior. In this view, the fan-like structure implied by Fig. 4 might indicate a crossover near a doping $p_{c}\sim 0.16$ that could be identified as a quantum critical point (QCP), as suggested by many authors Sachdev (2010); Keimer et al. (2015); Badoux et al. (2016); Damascelli et al. (2003); Kaminski et al. (2003); Ando et al. (2004); Hussey (2008); Sterpetti et al. (2017). This interpretation is problematic, since few expected signatures of quantum criticality are present in the density fluctuation spectra. For example, no soft collective mode, with energy falling to zero at $p_{c}$, is visible in the data. A spectral weight rearrangement is observed away from $p_{c}$ in both underdoped and overdoped samples, however its energy scale $\sim 0.5$ eV is more than an order of magnitude larger than the crossover temperature, $\sim 200$ K, over which this rearrangement takes place (Fig. 3(a)-(d)). Moreover, the response functions near $p_{c}$ do not exhibit any momentum dependence, which is expected in the usual Hertz-Millis picture of a quantum phase transition Sachdev (1999). The factoring of $\Pi^{\prime\prime}(\mathbf{q},\omega)$ we observe (Fig. 2) has been cited as evidence for local criticality, which has been argued to be a feature of an exotic QCP Zhu et al. (2015); Varma (2017). But this factoring is observed everywhere in the phase diagram, not just in the vicinity of $p_{c}$. Whether some kind of exotic critical point might explain the peculiar physics taking place remains, for now, an open question.

In summary, we have shown that Bi-2212, despite being a good conductor by most standards, does not exhibit well-defined plasmon excitations that are dispersing and long-lived anywhere in its phase diagram. Instead, this material exhibits a featureless, momentum-independent continuum in the density response throughout the fan-shaped, strange metal region of the phase diagram. Outside this fan, the response undergoes dramatic changes in spectral weight up to 0.5 eV exhibiting a sign-reversal at optimal doping. Our study establishes a featureless continuum as a defining property of the mysterious strange metal phase in Bi-2212 and places it at a crossover between two regimes with opposite trends in their charge susceptibility. Major open questions remain concerning the relationship between the density response and models based on quasiparticle scattering, disorder, or quantum critical fluctuations. A new kind of theory of interacting matter may be needed to explain the existence of this phase and its connection to other exotic phenomena such as high temperature superconductivity.

VII Acknowledgments

We thank D. van der Marel, P. W. Phillips, C. M. Varma, N. D. Goldenfeld, A. J. Leggett, W. E. Pickett, and J. Zaanen for enlightening discussions, M. Tran for supplying the ellipsometry data used to evaluate the $f$-sum rule, and I. El Baggari for useful sample preparation advice. This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences grant no. DEFG02-06ER46285. P.A. gratefully acknowledges support from the EPiQS program of the Gordon and Betty Moore Foundation, grant GBMF4542. Crystal growth was supported by DOE grant DE-SC0012704. B.U. acknowledges NSF CAREER grant DMR-1352604. M.M. acknowledges support by the Alexander von Humboldt Foundation through the Feodor Lynen Fellowship program. C.D. acknowledges use of the electron microscopy facilities in the Eyring Materials Center at Arizona State University.

Appendix A APPENDIX A: DETERMINING $\Pi(\mathbf{q},\omega)$

At large momentum, the Coulomb interaction $V(\mathbf{q})$ is small so the susceptibility, $\chi(\mathbf{q},\omega)$, and the polarizability, $\Pi(\mathbf{q},\omega)$, are nearly the same (Eq. 2). At small momenta, particularly $q<0.16$ r.l.u., relating them requires knowledge of $V(\mathbf{q})$. The precise functional form of $V(\mathbf{q})$ can depend on the material geometry, especially in quasi-2D systems Reed et al. (2010) or at the surface of a layered material. Understanding this relation for M-EELS is still a work in progress. Nevertheless, as argued in Ref. Mitrano et al. (2018), the approximate form shown in Eq. 3 is a sensible phenomenological starting point as it exhibits the same functional behavior as that of an infinite (i.e. not surface terminated) layered system Reed et al. (2010).

To determine the proportionality constant in Eq. 3, we follow a similar procedure to Ref. Mitrano et al. (2018). Noticing that in the limit of large momentum transfer, $\chi(\mathbf{q},\omega)\approx\Pi(\mathbf{q},\omega)/\epsilon_{\infty}$, we treat $V$ as a fit parameter, $V_{\textrm{fit}}$, and determine the values of $V_{\textrm{fit}}$ where $\Pi$ most closely resembles its value at the highest momentum measured, $q=0.5\,\textrm{r.l.u.}$, by minimizing the function,

[TABLE]

After obtaining $V_{\textrm{fit}}(q)$, we fit to it with the functional form of Eq. 3 using $d=15.62\textrm{ \AA}$ to determine the proportionality constant. The fits are shown in Fig. 5, along with curves with alternate values of $d$ shown for reference. The fitted proportionality constant was found to be $(7.9\pm 0.3)\cdot 10^{3}\,\textrm{eV}\cdot\textrm{\AA}^{2}$. Considering a rough estimate for the proportionality constant of $4\pi e^{2}d=2.83\cdot 10^{3}\,\textrm{eV}\cdot\textrm{\AA}^{2}$ is within a factor 3, the fit value is quite reasonable given the systematic uncertainties.

Appendix B APPENDIX B: CONSISTENCY BETWEEN M-EELS AND OTHER ELECTROMAGNETIC PROBES

Our M-EELS data are consistent, in most respects, with previous studies of the electromagnetic properties of cuprates. Like M-EELS Vig et al. (2017), spectroscopic ellipsometry Bozovic (1990); Levallois et al. (2016) and transmission EELS experiments with modest energy resolution (i.e., elastic linewidths of $\geq$ 0.3 eV) Wang et al. (1990); Nücker et al. (1989, 1991) also observe an overdamped plasmon excitation at $\sim$ 1 eV energy. At a small in-plane momentum, $q=0.05$ r.l.u., M-EELS and transmission EELS give nearly identical plasmon energy and lineshape Vig et al. (2017). Further, the plasmon energy measured with ellipsometry is independent of doping Terasaki, I. and Nakahashi, T. and Takebayashi, S. and Maeda, A. and Uchinokura, K. (1990); Uchida et al. (1991); Liu et al. (1999); Levallois et al. (2016), starkly contrasting with the behavior of a normal conductor Nozières and Pines (1999); Mahan (2000); Kogar et al. (2015), suggesting it may arise from a continuum of the sort we see.

Still, it is clear something qualitative must change in the charge response as $q\rightarrow 0$. For one, the plasmon lineshape observed with ellipsometry, which is strictly a $q=0$ probe, is 50% narrower than the M-EELS and transmission EELS data at the lowest momentum studied Vig et al. (2017). Further, the spectral weight in ellipsometry grows with decreasing temperature for all dopings by no larger than $\sim$3% Levallois et al. (2016), while the changes observed here are around 50% and may have either sign (Fig. 3(e)).

More fundamentally, the factoring of the polarizability, $\Pi^{\prime\prime}(q,\omega)=f(q)\cdot g(\omega)$, cannot persist to zero momentum because it violates the compressibility sum rule Mahan (2000); Nozières and Pines (1999),

[TABLE]

where $\kappa=-({\partial V}/{\partial P})_{N}/V$ is the compressibility. By definition, $\kappa=0$ for an insulator while for a metal $\kappa$ is a constant. If the factoring of $\Pi$ persisted all the way to $q\rightarrow 0$ the $f$-sum rule (Eq. 2) would require $\Pi(q,0)\sim q^{2}$ in the limit of small momentum, implying that the system is an insulator. However Bi-2212 is a metal, of course, meaning this factoring cannot persist as $q\rightarrow 0$. Full reconciliation between $q=0$ probes like ellipsometry and finite-$q$ techniques may require new theoretical ideas about non-commutativity of limits Setty et al. (2018). Encouragingly, however, our $\xi$, which represents integrated spectral weight change, exhibits the same trend with doping as the Coulomb energy reported in Ref. Levallois et al. (2016), changing sign at optimal doping in both studies. Forthcoming studies with higher momentum resolution will determine the nature of the crossover region between the two techniques.

We note that, at larger momenta, transmission EELS studies of Bi-2212 by different groups report conflicting results. Using 60 keV electrons in an EELS spectrometer employing Wien filters and a $30\,\textrm{meV}$ quasi-elastic line, Terauchi et al. reported a featureless continuum with a 1 eV cutoff energy similar to what we report here Terauchi et al. (1999). On the other hand, using 170 keV electrons with hemispherical analyzers and quasi-elastic line extending to $500\,\textrm{meV}$, Nücker et al. observed no continuum at all, but instead report conventional plasmon behavior exhibiting normal, Fermi liquid-like dispersion Nücker et al. (1989, 1991). More recently, transmission EELS studies by Schuster et al. of the underdoped cuprate Ca2-xNaxCuO2Cl2 also show evidence for a broad continuum, rather than conventional dispersing plasmons Schuster et al. (2009).

As an initial step towards establishing consistency between M-EELS and transmission EELS techniques, we have performed a preliminary measurement of OP91K Bi-2212 using a monochromated Nion UltraSTEM Scanning Transmission Electron Microscope EELS (STEM- EELS) with 10 meV resolution, as shown in Fig. 6. These measurements were performed at room-temperature with 60 keV electrons and convergence/acceptance semi-angles of 4 mrad. Samples were prepared using a “powder” method, where a single crystal of OP91K Bi-2212 is crushed in a mortar and pestle and dispersed onto a holey carbon grid using ethanol. A flake was found with the correct orientation and estimated thickness of below 400 Å. Sample crystallinity and ab-plane orientation was confirmed by verifying the well-known Bi-2212 supermodulations (see the inset of Figure 6). Note that the STEM-EELS measurements show precisely the same continuum with the same cutoff energy as M-EELS measurements. This indicates that the continuum is a bulk property of Bi-2212, and not a peculiarity of the surface.

We note that, because STEM-EELS instruments are designed for high spatial-resolution at the expense of momentum-resolution, the STEM-EELS spectrum in Fig. 6 is momentum-integrated azimuthally in the ab-plane for $q\lesssim 0.63$ r.l.u. (i.e. approximately the first Brillouin Zone). Nevertheless, because M-EELS measurements have established that the strange metal continuum is largely momentum-independent, Fig. 6 represents a strong evidence that the continuum is a bulk effect. More detailed transmission EELS studies with both high energy- and momentum-resolution would serve as an even more stringent test.