Universal features in the photoemission spectroscopy of high-temperature superconductors

The energy gap for electronic excitations is one of the most important characteristics of the superconducting state, as it directly reflects the pairing of electrons. In the copper–oxide high-temperature superconductors (HTSCs), a strongly anisotropic energy gap, which vanishes along high-symmetry directions, is a clear manifestation of the d-wave symmetry of the pairing. There is, however, a dramatic change in the form of the gap anisotropy with reduced carrier concentration (underdoping). Although the vanishing of the gap along the diagonal to the square Cu–O bond directions is robust, the doping dependence of the large gap along the Cu–O directions suggests that its origin might be different from pairing. It is thus tempting to associate the large gap with a second-order parameter distinct from superconductivity. We use angle-resolved photoemission spectroscopy to show that the two-gap behavior and the destruction of well-defined electronic excitations are not universal features of HTSCs, and depend sensitively on how the underdoped materials are prepared. Depending on cation substitution, underdoped samples either show two-gap behavior or not. In contrast, many other characteristics of HTSCs, such as the dome-like dependence of on doping, long-lived excitations along the diagonals to the Cu–O bonds, and an energy gap at the Brillouin zone boundary that decreases monotonically with doping while persisting above (the pseudogap), are present in all samples, irrespective of whether they exhibit two-gap behavior or not. Our results imply that universal aspects of high- superconductivity are relatively insensitive to differences in the electronic states along the Cu–O bond directions.Elucidating the mechanism of high-temperature superconductivity in the copper–oxide materials remains one of the most challenging open problems in physics. It has attracted the attention of scientists working in fields as diverse as materials science, condensed matter physics, cold atoms, and string theory. To clearly define the problem of high-temperature superconductors (HTSCs), it is essential to establish which of the plethora of observed features are universal, namely, qualitatively unaffected by material-specific details.An important early result concerns the universality of the symmetry of the order parameter for superconductivity. The order parameter was found to change sign under a 90° rotation (1, 2), which implies that the energy gap must vanish along the diagonal to the Cu–O bonds, i.e., the Brillouin zone diagonal. This sign change is consistent with early spectroscopic studies of near-optimally-doped samples (those with the highest in a given family), where a energy gap (3, 4) was observed (ϕ being the angle from the Cu–O bond direction), the simplest functional form consistent with d-wave pairing. More recently, there is considerable evidence (5–8) that, with underdoping, the anisotropy of the energy gap deviates markedly from the simple form. Although the gap node at is observed at all dopings, the gap near the antinode (near and 90°) is significantly larger than that expected from the simplest d-wave form. Further, the large gap continues to persist in underdoped (UD) materials as the normal-state pseudogap (9–11) above . This suggests that the small (near-nodal) and large (antinodal) gaps are of completely different origin, the former related to superconductivity and the latter to some other competing order parameter.This two-gap picture has attracted much attention (8), raising the possibility that multiple energy scales are involved in the HTSC problem. There is mounting evidence for additional broken symmetries (12–14) in UD cuprates, once superconductivity is weakened upon approaching the Mott insulating state. The central issue is the role of these additional order parameters in impacting the universal properties of high- superconductivity.In this paper we use angle-resolved photoemission (ARPES) to examine the universality of the two-gap scenario in HTSCs by addressing the following questions. To what extent are the observed deviations from a simple d-wave energy gap independent of material details? How does the observed gap anisotropy correlate, as a function of doping, with other spectroscopic features such as the size of the antinodal gap, and the spectral weights of the nodal and antinodal quasiparticle excitations?We systematically examine the electronic spectra of various families of cation-substituted Bi₂Sr₂CaCu₂O_8+δ single crystals as a function of carrier concentration to elucidate which properties are universal and which are not. We present ARPES data on four families of float-zone-grown Bi₂Sr₂CaCu₂O_8+δ single crystals, where was adjusted by both oxygen content and cation doping. As-grown samples, labeled Bi2212, have an optimal of 91 K. These crystals were UD to by varying the oxygen content. Ca-rich crystals (grown from material with a starting composition Bi_2.1Sr_1.4Ca_1.5Cu₂O_8+δ) with an optimal of 82 K are labeled Ca. Two Dy-doped families grown with starting compositions Bi_2.1Sr_1.9Ca_{1 − x}Dy_xCu₂O_8+δ with x = 0.1 and 0.3 are labeled Dy1 and Dy2, respectively. A full list of the samples used and their determined from magnetization measurements are shown in SI Text, where we also show high-resolution X-ray data that give evidence for the excellent structural quality of our samples.Our main result is that the Dy1 and Dy2 samples show clear evidence of a two-gap behavior in the UD regime , with loss of coherent quasiparticles in the antinodal region of k space where the gap deviates from a simple d-wave form. In marked contrast, the UD Bi2212 samples and the Ca samples show a simple d-wave gap in the superconducting state and sharp quasiparticles over the entire Fermi surface in a similar range of the UD regime. We conclude by discussing the implications of the nonuniversality of the two-gap behavior for the phenomenon of high superconductivity.We begin our comparison of the various families of samples by focusing in Fig. 1 on the superconducting state antinodal spectra as a function of underdoping. The antinode is the Fermi momentum k_F on the Brillouin zone boundary, where the energy gap is a maximum and, as we shall see, the differences between the various samples are the most striking. We show data at optimal doping, corresponding to the highest in each family, in Fig. 1A. Increasing Dy leads to a small suppression of the optimal compared with Bi2212, together with an increase in the antinodal gap and a significant reduction of the quasiparticle weight. This trend continues down to moderate underdoping, as seen in Fig. 1B, where we show UD Bi2212 and Dy2 samples with very similar . For more severely UD samples, with , spectral changes in the Dy-substituted samples are far more dramatic. In Fig. 1C, we see that quasiparticle peaks in the Dy samples are no longer visible, even well below , consistent with earlier work on Y-doped Bi2212 and also Bi2201 and La_1.85Sr_0.15CuO₄ (5, 15–18). In contrast, Bi2212 and Ca-doped samples with comparable continue to exhibit quasiparticle peaks. In this respect the latter two are similar to epitaxially grown thin-film samples that exhibit quasiparticle peaks all of the way down to the lowest (19).Open in a separate windowFig. 1.Superconducting state antinodal ARPES spectra. We use the label “Bi2212” for samples without cation doping, “Dy1” for 10% Dy, “Dy2” for 30% Dy, and “Ca” for Ca-doped samples. The temperature is indicated along with . OP denotes optimal doped, UD underdoped, and OD overdoped samples. (A) Antinodal spectra for OP samples of three different families: Bi2212 (blue), Dy1 (green), and Dy2 (red), showing an increase in gap and a decrease in quasiparticle weight with increasing Dy content. (B) Antinodal spectra for UD samples with similar (≃66 K) for Bi2212 (blue) and Dy2 (red). As in A, there is a larger gap and smaller coherent weight in the Dy-substituted sample. (C) Same as in B, but for four UD samples with near 55 K for Bi2212 (dark blue), Ca (light blue), Dy1 (green), and Dy2 (red). The Bi2212 and Ca spectra are very similar to each other and quite different from those of the Dy1 and Dy2 materials. (D) Doping evolution of the antinodal spectra of four Dy1 samples from OP to UD . (E) Doping evolution of the antinodal spectra of four Dy2 samples from OP to UD . We see in D and E the sudden loss of quasiparticle weight for below 60 K. (F) Doping evolution of the antinodal spectra of three Bi2212 samples and three Ca samples, showing well-defined quasiparticle peaks in all cases.A significant feature of the highly UD Dy samples in Fig. 1C is that, in addition to the strong suppression of the quasiparticle peak, there is severe loss of low-energy spectral weight. To clearly highlight this, we show the doping evolution of antinodal spectra for the Dy1 (Fig. 1D) and Dy2 (Fig. 1E) samples. These observations are in striking contrast with the Bi2212 and Ca-doped data in Fig. 1F, where we do see a systematic reduction of the quasiparticle peak with underdoping, but not a complete wipeout of the low-energy spectral weight. To the extent that the superconducting state peak–dip–hump line shape (20, 21) originates from one broad normal-state spectral peak, the changes in spectra of the Dy materials are not simply due to a loss of coherence, but more likely a loss of the entire spectral weight near the chemical potential.The doping evolution of the k-dependent gap is illustrated in Figs. 2 and​and 3. 3. In Fig. 2 we contrast the optimally doped Dy1 (T_c = 86 K) sample (Fig. 2 A and B) with a severely UD Dy1 (T_c = 38 K) sample (Fig. 2 C and D ), the spectra being particle–hole-symmetrized to better illustrate the gap. The OP 86 K sample shows a well-defined quasiparticle peak over the entire Fermi surface (Fig. 2A) with a simple d-wave gap of the form (blue curve in Fig. 2B). For the UD 38 K sample, we see in Fig. 2C well-defined quasiparticles near the node (red spectra), but not near the antinode (blue spectra). The near-nodal gaps (red triangles in Fig. 2D) are obtained from the energy of quasiparticle peaks and continue to follow a d-wave gap (blue curve in Fig. 2D). However, once the quasiparticle peak is lost closer to the antinode, one has to use some other definition of the gap scale. We identify a break in the slope of the spectrum, by locating the energy scale at which it deviates from the black straight lines (Fig. 2C), which leads to the gap estimates (blue squares) in Fig. 2D.Open in a separate windowFig. 2.Superconducting state spectra and energy gap for OD and highly UD Dy1 samples. (A) Symmetrized spectra at k_F, from the antinode (Upper) to the node (Lower) for an OP 86 K Dy1 sample. (B) Gap as a function of Fermi surface angle (0° is the antinode and 45° the node). The blue curve is a d-wave fit to the data. (C) Same as A for an UD 38 K Dy1 sample. Curves, near the node, with discernible quasiparticle peaks are shown in red; those near the antinode are shown in blue. (D) Gap along the Fermi surface from data of C.Open in a separate windowFig. 3.Energy gap anisotropies of various samples. (A) OD 79 K Ca (where ); (B) UD 54 K Ca; (C) OP 81 K Dy2; and (D) UD 59 K Dy2. The two near-optimal samples in A and C both show a simple d-wave gap. This behavior persists in the UD Ca sample of B, but the UD Dy2 sample of D has a two-gap behavior despite having a similar to the UD Ca sample.Despite the larger error bar associated with gap scale extraction in the absence of quasiparticles, it is nevertheless clear (Fig. 2D) that the UD 38 K Dy1 sample has an energy gap that deviates markedly from the simple d-wave form. This observation is called two-gap in the UD regime, in contrast with a single gap near optimality (Fig. 2B). It is easy to observe from Fig. 2 that the Fermi surface angle at which the energy gap starts to deviate from the form matches the one at which the spectral peak gets washed out. This is very similar to the two-gap behavior demonstrated in refs. 5, 15–18. From this, one might conclude that two-gap behavior is directly correlated with a loss of well-defined quasiparticle excitations in the antinodal region. However, we point to recent ARPES data on Y-doped Bi2212 (6, 7), where two-gap behavior has been observed despite the presence of small antinodal quasiparticle peaks.We next show that the two-gap behavior is not a universal feature of all UD samples. To make this point, we compare in Fig. 3 the gap anisotropies of the Ca-doped samples (Fig. 3 A and B) with the Dy2 samples (Fig. 3 C and D) with essentially identical , where both families have the same optimal . The near-optimal samples, OD 79 K Ca (Fig. 3A) and OP 81 K Dy2 (Fig. 3C) samples, both have a simple d-wave anisotropy (although different maximum gap values at the antinode). However, upon underdoping to similar values, the two have markedly different gap anisotropies. The UD 59 K Dy2 sample (Fig. 3D) shows two-gap behavior, and an absence of quasiparticles near the antinode (similar to the discussion in connection with Fig. 2 above). However, the UD 54 K Ca sample (Fig. 3B) continues to exhibit sharp spectral peaks and a single-gap, despite a very similar as the UD 59 K Dy2.Having established the qualitative differences in the gap anisotropies for various samples as a function of underdoping, we next summarize in Fig. 4 the doping evolution of various spectroscopic features. Instead of estimating the carrier concentration in our samples using an empirical equation (22) (that may or may not be valid for various cation substitutions), we prefer to use the measured to label the doping. In Fig. 4A we show the doping evolution of the antinodal energy gap, which is consistent with the known increase in the gap with underdoping.Open in a separate windowFig. 4.Antinodal gaps and quasiparticle weights. (A) Antinodal energy gap as a function of doping for various samples is seen to grow monotonically with underdoping. Here, and in B and C, the doping is characterized by the measured quantity , with UD samples shown to the left of and OD samples to the right. All results are at temperatures well below . (B) Coherent spectral weight for antinodal quasiparticles as a function of doping. Dy-doped samples exhibit a rapid suppression of this weight to zero for UD , whereas the Ca-doped samples show robust antinodal peaks even for . (C) Coherent spectral weight for nodal quasiparticles as a function of doping, which is seen to be much more robust than the antinodal one.The coherent spectral weight Z for antinodal quasiparticles is plotted in Fig. 4B (for details on the procedure used to estimate this weight, from a ratio of spectral areas, see SI Text). The Dy1 and Dy2 samples both show a sudden and complete loss of Z with underdoping (23), which coincides with the appearance of two-gap behavior. In marked contrast with the Dy samples, the Bi2212 and Ca samples that exhibit a single d-wave gap show a gradual drop in the antinodal Z. On the other hand, we find that the nodal excitations are much less sensitive to how the sample is UD compared with the antinodal ones. Similar sharp nodal excitations have been observed in Dy-doped Bi2212 samples in ref. 7 as well. The nodal quasiparticle weight Z in Fig. 4C decreases smoothly with underdoping for all families of samples, as expected for a doped Mott insulator (24).The two-gap behavior and the attendant loss of quasiparticle weight near the antinode imply a nodal–antinodal dichotomy, aspects of which have been recognized in k space (25–27) and in real space (28–30). Two possible, not mutually exclusive, causes of this behavior are disorder and competing orders.It is known that antinodal states are much more susceptible to impurity scattering, whereas near-nodal excitations are protected (31). However, it is not a priori clear why certain cation substitutions (Dy) should lead to more electronic disorder than others (Ca). As shown by our X-ray studies in SI Text, there is no difference in the structural disorder in Dy and Ca samples. One possibility is that Dy has a local moment, but there is no direct experimental evidence for this.The two-gap behavior in UD materials, with a large antinodal gap that persists above , is suggestive of an order parameter, distinct from d-wave superconductivity, which sets in at the pseudogap temperature . There are several experiments (12–14) that find evidence for a broken symmetry at . However, it is not understood how the observed small, and often subtle, order parameter(s) could lead to large antinodal gaps of , with a loss of spectral weight over a much larger energy range (Fig. 1 D and E).We now discuss the pertinence of competing order parameters based on our measurements. First, in our ARPES data, we have not found any direct evidence for density wave ordering (say, from zone folding). Second, our X-ray data did not provide any signature for additional diffraction peaks expected for long-range density wave ordering. However, none of these null results provide definitive evidence for the absence of a density wave ordering, particularly if it were short range. In contrast, in previously published work (5, 15–18), two-gap behavior has been conjectured to be a direct consequence of phase competition between d-wave superconductivity and some type of density wave ordering. As we have demonstrated, two-gap behavior in and of itself is a sample-specific issue and hence, even if we assume a linkage between competing order and two-gap behavior, it cannot be central to the question of superconductivity in HTSC systems.Whatever the mechanism leading to qualitatively different gap anisotropies for the UD Dy and Ca samples, it only produces relatively small, quantitative changes in key aspects of these materials, such as the dependence of on doping, the presence of sharp nodal quasiparticles, and the pseudogap. We thus conclude that antinodal states do not make a substantial contribution to the universal features of HTSCs. Clearly, two gaps are not necessary for high-temperature superconductivity.