• cuprates
• photoelectron spectroscopy

General features of the phase diagram of the copper oxide superconductors have been known for some time. The superconducting transition temperature T_c has a dome-like shape in the doping-temperature plane with a maximum near a doping δ ∼ 0.167 electrons per Cu atom. Although in conventional metals the electronic excitations for T > T_c are (i) gapless and (ii) sharply defined at the Fermi surface (1), the cuprates violate at least one of these conditions over much of their phase diagram. These deviations from conventional metallic behavior are most easily described in terms of two energy scales T^∗ (2, 3) and T_coh (4), which correspond to criteria (i) and (ii), respectively.

To address the role of these energy scales in defining the phase diagram, we concentrate on spectra where the superconducting energy gap is largest, the antinode [(π,0) → (ππ) Fermi crossing], where the spectral changes with doping and temperature are most pronounced (the SI Appendix has further details). Spectral changes at the node have been previously studied by Valla et al. (5) and such spectra remain gapless for all doping values (6). In Fig. 1, we show spectra at fixed temperature as a function of doping. Data points are indicated in Fig. 1A (See SI Appendix for experimental conditions and sample details). Initially, we show spectra at fixed momenta as a function of energy (energy distribution curves, or EDCs) that have been symmetrized (7) about the Fermi energy to remove the effects of the Fermi function. Later, we show that equivalent results are obtained from division of the EDCs by a resolution-broadened Fermi function. In the following figures, because two values of the doping can result in the same T_c, samples are labeled as OP for optimally doped, OD for overdoped, and UD for underdoped.

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Fig. 1.

Spectra at constant temperature as a function of doping. (A) Dots indicate the temperature and doping values of the spectra of the same color plotted in D–F. (B) Schematic phase diagram for a quantum critical point near optimal doping. (C) Schematic phase diagram for a doped Mott insulator. (D) Spectra at T ∼ 300 K for several samples measured at the antinode, where the d-wave superconducting gap below T_c is largest. The spectra are normalized to high binding energy and symmetrized in energy to eliminate the Fermi function. The doping values are indicated by the top row of dots in A. (E) Same as in D, but at T ∼ 150 K, with the dopings indicated by the middle row of dots in A. (F) Same as in D, but at T = 100 K, with the dopings indicated by the bottom row of dots in A.

The spectra at the antinode at the highest temperature (approximately 300 K) in Fig. 1D show two remarkable features: They are extremely broad in energy, exceeding any expected thermal broadening, and their line shapes, well described by a Lorentzian, are independent of doping. The large spectral widths indicate electronic excitations that cannot be characterized by a well-defined energy, implying that the electrons are strongly interacting.

The incoherent behavior of the spectra at 300 K is consistent with the strange metal regime in two model phase diagrams popular in the literature, shown schematically in Fig. 1 B and C. If Fig. 1B applies, there would be strong evidence for a single quantum critical point near optimal doping which dominates the behavior to high temperatures (8, 9). T^∗ would be the transition temperature for a competing order, with T_coh its “mirror” corresponding to where Fermi liquid behavior sets in. The non-Fermi liquid behavior in the strange metal phase above both scales would then arise from fluctuations in the quantum critical region (10). These same fluctuations presumably mediate superconducting pairing. On the other hand, if Fig. 1C applies, the phase diagram would arise from strong correlation theories based on doped Mott insulators (11–14). The T^∗ line is where spin excitations become gapped, whereas T_coh is the temperature below which doped carriers become coherent. Superconductivity emerges below both scales, where spin and charge excitations become gapped and coherent. Which of these two phase diagrams is the appropriate one has critical implications for our understanding of the cuprates. To study this, we reduce T. At approximately 150 K, the spectra show marked changes with doping, and three regions can be identified in Fig. 1E. At low δ, the spectrum (red curve) remains broad as in Fig. 1D, but now a spectral gap is present—the pseudogap. This results in a reduction of the low-energy spectral weight as probed by various experiments (15). On increasing δ, the spectral gap becomes less pronounced and disappears just below optimal doping (purple and brown curves), where the spectra now resemble those in Fig. 1D. Increasing δ beyond 0.17, the spectra exhibit a sharp peak centered at zero energy (E_F) (blue and green curves). It can be seen in Fig. 1E that the sharper portion of the latter two spectra rises above the Lorentzian part of the spectrum delineated by the purple curve. These sharp spectra are now similar to what one would expect for a more conventional metal (1, 16). The doping dependences near 150 K are again consistent with either Fig. 1 B or C.

A completely different behavior emerges at a lower temperature, 100 K (Fig. 1F). The pseudogap with no sharp peaks is still present for low δ (red and purple curves). But near optimal doping, the spectra change, now exhibiting sharp peaks separated by an energy gap (brown, blue, and light-blue curves). These sharp peaks indicate that the lifetimes of excitations have increased dramatically, in contrast to the spectra at 150 K for optimal doping (brown curve in Fig. 1E). For still higher δ, a single sharp peak centered at E_F appears (green and black curves). Notice that all the spectral changes are limited to an energy scale of less than 200 meV; outside this energy range, the spectra follow the same broad Lorentzian shape as in Fig. 1D.

Fig. 1F demonstrates that the spectral gap and coherence (manifested by sharp spectral peaks) coexist in the normal state near optimal doping, implying that the T^∗ and T_coh lines cross each other, as in Fig. 1C. To further illustrate this crossing, we plot spectra at fixed doping as a function of temperature, with the various data points indicated in Fig. 2A. Fig. 2B shows spectra for two optimally doped samples. At T = 90 K (light-blue curve), the sample is just emerging from the superconducting state. Increasing T, the sharp peaks at the edge of the gap decrease in intensity, whereas the gap magnitude remains constant. Finally, for T≥115 K (pink curve), the sharp peaks disappear, whereas the spectral gap remains. This indicates that the T_coh line has been crossed, but not the T^∗ line. For T≥138 K (red curve), the spectral gap has completely filled in, and the spectra have regained the broad, temperature-independent line shape characteristic of the strange metal phase of Fig. 1D.

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Fig. 2.

Spectra at constant doping as a function of temperature. (A) Dots indicate the temperature and doping values of the spectra of the same color plotted in B–D. (B) Symmetrized antinodal spectra for two optimally doped samples (δ = 0.16). The temperature values are indicated by the left row of dots in A. (C) Same as in B, but for a doping of δ = 0.183, with the temperatures indicated by the middle row of dots in A. (D) Same as in B, but for a doping of δ = 0.224, with the temperatures indicated by the right row of dots in A. Gray lines in B and C mark the location of the gap.

In contrast, upon increasing the doping, the crossing of the pseudogap and coherence temperatures are reversed, as illustrated in Fig. 2C. Starting in the superconducting state at 66 K (blue curve), one can see the same features as in Fig. 2B, but now the spectral gap is smaller. Once T_c is crossed at 90 K, the spectral gap and sharp peaks persist (green and brown curves). But at higher T, the gap disappears and we are left with a relatively sharp peak at E_F (red curve), in contrast to Fig. 2B. For higher temperatures, the peaks broaden as in Fig. 1D. If the doping is now increased even further (Fig. 2D), a spectral gap is no longer observed at any T > T_c. In this highly overdoped region, the superconducting transition is similar to that of conventional superconductors (1), as the spectral gap closes very near T_c. The peak at E_F initially remains sharp, but at high enough temperatures, the strange metal returns (purple, red, and violet curves).

In Fig. 3, we show that dividing the EDCs by a resolution-broadened Fermi function (17) gives equivalent results to symmetrizing them. To quantitatively determine the T^∗ line, we note that it is easily identified by where the spectral gap disappears (18). For T_coh, we need to identify where the sharp peak disappears. We find that we can model the broad, incoherent part of the spectrum with a Lorentzian centered at E_F, and the sharp, coherent piece with a Gaussian (for details, see SI Appendix). In Fig. 3D, we plot the height of the sharp component of the spectra above that of the constant Lorentzian. One clearly sees a linear decrease with T, from which we determine T_coh. T_coh can also be observed in plots of the angle-resolved photoemission (ARPES) signal as a function of momentum for a fixed energy, the momentum distribution curves shown in Fig. 3 E–G. In Fig. 3E, we show that a significant change in width occurs upon crossing T_coh, which clearly indicates that this is not a simple temperature broadening effect. The spectra remain relatively unchanged both below and above T_coh, with significant changes limited to temperatures close to T_coh. Furthermore, T_coh is strongly doping dependent. In Fig. 3F, we show spectra at similar T for an optimally doped sample and an overdoped one with T_c = 65 K, showing that the spectral widths depend on the region of the phase diagram, and not simply the temperature. This is emphasized in Fig. 3G, where no spectral changes are observed in the strange metal region over a wide range in temperature.

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Fig. 3.

Fermi function divided spectra. (A) Antinodal spectra for two optimally doped samples, δ = 0.16, showing sharp peaks with an energy gap (green curves) below T^∗, but broad gapless spectra (purple curve) above T^∗. Colored lines show Fermi function-divided data, with symmetrized data superimposed as sharp black lines. (B) Spectra for an overdoped sample, δ = 0.183, showing that, unlike in A, the spectral gap is lost above 100 K, whereas the sharp peak persists to higher temperature. (C) Data for an overdoped δ = 0.224 sample. The sharp spectral peak decreases in intensity with increasing temperature. By T = 250 K, the spectral line shape is broad and temperature independent. (D) Linearly decreasing intensity of the sharp spectral peak relative to the broad Lorentzian with increasing T for three values of δ. T_coh is where this intensity reaches zero. (E) Momentum distribution curves (MDCs) for an overdoped δ = 0.224 sample, showing that a qualitative change in spectral shape occurs near T_coh. (F) Comparison of the MDC of an OP doped sample, to that of an OD sample at a similar T. (G) T-independence of the spectral shape for an OP sample above T_coh.

The phase diagram shown in Fig. 4 summarizes our results. The solid dots are based on the antinodal spectra and are color coded to correspond to the four different regions in the normal state phase diagram. These correspond to antinodal spectra that are: (i) incoherent and gapped (brown dots), in the underdoped pseudogap region, (ii) incoherent and gapless (red dots), in the high-temperature strange metal, (iii) coherent and gapless (green dots), in the overdoped metal, and finally (iv) coherent and gapped (blue dots), in the triangular region above optimal doping formed as a result of the crossing of T^∗ and T_coh. In addition, we also plot T^∗ and T_coh as defined above by double triangles. We emphasize that, below T_c, we find coherent and gapped antinodal spectra for all doping values, even for very underdoped samples (6).

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Fig. 4.

Electronic phase diagram of Bi₂Sr₂CaCu₂O_8+δ versus hole doping, δ. Brown dots indicate incoherent gapped spectra, blue points coherent gapped spectra, green dots coherent gapless spectra, and red dots incoherent gapless spectra. The brown double triangles denote T^∗, and the green double triangles T_coh. T_c denotes the transition temperature into the superconducting state (shaded gray and labeled D-wave SC).

An earlier ARPES experiment showed the appearance of dichroism below a temperature equivalent to the T^∗ measured here (19), as did subsequent neutron scattering experiments that detected intraunit cell magnetic order (20, 21), both of which identify T^∗ as a phase transition. However, the present experiments do not measure an order parameter. Moreover, it is not clear that the large energy gap is due to magnetism. We therefore limit ourselves to calling T^∗ a “temperature scale.”

Although heat capacity (22) and more recent transport studies (23) have suggested Fig. 1B, transport represents a single static (dc) quantity. On the other hand, photoemission being an energy and momentum resolved probe, allows one to uniquely separate the influence of coherence, a spectral gap, and their momentum dependence. In further support of Fig. 4, we note that the high-doping side of the blue triangle near optimal doping, characterized by gapped and coherent spectra above T_c, has also been inferred from the T dependence of scanning tunneling spectra (24). To our knowledge, however, the full triangular region has not been identified before. Although at first sight this triangular region seems similar to the region where diamagnetism is observed above T_c (25), the latter has a larger extent over the phase diagram than the former. This is not a surprise because we are measuring single particle coherence, whereas the diamagnetism is a measure of superconducting fluctuations.

Our experimental finding that the two temperature scales intersect is not consistent with a single quantum critical point near optimal doping, although more complicated quantum critical scenarios cannot be ruled out. For instance, quantum critical points exist at the ends of the dome (26, 27). In our data, Fig. 4, superconductivity only emerges below both T^∗ and T_coh. And, optimal superconductivity emerges from a coherent, gapped, normal state. Hence, our results are more naturally consistent with theories of superconductivity for doped Mott insulators, as illustrated in Fig. 1C. We believe these results represent an important step forward in solving the highly challenging problem of high-temperature superconductivity.