Crossover of Charge Fluctuations across the Strange Metal Phase Diagram
A. A. Husain, M. Mitrano, M. S. Rak, S. I. Rubeck, B. Uchoa, J. Schneeloch, R. Zhong, G. D. Gu, P. Abbamonte
CCrossover of Charge Fluctuations across the Strange Metal Phase Diagram
Ali Husain, Matteo Mitrano, Melinda S. Rak, Samantha Rubeck, Bruno Uchoa, Katia March, Christian Dwyer, John Schneeloch, Ruidan Zhong, Genda D. Gu, and Peter Abbamonte Department of Physics and Materials Research Laboratory,University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA Department of Physics and Astronomy, University of Oklahoma, Norman, OK 73069, USA Department of Physics, Arizona State University, Tempe, AZ 85287, USA Condensed Matter Physics and Materials Science Department,Brookhaven National Laboratory, Upton, NY 11973, USA (Dated: November 13, 2019)A normal metal exhibits a valence plasmon, which is a sound wave in its conduction electrondensity. The mysterious strange metal is characterized by non-Boltzmann transport and violatesmost fundamental Fermi liquid scaling laws. A fundamental question is: Do strange metals haveplasmons? Using momentum-resolved inelastic electron scattering (M-EELS) we recently showedthat, rather than a plasmon, optimally-doped Bi . Sr . Ca . Cu . O x (Bi-2212) exhibits a feature-less, temperature-independent continuum with a power-law form over most energy and momentumscales [M. Mitrano, PNAS , 5392-5396 (2018)]. Here, we show that this continuum is presentthroughout the fan-shaped, strange metal region of the phase diagram. Outside this region, dra-matic changes in spectral weight are observed: In underdoped samples, spectral weight up to 0.5 eVis enhanced at low temperature, biasing the system towards a charge order instability. The situationis 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 be-tween these two opposite behaviors at which the response is temperature-independent. Our studysuggests that plasmons do not exist as well-defined excitations in Bi-2212, and that a featurelesscontinuum is a defining property of the strange metal, which is connected to a peculiar crossoverwhere the spectral weight change undergoes a sign reversal. INTRODUCTION
The enigmatic and poorly understood strange metalhas been found within the phase diagrams of manystrongly-correlated systems, including transition metaloxides, heavy fermion materials, organic molecular solids,and iron-pnictide superconductors [1–7]. This phase ischaracterized by violation of fundamental Fermi liquidscaling laws, and by its close proximity to other exoticphases such as unconventional superconductivity, chargeor spin density waves, and nematicity [1, 4, 6, 8]. Forexample, the prototypical copper-oxide strange metals,which are also high temperature superconductors, exhibita resistivity that is linear in temperature and exceedsthe Mott-Ioffe-Regel limit [2, 9], an optical conductiv-ity exhibiting an anomalous power law dependence onfrequency [10, 11], a magnetoresistance that is linear infield, violating Kohler’s rule [12], a quasiparticle decayrate that scales linearly with energy [13], and an NMRspin relaxation rate that violates the Korringa law [14].No generally accepted theory of matter can explain theseproperties, which appear to be incompatible with funda-mental assumptions of Boltzmann transport theory. Thestrange metal has thus become one of the great unsolvedproblems in condensed matter physics.The quasiparticle dynamics of strange metals havebeen studied extensively with angle-resolved photoemis-sion (ARPES) and scanning tunneling microscopy (STM)techniques, which directly measure the one-electron spec- tral function [15–20]. However, little is known about thetwo-particle charge response, which directly reveals thestrongly correlated nature of this phase [21].The fundamental charge collective mode of an ordi-nary metal is its plasmon, which is essentially a soundwave in its valence electron density [22]. On theother hand, we recently showed that in optimally dopedBi . Sr . Ca . Cu . O x (Bi-2212), a known strangemetal, the plasmon is overdamped and rapidly de-cays into a momentum-, energy-, and temperature-independent continuum extending up to an energy of ∼ × larger than thetemperature scale on which these changes take place, sug-gesting strong interactions are at play [24]. It is thereforecrucial to map out the exact region of the phase diagramwhere a featureless continuum is present, so its connec-tion to the strange metal and other neighboring phasescan be established.Here, we present a study of the density fluctuationsacross the doping-temperature phase diagram of thestrange metal Bi-2212 using momentum-resolved electronenergy-loss spectroscopy (M-EELS) [25]. This techniquemeasures the surface dynamic charge response of a ma-terial, χ ( q , ω ), and directly reveals the charged bosoniccollective modes of the system [25]. Note that, while gen-erally regarded as a surface technique, the probe depth a r X i v : . [ c ond - m a t . s t r- e l ] N ov of M-EELS is given by the inverse of the in-plane mo-mentum transfer, q − , making it somewhat more bulksensitive than single-particle spectroscopies like ARPESand STM [25]. In this study we focus on the energyregime 0.1 eV < ω < EXPERIMENT
Single crystals of Bi-2212 were grown using floatingzone methods described previously [26]. The currentstudy was done on underdoped crystals with T c =50 K(UD50K) and 70 K (UD70K), optimally doped crystalswith T c =91 K (OP91K), and overdoped crystals with T c =50 K (OD50K).Measurements of the charge fluctuation spectra wereperformed using meV-resolution, momentum-resolvedelectron energy-loss spectroscopy (M-EELS) [25]. M-EELS is a variant of surface HR-EELS [27] in whichthe momentum transfer of the probe electron is mea-sured with both high resolution and accuracy [25]. Mea-surements were performed on cleaved single crystals ofBi-2212 at 50 eV incoming electron beam energy and 4meV energy resolution at a fixed out-of-plane momen-tum transfer, q z = 4 .
10 ˚A − . We use Miller indices,( H, K ), to designate an in-plane momentum transfer q = (2 πH/a, πK/a ), where a = 3 .
81 ˚A is the tetragonalCu-Cu lattice parameter [15]. Unless otherwise specified,all momenta are along the (1,-1) crystallographic direc-tion, i.e., perpendicular to the structural supermodula-tion in this material [15]. The sample orientation matrixwas determined in situ using the (0,0) specular and (1,0)Bragg reflections, and verified by observing the (1,-1) re-flection, establishing a quantitative relationship betweenthe momentum, q , and the diffractometer angles. M-EELS spectra were taken from 0 to 2 eV energy loss andbinned into 30 meV intervals for improved statistics.M-EELS measures the dynamic structure factor of asurface, S ( q , ω ), which is proportional to the surface dy-namic charge susceptibility, χ (cid:48)(cid:48) ( q , ω ), by the fluctuation-dissipation theorem [25, 27, 28]. χ (cid:48)(cid:48) was determined fromthe raw data by dividing the M-EELS matrix elements,which depend on the momentum transfer [25], and an-tisymmetrizing to remove the Bose factor [23, 25]. Thedata were placed on an absolute scale by performing thepartial f -sum rule integral, (cid:90) ω χ (cid:48)(cid:48) ( q , ω ) dω = − π q m N eff , (1)where N eff was determined by integrating the q = 0 ellip-sometry data from Ref. [29] over the same energy intervalat the corresponding values of temperature and doping. q = 0.10 RT UD50K (300 K)UD70K (300 K)OP91K (300 K)OD50K (300 K) LT q = 0.240.00.51.0 q = 0.360 0.5 1.0 1.5012 q = 0.50 0 0.5 1.0 1.5 2.0 Energy Loss (eV) − χ ” ( q , ω ) ( – e V – Å – ) (a)(b)(c)(d) (e)(f)(g)(h) UD50K (300 K)UD70K (300 K)OP91K (300 K)OD50K (300 K)UD50K (300 K)UD70K (300 K)OP91K (300 K)OD50K (300 K)UD50K (300 K)UD70K (300 K)OP91K (300 K)OD50K (300 K) q = 0.10 UD50K (100 K)UD70K (100 K)OP91K (100 K)OD50K (115 K) q = 0.24 UD50K (100 K)UD70K (100 K)OP91K (100 K)OD50K (115 K) q = 0.36 UD50K (100 K)UD70K (100 K)OP91K (100 K)OD50K (115 K) q = 0.50 UD50K (100 K)UD70K (100 K)OP91K (100 K)OD50K (115 K)
FIG. 1. χ (cid:48)(cid:48) ( q , ω ) at room temperature (300 K) (a-d) and lowtemperature (either 100 K or 115 K) (e-h) for all four dopingsstudied. No significant q -dependence is seen for any dopingor temperature. At 300 K the spectra vary only slightly withdoping, compared to the dramatic doping dependence below ∼ M-EELS DATA
Figure 1 shows the M-EELS spectra for a selectionof momenta at room temperature (300 K) and at lowtemperature (100 K or 115 K, depending upon the dop-ing). The intensity rise below 0.1 eV in all spectra is dueto the well-known Bi-2212 optical phonons [25, 30, 31].Looking at the 300 K data, the lowest momentum (Fig.1(a)) shows a highly damped plasmon reported previ-ously [23],which appears in the spectrum as a local max-imum at ω ∼ ∼ planes, but that interband tran-sitions, potentially involving the BiO layers, contributeabove 1.5 eV and may create features around 2.3 eV and3.8 eV [33–35]. We therefore expect the M-EELS con-tinuum principally arises from the CuO planes but mayhave other contributions at high energy.At larger momenta (Fig. 1 (b)-(d)) the plasmon isno longer present, by which we mean that a local maxi-mum is no longer observed in the spectra. Instead, theplasmon decays into an energy-independent continuum,as reported previously [23]. As shown in Appendix B,this continuum appears also in transmission EELS ex-periments, and is thus a bulk property of Bi-2212. Thiscontinuum is not a subtle effect; it saturates the f -sumrule and is the primary feature of the charge response ofBi-2212. As reported in Ref. [23], the M-EELS spectraexhibit very little q -dependence at momenta greater than0.16 r.l.u., which implies that charge excitations barelypropagate in this material. Additionally, Fig. 1 (a)-(d)shows that, at T = 300 K, the continuum also has littledoping dependence across the composition range.Surprisingly, the spectra become dramatically doping-dependent at low temperature (Fig. 1(e)-(h)), which isthe main finding of this work. In overdoped samples, thespectral weight below 0.5 eV is greatly suppressed at lowtemperature [23]. By contrast, in underdoped materi-als, the weight in this energy range at low temperatureis enhanced. The 0.5 eV energy scale of this spectralweight rearrangement at all compositions is more thanan order of magnitude larger than the temperature scaleon which it occurs, indicating that the effect arises fromstrong electron-electron interactions. The enhancementin the low-energy susceptibility at low temperature sug-gests that underdoped Bi-2212 should have a tendencyto form charge density waves (CDW), which may be con-nected to recent observations from resonant x-ray scat-tering [36, 37] (note that we have not seen evidence fora true CDW in Bi-2212 with M-EELS, though furtherstudy may yet uncover such effects). All spectra aredoping- and temperature-independent above 1 eV wherethe system exhibits a universal ∼ /ω form. PROPERTIES OF THE POLARIZABILITY
The continuum is essentially momentum-independentat all doping values for q > .
16 r.l.u., where there ishardly any discernible difference between M-EELS spec-tra at different q values (Fig. 1). This implies that thesusceptibility, χ (cid:48)(cid:48) ( q , ω ), is constant over 90% of the Bril-louin zone, apart from an overall q dependence requiredby the f -sum rule (Eq. 1). Generally speaking, the mo-mentum dependence of the susceptibility χ ( q , ω ) is de-termined by two separate effects: the intrinsic polariz-ability of the system, Π( q , ω ), and the momentum de-pendence of the Coulomb interaction itself, V ( q ). It istherefore important to determine whether the deviationfrom a constant in the remaining portion of the Brillouinzone, q < .
16 r.l.u., is due to the Coulomb interaction,the polarizability, or the combination of the two.
UD50K UD70K
OP91K
OD50K
Energy Loss (eV) − Π ( q , ω ) / q ( – e V – Å – ) (a) (b)(c) (d) q=0.10q=0.16q=0.24q=0.36q=0.42q=0.50 q=0.10q=0.16q=0.24q=0.36q=0.42q=0.50q=0.10q=0.16q=0.24q=0.36q=0.42q=0.50q=0.10q=0.16q=0.24q=0.36q=0.42q=0.50 FIG. 2. Collapse of Π (cid:48)(cid:48) ( q , ω ) for all momenta for each of thefour doping values studied, achieved simply by dividing by q .The spectra were taken at T = 100K except for the OD50Ksample, which was measured at 115 K. Apart from the q scaling of the magnitude, which is required by the f -sum rule,the spectra are essentially momentum-independent. Physically, the polarizability Π( q , ω ) can be under-stood as the density response of the system with a com-pletely screened Coulomb interaction (i.e. the systemwith neutral rather than charged density fluctuations)[22, 38]. The polarizability Π( q , ω ) and susceptibility χ ( q , ω ) are related by [22, 38], χ ( q , ω ) = Π( q , ω ) (cid:15) ∞ − V ( q )Π( q , ω ) , (2)where V ( q ) is the Coulomb interaction and (cid:15) ∞ is thebackground dielectric constant, which for Bi-2212 is (cid:15) ∞ =4 . q , ω ) reveals the particle-hole excitation spectrumitself.We argued in Ref. [23] that Π( q , ω ) could be extractedfrom the M-EELS data by assuming a two-dimensionalform for the Coulomb interaction, V ( q ) ∝ e − qd q . (3)where d is on the order of the interlayer spacing in Bi-2212. Here, we apply this procedure to all four dopingvalues by extrapolating χ (cid:48)(cid:48) ( q , ω ) with a 1 /ω tail andKramers-Kronig transforming to acquire the real part, χ (cid:48) ( q , ω ). We then evaluate Π( q , ω ) for each doping fromEq. 2 using d = 15 .
62 ˚A, which is close to the bilayerspacing for Bi-2212, and a proportionality constant inEq. 3 for V ( q ) of 7 . · eV · ˚A (see Appendix A).The result is displayed in Fig. 2, which shows the scaledquantity, Π (cid:48)(cid:48) ( q , ω ) /q , at low temperature for each of thefour doping values. The spectra for all momenta collapseonto a single curve that is different for each doping value.This collapse implies that Π( q , ω ) is constant for all mo-menta (again, apart from an overall q factor requiredby the sum rule), even down to the lowest momentummeasured, q = 0 . ω > . q -dependence in the M-EELS data below 0.16 r.l.u., where the plasmon-like max-imum is visible in the data, is purely an effect of V ( q )in Eq. 2. The particle-hole spectrum itself, Π (cid:48)(cid:48) ( q , ω ),appears to be constant even down to the lowest momen-tum studied, q = 0 . ∼ . − χ (cid:48)(cid:48) ( q , ω ) in the low-temperatureoverdoped regime is thus inherited from Π (cid:48)(cid:48) ( q , ω ) and itis therefore more appropriate to think of it as a featureof the particle-hole excitation spectrum rather than as acollective, plasmon mode.A final, striking implication of Fig. 2 is that, at energy ω > . (cid:48)(cid:48) ( q , ω ) ∼ f ( q ) · g ( ω ), where f ( q ) ∼ q . Since the original literatureon the marginal Fermi liquid (MFL) phenomenology ofthe cuprates [40, 41], this factoring has been considered aa key signature of “local quantum criticality,” an exoticphase of matter in which the spatial correlation length ξ x ∼ ln ξ t , where ξ t is the temporal correlation length[23, 40–43]. One can think of such a phase as being char-acterized by a dynamical critical exponent z = ∞ . Thenew lesson we have learned here is that this factoringtakes place at all measured dopings and temperatures inthe phase diagram. It is therefore a general materials property of Bi-2212, and not a feature of a critical pointor a particular doping value. SPECTRAL WEIGHT TREND
The changes in spectral weight follow a distinct trendacross the phase diagram. The fine temperature de-pendence is displayed in Fig. 3, which shows spec-tra at a fixed momentum q ∗ = 0 .
24 r.l.u. at which χ (cid:48)(cid:48) ( q , ω ) ≈ Π (cid:48)(cid:48) ( q , ω ). At optimal doping the spectraare temperature-independent, but the overdoped mate-rial shows a suppression of spectral weight below 0.5 eVas the system is cooled, indicating the emergence of anenergy scale [23]. In the underdoped case, this trend re-verses: the weight below 0.5 eV is enhanced as the systemis cooled, exhibiting a power-law form at low tempera-ture (Fig. 3(a)-(b)). This enhancement may be a conse-quence of slowing CDW fluctuations in underdoped ma-terials [36, 37]. The optimally doped case corresponds toa turning point between regions with opposite behavior,where the resulting response is temperature-independent(Fig. 3(c)). The normal state at this doping correspondsto the center of the strange metal regime in which resis-tivity is linear over the widest temperature range [9, 19].A distinct trend is now clear: The M-EELS responseis featureless and doping-independent at room tempera-ture, but not at low temperature (Fig. 1). At the sametime, the response is temperature-independent at opti-mal doping, but not at other dopings (Fig. 3(a)-(d)).The overall behavior may be summarized using a param-eter that quantifies the deviation of the response from itsfeatureless form at high temperature, ξ = − (cid:90) . . [ χ (cid:48)(cid:48) ( q ∗ , ω ) − χ (cid:48)(cid:48) ref ( q ∗ , ω )] dω, (4)where the reference spectrum, χ (cid:48)(cid:48) ref ( q ∗ , ω ), is taken at op-timal doping at T = 150 K and q ∗ = 0 .
24 r.l.u. Thequantity ξ measures, as a function of temperature anddoping, the degree to which the spectra deviate from theintegrated spectral weight of χ (cid:48)(cid:48) ref ( q ∗ , ω ) which has a con-stant value of 12 . · − ˚A − . Note that this parameteris also a measure of the change in the Coulomb energy ofthe system [29].The behavior of ξ is summarized in Fig. 3(e). Its valueis small at 300 K for all four dopings. As the system iscooled, ξ becomes negative in the overdoped case, wherespectral weight is suppressed at low temperature, andpositive for underdoped materials, where the weight is en-hanced. At optimal doping, where the response is alwaysfeatureless, ξ remains small at all temperatures. Notethat the value of ξ as a function of doping at low tem-perature shows the same change in sign as the Coulombenergy determined from ellipsometry experiments [29],though the magnitude of the effect observed here is sig- Temperature (K) −4−202468
UD50KUD70KOP91KOD50K −1 −1 UD70K −1 OP91K −1 OD50K
Energy Loss (eV) (a) (b) (c) (d) (e) − χ ” ( q * , ω ) ( - e V – Å – ) ( - Å – ) S pe c t r a l W e i gh t, ξ FIG. 3. (a)-(d) Temperature dependence of χ (cid:48)(cid:48) ( q ∗ = 0 . , ω ) for each doping level studied. Note that the energy axis is plottedon a log-scale to emphasize the spectral weight change below 0.5 eV. (e) Doping and temperature dependence of the spectralweight change ξ , as defined in the main text to be the spectral weight between 0.1 eV to 0.5 eV referenced to optimal doping at150 K. The crossover from spectral weight accumulation to depletion with doping is evident, as is the sign-reversal at optimaldoping. nificantly larger (Appendix B). Hole doping p T e m pe r a t u r e ( K ) −3−2−10123 SM FL?PGAF SC ( - Å – ) S pe c t r a l W e i gh t, ξ FIG. 4. Evolution of the M-EELS continuum spectralweight across the Bi-2212 phase diagram, which was con-structed from the data in Refs. [18, 19, 44, 45]. HereAF=antiferromagnet, SC=superconductor, PG=pseudogap,FL=Fermi liquid, SM=strange metal. The colored points rep-resent the spectral weight change, ξ , in Fig. 3(e). ξ is smallthroughout the SM region in which the continuum remainsfeatureless. Outside the SM, the continuum spectral weightchanges rapidly with a different sign in the underdoped andoverdoped regimes. The trend is illustrated in another way in Fig. 4, whichshows the spectral weight change ξ as a function of dop-ing and temperature superposed on the known phase di-agram of Bi-2212 constructed from the phase boundariesin Refs. [18, 19, 44, 45]. The region over which ξ issmall, in which the response remains featureless, closelycoincides 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 nor-mal state resistivity. DISCUSSION
The highly unconventional behavior of the density re-sponse across the Bi-2212 phase diagram can be summa-rized as consisting of a flat continuum of density fluc-tuations within the fan-like, strange metal region, anddramatic changes in spectral weight up to 0.5 eV outsidethis region with a sign-reversal at optimal doping. Here,we discuss the implications of the presence of this con-tinuum for some other widely known properties of thecuprates.First, the continuum provides a natural, qualitative ex-planation for the normal state quasiparticle lifetimes inBi-2212 [15, 18, 19, 39]. Broadly speaking, the poorestquasiparticle coherence is observed in underdoped mate-rials at low temperature. The coherence increases withincreasing doping, with the longest lifetimes observed atlow temperature on the overdoped side [18, 19]. Thisbehavior can be understood by recognizing that the con-tinuum we observe should provide a decay path for quasi-particle damping, amplifying the imaginary part of theself-energy, Σ (cid:48)(cid:48) ( ω ) [15, 18, 19]. The strong damping ofquasiparticles in underdoped materials can then be un-derstood 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 coher-ence in overdoped materials arises because the continuumin this regime is suppressed (Fig. 3(d)).At optimal doping, in the strange metal phase, theimaginary part of the self-energy, Σ (cid:48)(cid:48) ( ω ), is linear in ω ,which has been shown to be consistent with an Eliash-berg function that is frequency-independent [17]. Thisbehavior is highly consistent with the observation thatthe continuum is frequency-independent in this regime(Fig. 3(c)), and suggests that the M-EELS data mayhave a direct relationship with the Eliashberg functionitself.It is critical to keep in mind, however, that while thecontinuum provides a clear decay channel for quasiparti-cles, the mechanism by which the quasiparticles in turngenerate the continuum is less clear. Because the quasi-particles are highly dispersive in Bi-2212 [18, 19], a stan-dard Lindhard calculation of the density response us-ing RPA [22, 38] would yield a continuum that is alsohighly momentum-dependent, which is inconsistent withthe M-EELS data at all compositions studied. It maybe the case that beyond-RPA effects, such as excitonic,local field, or vertex correction effects dominate the re-sponse properties of strange metals. New theoretical ap-proaches using nonperturbative techniques going beyondRPA, such as those based on the AdS-CFT correspon-dence, may provide progress in understanding the densityresponse of the strange metal [43, 46].It is worth considering whether the momentum-independence of the density fluctuations could be a con-sequence of strong disorder, which might eliminate mo-mentum conservation by explicitly breaking translationalsymmetry, resulting in q -integrated response functions inall measurements. Even in materials in which the degreeof disorder is low by traditional crystallographic stan-dards, translational symmetry could be broken by emer-gent electronic heterogeneity, for which there is ampleevidence in the cuprates [20]. Such heterogeneity haseven been proposed as the origin of the linear-in- T re-sistivity in strange metals [47]. Unfortunately, such aview of the M-EELS data is inconsistent with ARPESand STM studies, which report clearly dispersing quasi-particle excitations in Bi-2212 where the charge responseis q -independent [15–20]. Moreover, recent RIXS experi-ments on electron-doped cuprates, which are not strangemetals but have a similar degree of disorder [48], haveclearly shown conventional dispersing plasmon excita-tions, in agreement with Hubbard model-based RPA cal-culations [49]. Also, dispersing collective modes havebeen observed with M-EELS in other materials with sim-ilar degree of disorder [50]. So it seems unlikely that dis-order is the sole cause of the q -independence we see. Onthe other hand, it remains possible that the strong cou-pling physics of Bi-2212 conspires with disorder in suchas way as to make momentum irrelevant in the densityresponse, though not other observables.Finally, we consider the question of whether the contin-uum could be a sign of some kind of quantum critical be-havior. In this view, the fan-like structure implied by Fig.4 might indicate a crossover near a doping p c ∼ .
16 thatcould be identified as a quantum critical point (QCP), assuggested by many authors [9, 15, 39, 51–55]. This inter-pretation is problematic, since few expected signatures of quantum criticality are present in the density fluctuationspectra. For example, no soft collective mode, with en-ergy falling to zero at p c , is visible in the data. A spectralweight rearrangement is observed away from p c in bothunderdoped and overdoped samples, however its energyscale ∼ . ∼
200 K, over which thisrearrangement takes place (Fig. 3(a)-(d)). Moreover, theresponse functions near p c do not exhibit any momentumdependence, which is expected in the usual Hertz-Millispicture of a quantum phase transition [56]. The factor-ing of Π (cid:48)(cid:48) ( q , ω ) we observe (Fig. 2) has been cited asevidence for local criticality, which has been argued tobe a feature of an exotic QCP [40, 41]. But this fac-toring is observed everywhere in the phase diagram, notjust in the vicinity of p c . Whether some kind of exoticcritical point might explain the peculiar physics takingplace remains, for now, an open question.In summary, we have shown that Bi-2212, despite be-ing a good conductor by most standards, does not exhibitwell-defined plasmon excitations that are dispersing andlong-lived anywhere in its phase diagram. Instead, thismaterial exhibits a featureless, momentum-independentcontinuum in the density response throughout the fan-shaped, strange metal region of the phase diagram. Out-side this fan, the response undergoes dramatic changes inspectral weight up to 0.5 eV exhibiting a sign-reversal atoptimal doping. Our study establishes a featureless con-tinuum as a defining property of the mysterious strangemetal phase in Bi-2212 and places it at a crossover be-tween two regimes with opposite trends in their chargesusceptibility. Major open questions remain concerningthe relationship between the density response and modelsbased on quasiparticle scattering, disorder, or quantumcritical fluctuations. A new kind of theory of interact-ing matter may be needed to explain the existence ofthis phase and its connection to other exotic phenomenasuch as high temperature superconductivity. 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 forsupplying the ellipsometry data used to evaluate the f -sum rule, and I. El Baggari for useful sample preparationadvice. This work was supported by the U.S. Depart-ment of Energy, Office of Basic Energy Sciences grant no.DEFG02-06ER46285. P.A. gratefully acknowledges sup-port from the EPiQS program of the Gordon and BettyMoore Foundation, grant GBMF4542. Crystal growthwas supported by DOE grant de-sc0012704. B.U. ac-knowledges NSF CAREER grant DMR-1352604. M.M.acknowledges support by the Alexander von HumboldtFoundation through the Feodor Lynen Fellowship pro-gram. C.D. acknowledges use of the electron microscopyfacilities in the Eyring Materials Center at Arizona StateUniversity. APPENDIX A: DETERMINING Π( q , ω ) At large momentum, the Coulomb interaction V ( q ) issmall so the susceptibility, χ ( q , ω ), and the polarizability,Π( q , ω ), are nearly the same (Eq. 2). At small momenta,particularly q < .
16 r.l.u., relating them requires knowl-edge of V ( q ). The precise functional form of V ( q ) candepend on the material geometry, especially in quasi-2Dsystems [57] or at the surface of a layered material. Un-derstanding this relation for M-EELS is still a work inprogress. Nevertheless, as argued in Ref. [23], the ap-proximate form shown in Eq. 3 is a sensible phenomeno-logical starting point as it exhibits the same functionalbehavior as that of an infinite (i.e. not surface termi-nated) layered system [57].To determine the proportionality constant in Eq. 3,we follow a similar procedure to Ref. [23]. Noticingthat in the limit of large momentum transfer, χ ( q , ω ) ≈ Π( q , ω ) /(cid:15) ∞ , we treat V as a fit parameter, V fit , and deter-mine the values of V fit where Π most closely resembles itsvalue at the highest momentum measured, q = 0 . (cid:15) ∞ | Π( q , ω ) − Π(0 . , ω ) | = | χ ( q , ω )1 + V fit ( q ) χ ( q , ω ) − χ (0 . , ω ) | . (5)After obtaining V fit ( q ), we fit to it with the functionalform of Eq. 3 using d = 15 .
62 ˚A to determine the propor-tionality constant. The fits are shown in Fig. 5, alongwith curves with alternate values of d shown for refer-ence. The fitted proportionality constant was found tobe (7 . ± . · eV · ˚A . Considering a rough estimate forthe proportionality constant of 4 πe d = 2 . · eV · ˚A is within a factor 3, the fit value is quite reasonable giventhe systematic uncertainties. APPENDIX B: CONSISTENCY BETWEENM-EELS AND OTHER ELECTROMAGNETICPROBES
Our M-EELS data are consistent, in most respects,with previous studies of the electromagnetic propertiesof cuprates. Like M-EELS [25], spectroscopic ellipsom-etry [29, 34] and transmission EELS experiments withmodest energy resolution (i.e., elastic linewidths of ≥ ∼ q = 0 .
05 r.l.u., M-EELS and transmission EELSgive nearly identical plasmon energy and lineshape [25]. q (r.l.u.) V ( q ) ( e V ⋅ Å ) d = 13 Åd = 15.62 Åd = 19 ÅV fit (q) FIG. 5. V fit ( q ) determined by minimizing Eq. 5 for OD50K,and the subsequent fit using Eq. 3, obtaining a proportional-ity constant of (7 . ± . · eV · ˚A . Vertical error bars reflectstatistical fitting uncertainties, while horizontal error-bars re-flect the momentum resolution of the M-EELS instrument of0 .
03 ˚A − . For reference, curves evaluated with d = 13 ˚A and d = 19 ˚A are also shown. Further, the plasmon energy measured with ellipsometryis independent of doping [29, 60–62], starkly contrast-ing with the behavior of a normal conductor [22, 38, 63],suggesting it may arise from a continuum of the sort wesee.Still, it is clear something qualitative must change inthe charge response as q →
0. For one, the plasmonlineshape observed with ellipsometry, which is strictly a q = 0 probe, is 50% narrower than the M-EELS andtransmission EELS data at the lowest momentum stud-ied [25]. Further, the spectral weight in ellipsometrygrows with decreasing temperature for all dopings by nolarger than ∼
3% [29], while the changes observed hereare around 50% and may have either sign (Fig. 3(e)).More fundamentally, the factoring of the polarizability,Π (cid:48)(cid:48) ( q, ω ) = f ( q ) · g ( ω ), cannot persist to zero momentumbecause it violates the compressibility sum rule [22, 38],lim q → Π( q,
0) = − n κ, (6)where κ = − ( ∂V /∂P ) N /V is the compressibility. Bydefinition, κ = 0 for an insulator while for a metal κ is a constant. If the factoring of Π persisted all theway to q → f -sum rule (Eq. 2) would requireΠ( q, ∼ q in the limit of small momentum, implyingthat the system is an insulator. However Bi-2212 is ametal, of course, meaning this factoring cannot persist as q →
0. Full reconciliation between q = 0 probes like el-lipsometry and finite- q techniques may require new theo-retical ideas about non-commutativity of limits [64]. En-couragingly, however, our ξ , which represents integratedspectral weight change, exhibits the same trend with dop-ing as the Coulomb energy reported in Ref. [29], chang-ing sign at optimal doping in both studies. Forthcomingstudies with higher momentum resolution will determinethe nature of the crossover region between the two tech-niques.We note that, at larger momenta, transmission EELSstudies of Bi-2212 by different groups report conflictingresults. Using 60 keV electrons in an EELS spectrom-eter employing Wien filters and a 30 meV quasi-elasticline, Terauchi et al. reported a featureless continuumwith a 1 eV cutoff energy similar to what we report here[65]. On the other hand, using 170 keV electrons withhemispherical analyzers and quasi-elastic line extendingto 500 meV, N¨ucker et al. observed no continuum atall, but instead report conventional plasmon behaviorexhibiting normal, Fermi liquid-like dispersion [32, 59].More recently, transmission EELS studies by Schuster etal. of the underdoped cuprate Ca − x Na x CuO Cl alsoshow evidence for a broad continuum, rather than con-ventional dispersing plasmons [66].As an initial step towards establishing consistency be-tween M-EELS and transmission EELS techniques, wehave performed a preliminary measurement of OP91KBi-2212 using a monochromated Nion UltraSTEM Scan-ning Transmission Electron Microscope EELS (STEM-EELS) with 10 meV resolution, as shown in Fig. 6. Thesemeasurements were performed at room-temperature with60 keV electrons and convergence/acceptance semi-angles of 4 mrad. Samples were prepared using a “pow-der” method, where a single crystal of OP91K Bi-2212is crushed in a mortar and pestle and dispersed onto aholey carbon grid using ethanol. A flake was found withthe correct orientation and estimated thickness of be-low 400 ˚A. Sample crystallinity and ab-plane orientationwas confirmed by verifying the well-known Bi-2212 su-permodulations (see the inset of Figure 6). Note thatthe STEM-EELS measurements show precisely the samecontinuum with the same cutoff energy as M-EELS mea-surements. This indicates that the continuum is a bulkproperty of Bi-2212, and not a peculiarity of the surface.We note that, because STEM-EELS instruments aredesigned for high spatial-resolution at the expense ofmomentum-resolution, the STEM-EELS spectrum inFig. 6 is momentum-integrated azimuthally in the ab-plane for q (cid:46) .
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Energy Loss (eV) EE L S I n t en s i t y ( a r b . ) STEM-EELS (raw)STEM-EELS (antisymm.)M-EELS
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