Ubiquitous antinodal quasiparticles and deviation from simple d-wave form in underdoped Bi-2212
I. M. Vishik, M. Hashimoto, W.-S. Lee, T. P. Devereaux, Z.-X. Shen
aa r X i v : . [ c ond - m a t . s up r- c on ] A ug Ubiquitous antinodal quasiparticles and deviation from simple d -wave form in underdoped Bi-2212: Supplementary materials
1o assess the effects of instrument resolution on the measured gap function, we simulatespectra using a model spectral function proposed by Kordyuk et al. [S1]:Σ”( k , ω ) = q ( αω ) + ( βT ) (1)The simulations in Fig. S1 use the values ( α, β, T ) = (0 . , . , K ) which produces spectracomparable to experiments. An additional energy-independent momentum broadening of0.006 ˚ A − is included for better agreement with data. The band structure is taken to be atight binding model with parameters( µ, t, t ′ , t ′′ , t ′′′ , t ′′′′ ) = (0 . , − . , . , − . , − . , . F in a low-temperature experiment, applying a convolution operation to a band which crossesE F will tend to push weight asymmetrically near E F . In the cuprates, the near-nodal regionis most strongly affected by resolution effects because the Fermi velocity is larger. Thisis seen in Fig. S1(h)-(j) which shows symmetrized EDCs at k F for three cuts indicatedin panel (k). Closer to the node, instrumental energy resolution pushes the EDC peak tohigher binding energy, but this effect is less pronounced at the antinode where bands areless dispersive.The simulations are summarized in Fig. S1(a)-(d) which shows gaps around the Fermisurface for different choices of instrument resolution. The gap function is deliberately chosento deviate from a simple d -wave form, and is instead described by a form with a higherharmonic: ∆( θ ) = ∆ ∗ [cos(2 θ )+(1 − B ) ∗ cos(6 θ )]. The parameter B quantifies the degree ofdeviation from a simple d -wave form, with B=1 signifying a simple d -wave form and smallervalues indicating greater deviation from a simple d -wave form. With increasing energyresolution and identical momentum resolution, the B parameter approaches 1, indicatingdecreasing deviation from a simple d -wave form. This may be one of the reasons Ref. S2reports a simple d -wave form for all dopings. When the deviation from a simple d -wave formis subtle, poorer resolution can obscure this effect. Increasing the momentum resolution (Fig.S1(d)) can also smooth out subtle features in the gap function.2 G a p ( m e V ) θ (°) no resolution ∆ =39 meVB=0.88 50403020100 G a p ( m e V ) θ (°) ∆ E=8 meV ∆ k=0.01Å -1 ∆ =41 meVB=0.91 50403020100 G a p ( m e V ) θ (°) ∆ E=20 meV ∆ k=0.01Å -1 ∆ =43 meVB=0.94 G a p ( m e V ) θ (°) ∆ E=8 meV ∆ k=0.035Å -1 ∆ =45 meVB=1 -0.3-0.2-0.10.00.1 E - E F ( e V ) -0.4 -0.2 0.0 0.2 0.4k ( π /a) no resolution 1 -0.3-0.2-0.10.00.1 E - E F ( e V ) -0.4 -0.2 0.0 0.2 0.4k ( π /a) -0.3-0.2-0.10.00.1 E - E F ( e V ) -0.4 -0.2 0.0 0.2 0.4k ( π /a) -0.3-0.2-0.10.00.1 E - E F ( e V ) -0.4 -0.2 0.0 0.2 0.4k ( π /a) ∆ E=8meV, ∆ k=.01Å -1 -0.3-0.2-0.10.00.1 E - E F ( e V ) -0.4 -0.2 0.0 0.2 0.4k ( π /a)-0.3-0.2-0.10.00.1 E - E F ( e V ) -0.4 -0.2 0.0 0.2 0.4k ( π /a) -0.3-0.2-0.10.00.1 E - E F ( e V ) -0.4 -0.2 0.0 0.2 0.4k ( π /a)-0.3-0.2-0.10.00.1 E - E F ( e V ) -0.4 -0.2 0.0 0.2 0.4k ( π /a)-0.3-0.2-0.10.00.1 E - E F ( e V ) -0.4 -0.2 0.0 0.2 0.4k ( π /a) ∆ E=20meV, ∆ k=.01Å -1 I n t e n s i t y ( A r b ) -100 -50 0 50 100E-E F (meV) No resolution ∆ E=8meV ∆ E=20meV -100 -50 0 50 100E-E F (meV) I n t e n s i t y ( A r b ) -100 -50 0 50 100E-E F (meV) (e1)(e2)(e3) (f1)(f2)(f3) (g1)(g2)(g3) (h) (i)(j) -101 k y ( π / a ) π /a) (k) FIG. S1. (a)-(d) Gap vs Fermi surface angle for various choices of energy and momentum resolutionin simulation. Fits are to a d -wave gap with a higher harmonic term: ∆( θ ) = ∆ ∗ [cos(2 θ ) + (1 − B ) ∗ cos(6 θ )]. Panel (c) marks the three cuts which are detailed in subsequent panels. (e)-(g)simulated data with cuts taken parallel to the Brilliouin zone boundary. In (f)-(g), data has beenconvolved with a resolution function. (h)-(j) EDCs at k F with varying resolution. (k) Fermi surfaceschematic, Fermi crossings used in (a)-(d), and cuts used in (e)-(g). S1] A. A. Kordyuk, S. V. Borisenko, M. Knupfer, and J. Fink, Phys. Rev. B , 064504 (2003).[S2] J. Zhao, U. Chatterjee, D. Ai, D. Hinks, H. Zheng, G. Gu, S. Rosenkranz, J.-P. Castellan,H. Claus, M. R. Norman, M. Randeria, and J. C. Campuzano, Proc. Nat. Acad. Sci. ,17774 (2013). r X i v : . [ c ond - m a t . s up r- c on ] A ug Ubiquitous antinodal quasiparticles and deviation from simple d -wave form in underdoped Bi-2212 I. M. Vishik, Makoto Hashimoto, W. S. Lee, T. P. Devereaux, and Z. X. Shen
3, 41
Massachusetts Institute of Technology,Department of Physics, Cambridge, MA, 02139, USA Stanford Synchrotron Radiation Lightsource,SLAC National Accelerator Laboratory,Menlo Park, California 94025, USA Stanford Institute for Materials and Energy Sciences,SLAC National Accelerator Laboratory,2575 Sand Hill Road, Menlo Park, CA 94025, USA Geballe Laboratory for Advanced Materials,Stanford University, Stanford, CA 94305, USA (Dated: August 4, 2018)
Abstract
The momentum dependence of the superconducting gap in the cuprates has been debated, withmost experiments reporting a deviation from a simple d x − y form in the underdoped regime anda few experiments claiming that a simple d x − y form persists down to the lowest dopings. Weaffirm that the superconducting gap function in sufficiently underdoped Bi Sr CaCu O δ (Bi-2212) deviates from a simple d -wave form near the antinode. This is observed in samples wheredoping is controlled only by oxygen annealing, in contrast to claims that this effect is only seenin cation-substituted samples. Moreover, a quasiparticle peak is present at the antinode down top=0.08, refuting claims that a deviation from a simple d -wave form is a data analysis artifactstemming from difficulty in assessing a gap in the absence of a quasiparticle. ignificance Statement: The origin of high temperature superconductivity in the cupratesis an enduring question in condensed matter physics. A key difficulty is that the electronicphase existing at temperatures above the superconducting transition temperature is poorlyunderstood. This pseudogap phase is alternately attributed to fluctuating superconductivityor to a state which is distinct from superconductivity, and these competing scenarios havedifferent implications for the origin of superconductivity. We report evidence in support forthe latter scenario, which manifests in the angular dependence of the superconducting gap.
I. INTRODUCTION
Angle-resolved photoemission spectroscopy (ARPES) can measure the spectral gap in thesuperconducting state to great precision, but yet there are still disputes about the details ofthe momentum dependence. In overdoped Bi-2212, there is wide agreement that the super-conducting gap function follows a simple d -wave form, expressed at 0.5 ∗| cos( k x ) − cos( k y ) | or cos(2 θ ) [25–27]. We use the former convention in this paper. In the underdoped regime,there are experiments that find that the gap function deviates from a simple d -wave form[25, 26, 28, 29], with the antinodal region exhibiting a larger gap than implied by the simple d -wave trend in the near nodal region. This is reported in Bi-2212, (Bi,Pb) (Sr,La) CuO δ (Bi-2201) [30], La − x Sr x CuO δ (LSCO) [31, 32], the inner plane of Bi-2223 which is moreunderdoped than the outer planes [33], and YBa Cu O y (YBCO) [34]. Recently, Zhao etal have reported that the reported deviation from a simple d -wave form in Bi-2212 is anartifact of cation substitution which is used to achieve stable underdoping [27]. They claimthat cation substitution (Dy or Y) on the Ca site suppresses the antinodal quasiparticle, andwhen antinodal quasiparticles are present, the gap function always follows a simple d -waveform. They also claim that without cation substitution, the gap function always follows asimple d -wave form.In the ARPES literature, a deviation of the superconducting gap function from a simple d -wave form is often taken as evidence that the pseudogap coexists with superconductivitybelow T c , which implies that it is a distinct phase. However, the purpose of this paper isnot to discuss the relationship between the pseudogap and superconductivity. Rather, wesimply aim to clarify the experimental facts. We show data demonstrating a deviation froma simple d -wave form in a sample which was underdoped only by oxygen annealing. Then,2 ample Composition Photon energy Cut geometry Resolution Temperature FigureUD50 Bi Sr (Ca,Y)Cu O δ Sr (Ca,Dy)Cu O δ x Sr − x CaCu O δ Sr CaCu O δ Sr CaCu O δ π , π ) line and ΓM refers to cuts taken parallelto ( π ,0)-( π , π ). UD(OD) refers to underdoped (overdoped) samples and the number which followsgives the T c . we demonstrate that antinodal quasiparticles are clearly observed in samples with variouscompositions down to a T c of 50K (p ≈ d -wave form in underdoped samples with a T c of 75K or smaller. We explore variousreasons for the discrepancy between our result and that of Zhao et al including matrixelements and resolution.ARPES experiments were performed at Stanford Synchrotron Radiation Lightsource(SSRL). Samples were cleaved in situ at the measurement temperature (T < T c ) at a pressurebetter than 5 × − Torr. The Fermi energy, E F , was determined from the Fermi edge ofpolycrystalline gold which is electrically connected to the sample. Table I shows the com-position and experimental configuration for all of the samples presented in this paper. Mostsamples were measured with a SCIENTA R4000 analyzer (angular resolution ≈ . ◦ ), ex-cept for UD50 which was measured with a SCIENTA SES-200 analyzer (angular resolution ≈ . ◦ ). Energy resolutions are also listed in Table I. II. DATA
Fig. 1 shows energy distribution curves (EDCs) at the Fermi momentum, k F , for UD75,as well as gaps derived from those data. Notably, the T c of this sample was achievedonly by oxygen annealing, and no cation substitution. Quasiparticles (QPs) are observedover the entire Fermi surface (FS), from the node to the antinode. Symmetrization, given3 n t e n s i t y ( a r b , o ff s e t ) -0.10 0.00E-E F (eV)UD75 -0.1 0.0 0.1E-E F (eV) θ G a p ( m e V ) EDC peak position Fitted gap x )-cos(k y )| ∆ G a p ( m e V ) k y k x θ FS EDC momenta Cut (c)(d) I n t e n s i t y ( A r b ) -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15E-E F (eV) E D C w i d t h ( m e V ) x )-cos(k y )| Norman model, Γ Lorentzian FWHM
Symmetrized EDCFitting: Lorentzian + Linear BG Linear BG Lorentzian Lorentzian + Linear BGFitting: Norman model Norman model (e)(f)
FIG. 1. Antinodal EDCs and deviation from simple d -wave form in sample whose doping iscontrolled only by oxygen annealing without any cation substitution. (a) Raw EDCs at k F forUD75 from near the node (1, top) to the antinode (bottom) (b) Symmetrized EDCs at k F , samecuts as (a). (c) Gap as a function of the simple d -wave form, represented by 0.5* | cos( k x ) − cos( k y ) | .Gap defined from symmetrized EDCs in two ways: 1) fitting to assumed model (green) [35] and 2)from energy of EDC peak (cyan). Dashed lines are linear fits to the first 5 data points. (d) Gap asa function of Fermi surface angle, θ , defined in inset. Dashed lines are fits of first 5 data points to∆( θ ) = ∆ cos(2 θ ) and color definitions are same as panel (c). Inset: Fermi crossings for EDCs in(a)-(b) and gap fits in (c)-(d). Dashed lines represent cut geometry of experiment. (e) EDC widthas a function of simple d -wave form, determined by two different methods: (1)the Γ parameter inthe Norman model [35] which defines the EDC width (orange) and (2) Lorentzian FWHM whena portion of the EDC is fit to a Lorentzian plus linear background (green). (e) Both EDC peakwidth fitting methods, shown for cut 5. by I ( k F , ω ) + I ( k F , − ω ), is used to remove the Fermi-Dirac cutoff, assuming particle-holesymmetry, which should be valid assumption in the superconducting state. The gap at eachmomentum is determined in two ways in Fig. 1(d): by fitting symmetrized EDCs to aminimal model [35] and from the peak positions of symmetrized EDCs. Both yield resultswithin error bars of one another. The momentum dependence of the superconducting gap isillustrated by plotting as a function of the simple d -wave form, expressed as 0.5* | cos( k x ) − k y ) | . This expression is equal to zero at the node (along the (0,0) to ( π , π ) line) and closeto one at the antinode (where the FS meets the Brillouin zone boundary). A superconductinggap is said to follow a simple d -wave form if all data points fall on a straight line whenplotted in terms of 0.5* | cos( k x ) − cos( k y ) | . By this criterion, the gap function in Fig.1(d) does not follow a simple d -wave form over the entire FS. In the near-nodal region,the gap function does follow a simple d -wave form, and the linear fit to the first 5 datapoints is shown by dashed lines for both methods of determining gaps. In the antinodalregion, measured gaps are larger than the trend implied by the dotted lines, indicating adeviation of the gap function from a simple d -wave form. Another expression for the simple d -wave form is cos(2 θ ), and gaps are plotted in terms of the FS angle, θ in Fig. 1(d).Again, fitting the five data points closest to the node to a simple d -wave form yields atrend from which the antinodal points deviate. These data refute the claim in Ref. 27that a deviation from a simple d -wave form is only observed in cation substituted samples.To emphasize the robustness of antinodal quasiparticles, the EDC width is plotted as afunction of 0.5* | cos( k x ) − cos( k y ) | in Fig. 1(e). Two methods of quantifying the EDC widthare shown. The simplest method is to assume a Lorentzian peak on a background whichvaries linearly with binding energy. The second method is from the so called single-particlescattering rate, Γ , from the same phenomenological model from which the gap was fit inpanels (c)-(d). The EDC width derived by both methods varies approximately by a factorof two around the Fermi surface, demonstrating that the EDC width does not diverge in theantinodal region. This verifies that the criterion used to assess the gap is the same in thenear-nodal and near-antinodal regions. Notably, the EDC width in the antinodal region issmaller than the gap energy. We note that the increase in EDC width near the node arisesfrom momentum resolution which can broaden lineshapes in energy for a dispersive band.Fig. 2 shows EDCs around the Fermi surface for several samples of varying compositionstogether with their fitted gaps. In the most underdoped sample, UD50, doping is achievedby substituting Y on the Ca site. The the second most underdoped sample, UD65, an excessof Bi achieves lower T c . UD92 represents the as-grown doping of Bi Sr CaCu O δ . In Fig.2, the two most underdoped samples clearly show QPs all around the Fermi surface togetherwith a deviation of the gap function from a simple d -wave form. This together with anidentical result for UD75 in Fig. 2 demonstrates that the conclusion of a universal simple d -wave form of the superconducting gap in Ref. 27 is simply not correct. We note that5 a) UD50 -0.15 -0.10 -0.05 0.00 0.05E-E F (eV) -0.10 -0.05 0.00 0.05E-E F (eV) (c) UD92 node (N)antinode (AN) N AN ( π,π )( π , ) (0,0) (b) UD65 -0.10 -0.05 0.00 0.05E-E F (eV) G a p ( m e V ) G a p ( m e V ) G a p ( m e V ) (d) UD50 (e) UD65 (f) UD92 I n t en s i t y ( A r b . U n i t s ) FIG. 2. Ubiquitous QPs all around the FS for various compositions and corresponding gap functions(a)-(c) EDCs at k F from the node(top) to the antinode (bottom) for UD50, UD65, and UD92.Corresponding fitted gaps are plotted in (d)-(f). Dashed lines in (d)-(f) represents simple d -waveform. in the inner-plane of a trilayer cuprate, Bi Sr Ca2Cu O δ (Bi-2223), a deviation of thegap function from a simple d -wave form is also accompanied by quasiparticles [33]. As anaside, recent high-resolution experiments showed subtle deviations from a simple d -waveform in UD92 in the portion of the FS intermediate between the node and antinode, andthis behavior persisted up to p=0.19 [25, 26].The ARPES results shown in Figs. 1 and 2 are in agreement with other spectroscopies,including Raman and scanning tunneling spectroscopy (STS) which show different energyscales in the near-nodal and near-antinodal regions of momentum space implying a deviationfrom a simple d -wave form [36–38]. This is illustrated with STS data in Fig. 3, showingspectra for a UD58 sample. Spectra averaged over a large field of view clearly show twoenergy scales, marked with black and red arrows. If the gap function followed a simple d -wave form, STS would only show one energy scale, and this is indeed what is observedin overdoped samples [36]. In the d -wave superconducting state, energies close to zero6 ias (mV) N o r m a li z e d C ondu c t a n ce −150 −100 −50 0 50 100 15000.511.52 UD58 avg Bias (mV) N o r m a li z e d C ondu c t a n ce −150 −100 −50 0 50 100 15000.511.52 ∆ = 77 mV ∆ = 87 mV ∆ = 94 mV ∆ = 98 mV ∆ = 102 mV ∆ = 107 mV ∆ = 114 mV UD58, LDOS
FIG. 3. STS spectrum, UD58 (Bi Sr (Ca,Dy)Cu O δ ), averaged over 25nm × bias voltage correspond to momenta near the node, so the lower energy scale (black arrow)corresponds to the near-nodal gap and the higher energy scale (red arrow) corresponds to theantinodal gap. At higher bias voltage, STS shows local inhomogeneity on several-nanometerlength scales [39, 40], but crucially, the two distinct energy scales are clearly visible in localspectra as well as the spectrum averaged over a large field of view. The averages spectrumis most relevant for comparisons to ARPES where the beam spot size is usually larger than100 µ m. STS data suggest that a deviation from a simple d -wave form is a generic feature ofunderdoped cuprates, rather than an anomaly observed by a single experimental technique. III. DISCUSSION
What might be the cause of the differing conclusions from our data and from Ref. 27? Akey point which must be clarified is that the absence of a quasiparticle is not as conclusiveas the presence of a quasiparticle. In particular, matrix element effects [41–43] arising froma poor choice of experimental configuration (polarization, cut geometry, photon energy) canmake a quasiparticle become less apparent in the spectrum. To illustrate the effects ofmatrix elements, Fig. 4 shows EDCs at the bonding band k F for UD55 taken with several7 n t e n s i t y ( A r b , o ff s e t ) -0.3 -0.2 -0.1 0.0 0.1E-E F (eV)Bonding band 22.7 eV 21 eV 19 eV 18.4 eV FIG. 4. EDCs at k F at the antinode (bonding band) for UD55. Curves for different incident photonenergies are offset vertically for clarity. different photon energies. Data were taken on a single sample and a single cleave with beampolarization parallel to the Cu-O bond direction. The quasiparticle peak is visible with 22.7eV photons, enhanced with 21 eV photons, and greatly suppressed when measurements aredone with 19 eV photons or 18.4 eV photons. For the UD75 and UD92 samples, 22.7 eVphoton energy enhances the intensity of the bonding band relative to the antibonding band,which is why this photon energy is commonly chosen for for experiments on Bi-2212 nearoptimal doping [29]. However, for UD55, 22.7 eV does not yield the optimal cross section.Similarly, 18.4 eV photon energy is a common choice to enhance the antibonding band inoverdoped Pb-doped samples, but for UD55, this photon energy yields poor cross sectionfor both the bonding and antibonding bands. In deeply underdoped Y-Bi-2212, the optimalcross-section was found at 19eV photon energy measured in the second Brillouin zone [28].Different experimental configurations must be thoroughly explored before concluding that aquasiparticle is absent.Another difference between experiments is that the ones highlighted in this manuscriptwere typically performed with energy resolutions of better than 10 meV, while those ofZhao et al were performed with energy resolutions of 15-20 meV. A poorer resolution can8ffect the EDC peak position, particularly in the near-nodal region where bands are moredispersive and gaps are smaller. This can diminish the visibility of details of the momentumdependence of the gap, such as a deviation from a simple d -wave form, in the superconductingstate which are clearly visible in experiments performed with better resolution. Simulationsof energy and momentum resolution effects are shown in Supplementary Information.The work in Ref. 27 brought up an important question about the effects of chemistryon the momentum-resolved electronic structure of cuprates, and it is crucial that futureARPES experiments address this with adequate rigor. This is an important avenues for un-derstanding the mechanism of high temperature superconductivity: in single-layer cuprates.T c,max varies by more than a factor of two depending on the composition of the chargereservoir layers between the CuO planes and the types of disorder which are present [44].ARPES has addressed this issue in single-layer Bi Sr . L . CuO δ (L = La, Nd, Eu, Gd),and Gd doping has been shown to lower T c substantially [45], accompanied by a loss ofantinodal coherence and an enhanced antinodal gap [46]. Eu doping also suppresses T c , andhas been shown to increase T* [47]. In Bi-2212, crystal-growth studies have shown thatSr-site disorder, particularly, excess Bi on the Sr site, suppresses T c , although not as acutelyas in Bi-2201 [48]. Bi-2212 samples with a stochiometric composition have a maximum T c of 94K, but typical samples have an excess of Bi, which brings the maximum T c down to89-92K [44]. By doping 8% Y on the Ca site, it is possible to raise the maximum T c ofBi-2212 to 96K, by stabilizing growth of samples with stochiometric Bi and Sr concentra-tions. The fact that cation substitution is required to maximize the T c of Bi-2212 refutesclaims that cation substituted samples are more disordered. However, the effect this has onARPES observables–gaps, dispersions, and spectral weight–has not been explored system-atically. This is crucial for interpreting ARPES data going forward. However, this shouldnot be confused by the claims of Ref. 27 that attribute the shape of the gap function tochemistry effects. Data collected from similar samples as in Ref. 27 with better resolutionand optimized matrix element detected clear QP peak, questioning the claim that the QPare suppressed in deeply underdoped samples due to this chemistry effect. Additionally,similar results are observed in other cuprates by a number of research groups [30, 33, 34].The work presented in this manuscript reaffirms the intrinsic nature of a gap function thatdeviates from simple d -wave form in underdoped cuprates, suggesting a pseudogap origin ofthis observation. 9 CKNOWLEDGMENTS
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