⇦Chinese Physics Letters, 2018, Vol. 35, No. 1, Article code 017401 Temperature Evolution of Energy Gap and Band Structure in the Superconducting and Pseudogap States of Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ Superconductor Revealed by Laser-Based Angle-Resolved Photoemission Spectroscopy * Xuan Sun(孙璇)1,2, Wen-Tao Zhang(张文涛)1, Lin Zhao(赵林)1, Guo-Dong Liu(刘国东)1, Gen-Da Gu(顾根大)3, Qin-Jun Peng(彭钦军)4, Zhi-Min Wang(王志敏)4, Shen-Jin Zhang(张申金)4, Feng Yang(杨峰)4, Chuang-Tian Chen(陈创天)4, Zu-Yan Xu(许祖彦)4, Xing-Jiang Zhou(周兴江)1,2,5** Affiliations 1National Laboratory for Superconductivity, Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190 2University of Chinese Academy of Sciences, Beijing 100049 3Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973, USA 4Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, Beijing 100190 5Collaborative Innovation Center of Quantum Matter, Beijing 100871 Received 20 November 2017 *Supported by the National Key Research and Development Program of China under Grant No 2016YFA0300300, the National Natural Science Foundation of China under Grant No 11334010, the National Basic Research Program of China under Grant No 2015CB921300, and the Strategic Priority Research Program (B) of the Chinese Academy of Sciences under Grant No XDB07020300.
**Corresponding author. Email: XJZhou@iphy.ac.cn Citation Text: Sun X, Zhang W T, Zhao L, Liu G D and Gu G D et al 2018 Chin. Phys. Lett. 35 017401 Abstract We carry out detailed momentum-dependent and temperature-dependent measurements on Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (Bi2212) superconductor in the superconducting and pseudogap states by super-high resolution laser-based angle-resolved photoemission spectroscopy. The precise determination of the superconducting gap for the nearly optimally doped Bi2212 ($T_{\rm c}=91$ K) at low temperature indicates that the momentum-dependence of the superconducting gap deviates from the standard $d$-wave form ($\cos(2{\it \Phi}$)). It can be alternatively fitted by including a high-order term ($\cos(6{\it \Phi}$)) in which the next nearest-neighbor interaction is considered. We find that the band structure near the antinodal region smoothly evolves across the pseudogap temperature without a signature of band reorganization which is distinct from that found in Bi$_2$Sr$_2$CuO$_{6+\delta}$ superconductors. This indicates that the band reorganization across the pseudogap temperature is not a universal behavior in cuprate superconductors. These results provide new insights in understanding the nature of the superconducting gap and pseudogap in high-temperature cuprate superconductors. DOI:10.1088/0256-307X/35/1/017401 PACS:74.25.Jb, 74.72.-h, 79.60.-i, 74.72.Kf © 2018 Chinese Physics Society Article Text High-temperature cuprate superconductors are well-known by their unusual superconducting state and anomalous normal state.[1-5] It is generally believed that the superconducting gap shows a distinct $d$-wave form[1,6-9] while its normal state exhibits anomalous behaviors including most notably the existence of the pseudogap.[2,8-12] The pairing mechanism of superconductivity, the nature of the pseudogap, and the relationship between the pseudogap and superconductivity remain unclear after three decades of intensive research.[5] It is still under debate on the origin of the pseudogap,[2,9] in particular, whether it represents a crossover or a phase transition across the pseudogap temperature $T^{*}$. Angle-resolved photoemission (ARPES) measurements on Bi$_2$Sr$_2$CuO$_{6+\delta}$ (Bi2201) find that the band structure near the antinodal region experiences a dramatic band reorganization at the pseudogap transition temperature; the bands show different Fermi momenta across $T^{*}$.[9,13,14] These observations signal a phase transition at $T^{*}$.[13,14] It is important to investigate whether such a band reorganization across $T^{*}$ is universal or not in cuprate superconductors, which will be crucial for understanding the nature of the pseudogap. In this Letter, we report detailed momentum- and temperature-dependent ARPES measurements on Bi2212 using a super-high resolution laser-based ARPES system. We precisely determine the momentum dependence of the superconducting gap in a nearly optimally doped Bi2212 with a $T_{\rm c}$ at 91 K. It is found that the superconducting gap deviates from the standard $d$-wave form ($\cos(2{\it \Phi})$) and it can be better fitted by including high-order term ($\cos(6{\it \Phi})$). We also measure the temperature dependence of the band structure near the antinodal region in both the superconducting and the pseudogap states. It is shown that the antinodal bands evolve smoothly with temperature and no signature of band reorganization occurs across the pseudogap transition $T^{*}$. Our results provide new information in understanding the pairing mechanism and the origin of the pseudogap in high-temperature cuprate superconductors. In experiment, angle-resolved photoemission measurements were carried out on a home-built vacuum ultraviolet (VUV) laser-based ARPES system.[15] The photon energy of the laser light source is 6.994 eV with a bandwidth of 0.26 meV. The energy resolution of the electron energy analyzer (Scienta R4000) was set at 1 meV, giving rise to an overall energy resolution of $\sim$1.0 meV, which is more significantly improved than some previous synchrotron-based ARPES measurements.[6-9] The angular resolution is $\sim$0.3$^{\circ}$, corresponding to a momentum resolution $\sim$0.004 Å at the photon energy of 6.994 eV, more than twice improved from 0.009 Å at a regular photon energy of 21.2 eV for the same angular resolution. The Fermi level is referenced by measuring clean polycrystalline gold that is electrically connected to the sample. High-quality Bi2212 single crystals were grown by the floating zone method. Nearly optimally doped Bi2212 single crystals with a $T_{\rm c}$ at 91 K (denoted as OP91K hereafter) were cleaved in situ and measured in vacuum with a base pressure better than $5\times10^{-11}$ Torr.
cpl-35-1-017401-fig1.png
Fig. 1. Fermi surface and band structure of optimally doped Bi2212 ($T_{\rm c}=91$ K) measured by laser ARPES at 15 K. (a) Spectral weight as a function of two-dimensional momentum $(k_x,k_y)$ integrated over [$-$1 meV,1 meV] energy window with respect to the Fermi level $E_{\rm F}$. The antibonding and bonding Fermi surface sheets are marked by the gray line and the dotted line, respectively. (b) Band structure at different momentum cuts. The corresponding location of the momentum cuts are marked in (a) as red lines. Constant energy contours at Fermi level (c), binding energies of 10 meV (d) and 25 meV (e).
Figure 1 shows the Fermi surface and band structure of the OP91K Bi2212 measured at 15 K. For the laser ARPES system with the photon energy of 6.994 eV, to the best of our knowledge, it is the first time to approach very close to the antinodal (0, $\pi$) point. This provides us a good opportunity to study the band structure and energy gap on the entire Fermi surface with super-high energy and momentum resolutions. Figure 1(a) shows the measured Fermi surface overlaid with the bonding and the antibonding Fermi surface sheets as reported before.[16-18] Antibonding bands dominate in our measurements and the bonding bands are weak that are only observed near the antinodal region at low temperatures. The spectral weight in our Fermi surface mapping (Fig. 1(a)) is mainly confined near the nodal region because of the increased gap opening when moving the momentum to the antinodal region. Figure 1(b) shows the band structure evolution from the nodal to the antinodal regions. The total band width shrinks while the bands near the Fermi level shift downwards when moving from the nodal to the antinodal regions because of gap opening. Figures 1(c)–1(e) show the constant energy contours at the Fermi level and binding energies of 10 meV and 25 meV, respectively. The spectral weight distribution extends to antinodal region with increasing binding energy and eventually the entire contour shows up at the binding energy of 25 meV (Fig. 1(e)). The nearly complete coverage of the nodal and antinodal regions makes it possible to investigate the detailed momentum dependence and temperature dependence of the band structure and the energy gap as we will show in the following. Figure 2 shows the high-resolution measurement of the superconducting gap for the OP91K Bi2212 sample. Figures 2(a) and 2(b) show, respectively, the original photoemission spectra (energy distribution curves, EDCs) and the symmetrized EDCs measured at 15 K on the Fermi surface (Fig. 2(c)). Well-defined superconducting coherence peaks are observed on the entire Fermi surface (Fig. 2(a)). The EDC symmetrization with respect to the Fermi level is a standard procedure to remove the Fermi–Dirac function and to extract the energy gap. The energy gap is characterized by a dip at the Fermi level and the gap size 2${\it \Delta}$ is determined from the distance between the two peaks. The gap size can be obtained quantitatively by fitting the symmetrized EDCs to a phenomenological formula.[19] The obtained superconducting gap is shown in Fig. 2(d) (solid circles). This is the first laser ARPES measurement of the superconducting gap covering nearly the entire Fermi surface with a super-high energy resolution (1 meV). This makes it possible to examine the exact gap form for Bi2212 in the superconducting state. We first fitted the measured superconducting gap with the standard $d$-wave form, $\cos(2{\it \Phi})$ (blue and red lines in Fig. 2(d)), and found that the fitted lines cannot match the measured data well. This clearly indicates that the superconducting gap of OP91K Bi2212 does not follow the standard $d$-wave form. The same is true when we plot the superconducting gap as a function of $|\cos(k_xa)-\cos(k_ya)/2|$, which is another way of $d$-wave expression. The nodal region can be well-fitted by a standard $d$-wave gap form, while the antinodal region deviates significantly from the fitted line (Fig. 2(e)). Similar behaviors were also reported before.[20,21] Such a deviation has been attributed to either the pseudogap opening[21] or consideration of high order term that is related to the next nearest neighbor interactions.[20] By adding a high order term ($\cos(6{\it \Phi})$) into the gap form, we find that our measured gap can be fitted rather well (black lines in Figs. 2(d) and 2(f)). In particular, when we decomposed the fitted result into two components, $\cos(2{\it \Phi})$ (purple line in Fig. 2(f)) and $\cos(6{\it \Phi})$ (green line in Fig. 2(f)), we find that the minimum of the $\cos(6{\it \Phi})$ term near 30$^{\circ}$ seems to have some hint in the measured data (Fig. 2(f)). Our data support a $d$-wave superconducting gap with the next nearest-neighbor interactions considered in optimally doped Bi2212.
cpl-35-1-017401-fig2.png
Fig. 2. Precise determination of the superconducting gap for an optimally doped Bi2212 at 15 K. (a) Original EDCs along the Fermi surface. (b) The corresponding symmetrized EDCs along the Fermi surface. The location of the Fermi momenta along the Fermi surface is shown in (c). The original EDCs (a) and the symmetrized EDCs are normalized in intensity near the Fermi level region for easy comparison. (d) Extracted superconducting gap as a function of the momentum angle (${\it \Phi}$) that is defined in (c). The blue curve and red one represent standard $d$-wave form (${\it \Delta}_0\cos(2{\it \Phi})$) with ${\it \Delta}_0=26.3$ meV and 32.3 meV, respectively. The black curve is obtained by fitting the measured gap with the form ${\it \Delta}_0|B\cos(2{\it \Phi})+(1-B)\cos(6{\it \Phi})|$. (e) Superconducting gap as a function of $|\cos(k_xa)-\cos(k_ya)|/2$. The blue line represents 27.9$|\cos(k_xa)-\cos(k_ya)|/2$. (f) Fitting of the measured superconducting gap (red circles) with the form ${\it \Delta}_0|B\cos(2{\it \Phi})+(1-B)\cos(6{\it \Phi})|$. The obtained fitting parameters are ${\it \Delta}_0=32.0$ meV and $B=0.92$. The purple and green curves show the two components in the gap fitting: a standard $d$-wave term $32.0\times0.92\cos(2{\it \Phi})$ (purple) and a high order term $32.0\times0.08\cos(6{\it \Phi})$ (green).
Figure 3(a) shows the temperature-dependent band structure for a momentum cut near the antinodal region in the OP91K Bi2212 sample over a large temperature range between 15 K and 180 K. The location of the momentum cut is marked in the upper-left inset as the red line in Fig. 3(b). Figure 3(b) shows the extracted photoemission spectra (EDCs) measured at different temperatures for the Fermi momentum near the antinodal region (red circle in the inset of Fig. 3(b)). At low temperature, sharp superconducting coherence peaks have developed in the superconducting state from incoherent spectra at high temperature. The symmetrized EDCs (Fig. 3(c)) are obtained from the original EDCs (Fig. 3(b)) to extract the energy gap and the gap closing temperature. From Fig. 3(c) and the extracted gap size shown in Fig. 3(d), the gap closing temperature is about 140 K, which is consistent with the previous report that the pseudogap temperature for the optimally doped Bi2212 is around 160 K.[8] Figure 3(e) zooms in on the band structure evolution with temperature in both the superconducting state and the pseudogap state. We find that the Fermi momentum determined at 180 K is exactly at the same position as that determined at low temperatures when the energy gap opens. This indicates that this band does not experience a band reorganization across the pseudogap temperature $T^{*}$, which is different from the case in Bi2201.[9,13,14]
cpl-35-1-017401-fig3.png
Fig. 3. Evolution of band structure with temperature near the antinodal region for the optimally doped Bi2212 with a $T_{\rm c}=91$ K. (a) Band structure measured at different temperatures between 15 K and 180 K. The location of the momentum cut is marked in the inset of (b) by the red line. (b) Photoemission spectra (EDCs) measured at different temperatures. The corresponding Fermi momentum is marked as the red circle in the inset. (c) Symmetrized EDCs at different temperatures. (d) Energy gap as a function of temperature determined from the symmetrized EDCs in (c). (e) Zoomed-in band structure at different temperatures. The images represent the second derivative of the bands in (a) with respect to energy. The arrows mark the position of the Fermi momentum for each band.
cpl-35-1-017401-fig4.png
Fig. 4. Evolution of band structure with temperature close to the antinodal region for Bi2212. The measured sample surface corresponds to an overdoped case with a $T_{\rm c}\sim$77 K because of sample aging. (a) Band structure measured at different temperatures between 25 K and 180 K. After the sample was warmed up to 180 K, it was cooled down to 25 K again. The measured band (right-most panel in (a)) is identical to the initial one (left-most panel in (a)), indicating slight sample change during the cycle of the temperature-dependent measurement. (b) Second derivative images with respect to momentum for the four images in (a) measured at 25 K, 80 K, 120 K and 160 K. The corresponding momentum cut for the bands in (a) and (b) are marked by the red line in the inset of (c). This momentum cut crosses two antibonding Fermi surface sheets and a bonding Fermi surface sheet. (c) Photoemission spectra (EDCs) at the band bottom ($k_{\rm B}$ in the inset in (c)) measured at different temperatures. (d) EDCs at the Fermi momentum ($k_{\rm R}$ in (b)) measured at different temperatures. (e) Symmetrized EDCs at different temperatures corresponding to spectra in (e). (f) Zoomed-in band structure at different temperatures. The images are second derivative of the ones in (a) with respect to energy. The arrows mark the position of the Fermi momentum for the right antibonding band. (g) MDCs near the Fermi level for the measured bands (a) at different temperatures. The two arrows mark the position of the Fermi momentum for the left ($k_{\rm L}$) and the right ($k_{\rm R}$) antibonding bands.
Figure 4(a) shows another temperature-dependent measurement for Bi2212 along a momentum cut that is very close to the (0, $\pi$) antinodal point. The location of the momentum cut is marked in the inset of Fig. 4(c) as the red line which crosses two antibonding Fermi surface sheets ($k_{\rm L}$ and $k_{\rm R}$) and the (0,0)–(0,$\pi$) line ($k_{\rm B}$). The bonding bands can also be observed at low temperatures in Fig. 4(a). The bottom of the antibonding band shifts downwards with increasing temperature (Fig. 4(a)), which is more obvious in the corresponding second derivative images (Fig. 4(b)). This evolution of the band bottom with temperature is opposite to that reported in Bi2201.[13,14] The photoemission spectra (EDCs) at the band bottom ($k_{\rm B}$) are plotted in Fig. 4(c). Figure 4(d) plots EDCs at different temperatures at the Fermi momentum $k_{\rm R}$ and the corresponding symmetrized EDCs are shown in Fig. 4(e). The gap closing temperature for this Fermi momentum is $\sim$95 K (Fig. 4(e)). We note that this optimally doped Bi2212 sample with a $T_{\rm c}=91$ K shows an aging effect after a long-time ARPES measurement which tends to increase the hole doping. The band structure (Fig. 4(a)), the gap size (Fig. 4(e)) and the pseudogap temperature are both in good agreement with those in the overdoped Bi2212 sample with a $T_{\rm c} \sim 77$ K as we have checked by our own measurements on overdoped Bi2212 samples and compared with the previous results.[8] The relatively low pseudogap temperature and the momentum cut closer to the antinodal point provide us a better chance to investigate the band evolution with temperature in the pseudogap state. Figure 4(f) shows the detailed band structure measured above $T^{*}$ (160 K and 120 K), in the pseudogap state (80 K) and in the superconducting state (25 K). The corresponding momentum distribution curves (MDCs) for the measurement in Fig. 4(a) near the Fermi level are shown in Fig. 4(g). It is clear from Figs. 4(f) and 4(g) that the Fermi momentum keeps the same across the pseudogap transition and the superconducting transition temperatures. No sign of band reorganization is detected across the pseudogap temperature $T^{*}$. In summary, from our super-high resolution laser ARPES measurements on Bi2212, we find that the superconducting gap does not follow a standard $d$-wave form and a high order term needs to be considered. We also find that the Fermi momentum of the bands near the antinodal region keeps the same when crossing the pseudogap transition and the superconducting transition. This is different from that observed in Bi2201 where band reorganization was observed across $T^{*}$ over a large momentum space. Our observations indicate that the band reorganization across $T^{*}$ is not universal in cuprate superconductors. The origin of the pseudogap in cuprate superconductors, whether it represents a phase transition or a crossover at $T^{*}$, needs to be further investigated. References Pairing symmetry in cuprate superconductorsThe pseudogap in high-temperature superconductors: an experimental surveyAngle-resolved photoemission studies of the cuprate superconductorsDoping a Mott insulator: Physics of high-temperature superconductivityFrom quantum matter to high-temperature superconductivity in copper oxidesAnomalously large gap anisotropy in the a - b plane of Bi 2 Sr 2 CaCu 2 O 8 + δ Angle-resolved photoemission spectroscopy study of the superconducting gap anisotropy in Bi 2 Sr 2 Ca Cu 2 O 8 + x Phase competition in trisected superconducting domeEnergy gaps in high-transition-temperature cuprate superconductorsUnconventional Electronic Structure Evolution with Hole Doping in Bi 2 Sr 2 CaCu 2 O 8 + δ : Angle-Resolved Photoemission ResultsExcitation Gap in the Normal State of Underdoped Bi2Sr2CaCu2O8+deltaFormation of asteroid satellites and doublet craters by planetary tidal forcesParticle–hole symmetry breaking in the pseudogap state of Bi2201From a Single-Band Metal to a High-Temperature Superconductor via Two Thermal Phase TransitionsDevelopment of a vacuum ultraviolet laser-based angle-resolved photoemission system with a superhigh energy resolution better than 1 meVImportance of Matrix Elements in the ARPES Spectra of BISCOBilayer Splitting in the Electronic Structure of Heavily Overdoped Bi 2 Sr 2 CaCu 2 O 8 + δ Photoemission study of Pb doped Bi 2 Sr 2 CaCu 2 O 8 : A Fermi surface picturePhenomenology of the low-energy spectral function in high- T c superconductorsSuperconducting Gap Anisotropy and Quasiparticle Interactions: A Doping Dependent Photoemission StudyDistinct Fermi-Momentum-Dependent Energy Gaps in Deeply Underdoped Bi2212
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