The simplest and most ideal systems to study are YBa₂Cu₃O_7−δ (Y123) [8] and YBa₂Cu₄O₈ (Y124) [12], the first because it can be doped over a wide range in very small increments and the second because it is perhaps the most defect-free structure amongst the cuprates. Figure 1 shows the temperature dependence of (a) the electronic specific heat coefficient, γ ≡ C _P /T, and (b) the electronic entropy, S ≡ ∫ 0 T γ d T , for both of these systems [12]. From here on, we omit the identifier “electronic” as all the thermodynamic functions quoted in the following are electronic and do not include any phonon component. For Y123, we show the data just for oxygen contents of 6.97 and 6.92, that is, nearly fully oxygenated Y123. These plots capture the essential pseudogap physics of the cuprates which will be seen to be canonical. Fully oxygenated Y123 happens to lie just at the doping point where the pseudogap closes, while Y124 is like under-doped Y123 in which the pseudogap is present. Three key features can be observed in these data:

(i) Fully oxygenated Y123 exhibits a large jump in γ at T _c while, in Y124, the jump is strongly suppressed. This shows that compared with Y123, the density of pairs in the Y124 condensate is strongly diminished by the removal of the antinodal states by the pseudogap.

(ii) It is possible to infer the normal-state behavior in the absence of superconductivity using the method of entropy balance, or equal-area construction. Thus, in Figure 1A, the red dashed curve is the normal-state specific-heat coefficient, γ _n(T), corresponding to the solid red curve. The pink-shaded area between the two curves must be the same above and below the crossing point. There is some latitude in the construction, but it is quite constrained as it must be continuous with the measured behavior above T _c. For the fully oxygenated sample, γ _n(T) is constant, reflecting a fairly flat DOS in the neighborhood of E _F. This reveals that there is no pseudogap present in fully oxygenated Y123, but a small gap has already opened at oxygen 6.92 (red dashed curve) as reflected by the low-temperature downturn. Now, when we turn to Y124, the same construction (blue shading) shows γ _n(T) (blue dashed curve) to be rather strongly suppressed at low-T, thus highlighting a fully developed pseudogap in this more under-doped sample. Even so, there remains a small but finite residual γ _n at T = 0 and this reflects the Fermi arcs or pockets present in the under-doped cuprates which have been revealed in ARPES measurements on Bi2212 and La214 [13]. Integration of γ _n(T) gives the normal-state entropy S _n(T) shown by the red and black dashed curves in Figure 1B. The integrated area between S _n(T) and S(T) gives the superconducting condensation energy, U ₀, which will be discussed later.

(iii) Perhaps, most notably, the data show that the γ(T) curves for Y123 and Y124 come together at high temperature, whereas the S(T) curve for Y124 is displaced downward in parallel fashion relative to that for Y123. This arises from the presence of the pseudogap and its persistence to very high temperature. To see this, the entropy for a weakly interacting Fermi liquid is considered [14]: S n = − 2 k B ∫ − ∞ ∞ f ⁡ ln f + 1 − f ln 1 − f N E d E , (1) where f(E) is the Fermi function and N(E) is the electronic DOS for one spin direction. This is just a weighted integral of the DOS with the “Fermi window” [f  ln(f) + (1 − f) ln (1 − f)]. This Fermi window for S(T) is shown as a function of E in the insert of Figure 1B for three different temperatures. It is single peaked at E _F and so a gap at E _F (illustrated by the triangular gap in the figure) is always seen by the entropy, even at high T so long as the gap remains. In contrast, for γ(T) ≡ ∂S/∂T, the Fermi window is necessarily double peaked about E _F, and at high enough temperature, it does not see the gap at E _F. As a consequence, γ(T) returns to its bare value, while S(T) is suppressed until, and only if, the gap closes. The fact that S(T) remains suppressed in Figure 1B shows that the gap persists to the highest temperature investigated (nearly 400 K). The magnitude of the gap, E*, can be read off from where the extrapolation of the high-T linear section of S(T) intercepts the y-axis. For the triangular gap shown, the intercept is 2 ln(2) k _B N ₀ E*V _M, where V _M is the molar volume. If the DOS is non-zero at E = 0 with value N ₁, then the intercept is 2 ln(2) k _B(N ₀ − N ₁)E*V _M. As noted, such a finite value would reflect the presence of a small Fermi surface (arcs or pockets) [13], and this is confirmed by the non-zero value of γ _n (T = 0) for both Y123 and Y124 seen in Figure 1.

Ca doping in YBa₂Cu₃O_7−δ allows this system to be significantly over-doped at full oxygenation, with over-doped T _c reduced to about 45 K. This means that much of the superconducting phase diagram can be explored simply by changing δ by annealing and quenching [11]. For a 6-g sample, the mass change from deoxygenation is 144 mg, so δ can be measured rather accurately and even more accurately from the change in the residual phonon term of the specific heat. This system also benefits from the fact that the van Hove singularity (vHs) lies far below E _F and is not traversed until beyond the superconducting domain [15]. This is important as the proximity of the vHs in other cuprates somewhat confuses the interpretation of the entropy and its parallel shift at high T, as we will see. Figure 2 shows (a) γ(T) and (b) S(T) for YBa₂Cu₃O_7−δ over a range of doping states spanning the under- and over-doped regions [11].

The same features noted previously for YBa₂Cu₃O_7−δ and YBa₂Cu₄O₈ are evident, but now, the systematic effect of doping reveals a progressive reduction in the jump Δγ/γ _c with under-doping and a progressive downward parallel shift in entropy at high T, indicating a growing pseudogap with under-doping. At the same time, the value of γ at high T remains essentially doping independent. At no point does the entropy recover to its bare value at high T, again showing that the pseudogap does not close with increasing T. γ(T) exhibits a linear slope at low T, consistent with d-wave pairing. In the under-doped region at low-T, these curves stack on top of each other, while in the over-doped region, they become increasingly steeper. This slope is proportional to 1/Δ₀, where Δ₀ is the amplitude of the d-wave gap as T → 0. Accordingly, it can be seen that the gap amplitude remains more or less constant in the under-doped region and falls in the over-doped region, just as found in systematic ARPES studies [16]. Values of Δ₀ (divided by 2.5) obtained in this way are summarized in Figure 3. The values of E* obtained from the y-axis intercept of the high-T linear entropy are also plotted. E* descends more or less linearly with doping, vanishing at p ≈ 0.19. Additionally, at this critical doping level, the values of Δγ/γ _c abruptly begin to fall as the pseudogap opens.

Examination of the transition steps in γ(T) at T _c shown in Figures 1, 2 reveals that these transitions are not completely abrupt but show the effect of fluctuations above and below T _c. These arise because of the low superfluid density found in the cuprates which results in strong fluctuations in both amplitude and phase, setting in simultaneously well above T _c [17]. As a consequence, T _c is pushed significantly below its mean-field value, T c mf . These perturbations to γ(T) near T _c can be analyzed to calculate T c mf , and the method is illustrated in Figure 4 [17].

Figure 4 shows the specific heat coefficient, γ(T), for Y_0.8Ca_0.2Ba₂Cu₃O_6.75. At this doping (p = 0.185), the pseudogap has just closed and the normal-state specific-heat coefficient γ _n(T) is essentially constant (dashed line). γ s mf is the mean-field γ in the superconducting state deduced by entropy balance, namely, the area abc equals the area cde. Additionally, by entropy balance, the gray shaded area under the fluctuation contribution must equal the hatched area which therefore defines both T c mf and Δγ ^mf. This illustrates the method for extracting the mean-field parameters. For other doping states, γ _n(T) is no longer flat but the same construction equally applies. The values of T c mf for different oxygen contents are plotted [17] in Figure 3 by the blue data points and error bars as annotated. In the under-doped region, T _c is depressed below T c mf by up to 30 K. For Bi2212, the depression is higher still—up to 45 K. This is quantitatively consistent with conclusions drawn from both ARPES [18] and scanning tunneling spectroscopy measurements [19]. In contrast, a fluctuation analysis of intrinsic tunneling data in over-doped Bi2212 single crystals suggests somewhat smaller depression of 7–13 K [20]. It is notable that T c mf tracks Δ₀/2.5 across the over-doped region, that is, nearly equal to the weak-coupling mean-field BCS value of Δ₀/2.14.

By integrating γ(T) − γ _n (T) in Figure 4A, we obtain ΔS = S _s − S _n and similarly for Δ S mf = ∫ 0 T ( γ s mf − γ n ) d T . Values of ΔS and ΔS ^mf are plotted in Figure 4C by the solid and dashed curves, respectively, where only every 4^th data point is shown from the experimental data. These may in turn be integrated to generate the condensation free energies ΔF(T) and ΔF ^mf(T) [21]. The former, ΔF(T), at T = 0 is the condensation internal energy, U ₀. Finally, we wish to construct the BCS ratio U 0 / γ T c 2 , but as BCS is a mean-field theory, the appropriate T _c to use is the mean-field value which we have calculated. For d-wave weak-coupling superconductivity, this ratio should be 0.17.

These integrals were evaluated for all the doping states and the false-color plot in Figure 3 is Δ F / γ T c mf 2 plotted as a function of p and T. As shown previously [21], this BCS ratio sits close to 0.17 across the entire over-doped regime until p falls below p* ≈ 0.19 when the pseudogap opens and the ratio falls abruptly as spectral weight is removed by the pseudogap.

Elsewhere [21], we have shown that starting from the BCS Hamiltonian one can deduce ζ N 0 Δ T 2 = 2 Δ F T + T Δ S T ≡ 2 Δ U T − T Δ S T , (2) where ζ = 1 for s-wave and 1/2 for d-wave and N (0) is obtained from the usual expression for the Sommerfeld “constant” γ n = 2 3 π 2 k B 2 N 0 1 + λ , (3) where λ is the usual electron–boson coupling parameter in the Eliashberg theory [22] and N ₀ is the bare band DOS, un-renormalized by electron-boson or Coulomb effects. Thus, Eq. 2 expresses Δ(T) directly in terms of thermodynamic functions ΔF, ΔS = S _n − S _s, and ΔU = U _n − U _s, which we have calculated. We do not know the magnitude of λ in Eq. 3, so we relate N (0) to γ _n using Eq. 3 with λ = 0 recognizing then that deduced values of Δ(T) should be larger by the factor 1 + λ . In this way, we have evaluated Δ(T) and Δ^mf(T), and these are plotted in Figure 4B.

Several points are worth noting: (i) first, Δ^mf(T) is found to follow almost precisely the BCS d-wave temperature dependence. This means that the cuprates are close to weak-coupling behavior as we have previously deduced [17], thus justifying the basic assumptions of our analysis. (ii) Second, with increasing temperature, Δ(T) starts to fall below Δ^mf(T) at the onset of SC fluctuations below T _c. At T _c, there is an inflection in Δ(T) which then remains finite and falls only slowly to zero above T _c, and indeed well above T c mf . As it does, it becomes less well defined due to the implicit square root in Eq. 2. So, the terminal point for a finite non-zero gap is not sharply defined. (iii) Third, at T _c, the coherent superconducting state vanishes and this finite residual “gap” reflects a fluctuation-induced loss, above T _c, of spectral weight in the DOS at E _F—as described in figure 10.2 in Larkin and Varlamov [23]. Here, it is only a partial gap but easily distinguishable from the partial gap that is the pseudogap. Similar results were also obtained for Bi2212, but the fluctuation domain was found to extend somewhat higher than in Y_0.8Ca_0.2Ba₂Cu₃O_7−δ [21].

Turning now to La_2−x Sr _x CuO₄, we illustrate the complicating role of a proximate vHs [24]. Figure 5 shows (a) γ(T) and (b) S(T) for this system over a rather broad range of doping values (annotated) [11], including some unpublished data. The most extremely over-doped data (x = 0.30, 0.35, and 0.45) are depicted by the dashed curves for ease of identification. As found for Y_0.8Ca_0.2Ba₂Cu₃O_7−δ , the γ(T) data come together at high temperature, although the convergence seems slower because of the broader doping range, and likewise the entropy curves, S(T), remain spread out at high T as a set of parallel straight lines which do not close at any temperature, consistent with a normal-state gap which never closes at elevated temperature. But at what doping level does the pseudogap finally close? The high-T linear portion is linear to the origin just above x = p = 0.17, so this would seem to be the general location of p _crit where the pseudogap closes. But, above this doping level, the curves continue to spread out in a parallel manner but now with negative curvature. This arises from the close proximity of the vHs, observed in ARPES to lie at x ≈ 0.21 [24]. The approach of the vHs is evident in γ(T) from the peaks (arrowed in Figure 5A) which migrate progressively toward T = 0 at the vHs crossing, which in these polycrystalline samples lies at 0.23. It is also evident in S(T) by the entropy rising to a maximum at x = 0.23 before falling again once the vHs moves into the unoccupied states above E _F. This peak should not be confused with the peak expected from normal-state pair dissociation above T _c within the so-called “pairon” model [4] of the pseudogap. Such a pair-dissociation peak would result in S(T) curves recovering their bare values at high temperature, which clearly they do not. It must be recalled that this peak is absent in Y_0.8Ca_0.2Ba₂Cu₃O_7−δ where the vHs is much further removed below E _F [15].

The vHs crossing point is easily identified in plots of γ(p, T) vs p at fixed T (such as 40 or 50 K). This passes through a more or less symmetric peak at x = 0.23 (as does the susceptibility [11]), the location of the vHs. Zhong et al. have calculated γ(p, 0) vs p from the ARPES-derived dispersion and the curves are essentially identical to the experimental data described previously, except theirs peak at x = 0.21. The small difference, 0.21 vs 0.23, for bulk samples is probably attributable to the use of epitaxial thin films in the former. This could, for example, be due to lattice strain or incomplete oxygenation in the latter.

Similar behavior is observed in Bi2212. Differential specific heat measurements were carried out on three samples, namely, pure, Pb-substituted to enable excess doping, and Y-substituted to enable increased under-doping. Where the sample dopings overlapped, the data were in remarkable agreement [11]. We focus on the pure Bi₂Sr₂CaCu₂O_8+δ data which were fitted [25] in the normal state using a rigid ARPES-derived anti-bonding band dispersion reported by Kaminski et al. [26] together with the standard non-interacting Fermion model, Eq. 1, for calculating the entropy. The fits are excellent [25], but, for the sake of clarity and because they are so instructive, we show here only the calculated normal-state curves. Figure 6 shows these normal-state fits for (a) γ(T) and (b) S(T) plotted for the different doping levels investigated ranging from p = 0.129 to 0.209 as annotated in the legend. For Bi2212, the anti-bonding band vHs crossing occurs at p ≈ 0.23 as was found in the ARPES study by Kaminski et al. [26]. The fits of course quantify the separation, ΔE _vHs, of the vHs below E _F. The legend includes ΔE _vHs values for each doping level.

The plots show, at high temperature, the same generic convergence of γ(T), combined with a parallel shift of S(T) that never closes. The entropy is linear to the origin at a doping of 0.18 or 0.19. This is where the pseudogap closes. Below this doping, the transition jump, Δγ _c, collapses rapidly confirming the opening of the pseudogap. The opening of the pseudogap is also seen in the onset of a suppression of γ(T) at low T. However, like La214, at higher doping, the entropy curves develop negative curvature and continue to shift to higher values in a parallel manner. Again, this is due to the approaching vHs which is seen more clearly in γ(T) by the local peaks which are arrowed in the figure and which push to increasingly lower temperature as the vHs is approached. The distance from the vHs is given in terms of the peak location, T _peak, by Δ_vHs ≈ 4.3k _B T _peak as shown in the inset to Figure 6A.

Figure 7 shows (a) the experimental values of γ(T) for the same range of doping states as shown in Figure 6B, and (b) the difference, Δγ(13, T) = γ(13, T), − γ(0, T) in γ(T) at 13 T and that at 0 T. The abrupt effect of the opening of the pseudogap can be seen in both datasets by the collapse of the jump in γ at T _c starting at p = p* = 0.19. In the over-doped region above this doping level, the jump remains constant in height for both. This clearly shows that the sudden fall in Δγ _c at p* is not an artifact of the subtraction of the phonon contribution but is fundamental to the electronic behavior.

Δγ(H,T) may be integrated to obtain the field-dependent entropy and the free energy, ΔF(H, T), by integration again of the entropy. We find it preferable to perform just a single integration and obtain the free energy using the following generic thermodynamic relation: − Δ F H , T = ∫ τ T T Δ γ H , T d T − T ∫ τ T Δ γ H , T d T , (7) where the first term is the electronic internal energy, − ΔU(H, T), and the second term is the electronic entropy term, TΔS(H, T). Ideally, the integrals in Eq. 7 are from τ = T _c down to some value of T < T _c. However, as noted, the cuprates are distinguished by strong fluctuations over quite a broad range around T _c, and as a consequence, the integrals must be performed from some temperature, τ, well above T _c, sufficiently above the range of superconducting fluctuations, where Δγ(H, τ), ΔF(H, τ), and ΔS(H, τ) have fallen to zero.

In Figure 7B, we linearly extrapolate Δγ(13, T) to zero at T = 0 from 20 K, integrate as in Eq. 7 to obtain the field-dependent state functions, and these are plotted in Figure 8.

It is interesting that the effect of superconducting fluctuations in ΔU and TΔS extend to rather high temperatures, while these largely cancel in ΔF and thus extend over a much smaller range above T _c.

We use these data to calculate the superfluid density ρ s ≡ λ a b − 2 at T = 0 for each doping level, and these values are summarized in Figure 9 along with the wider set of results obtained previously for Bi₂Sr₂CaCu₂O_8+δ . In Figure 9A, we show values for the jump in γ at T _c for three different compositions of Bi2212 as annotated: Bi_2.1Sr_1.9CaCu₂O_8+δ , Bi_2.1Sr_1.9Ca_0.85Y_0.15Cu₂O_8+δ to extend to lower doping, and Bi_1.9Pb_0.2Sr_1.9CaCu₂O_8+δ to extend to higher doping. The green stars are Δγ ^mf obtained from the entropy-balance analysis, while the orange data points are the magnetic jump Δγ [13]. All reveal an essentially constant value above p*, and all collapse abruptly as p falls below p* due to the opening of the pseudogap. The value of Δγ ^mf/γ _n at T _c is 1.15, not much above the weak-coupling mean-field d-wave value of 0.95 and consistent with (using a strong-coupling Padamsee calculation [14] for d-wave superconductivity) a gap ratio of Δ 0 / k B T c mf = 2.34 , just a little higher than the BCS weak-coupling ratio of 2.14.

Next, in Figure 9B, we show the calculated ground-state values of λ 0 − 2 along with ΔF [13] and the condensation free energy ΔF _ns (0) at T = 0. The latter is, of course, identical to the condensation internal energy, U ₀. Again, all parameters remain rather constant on the over-doped side and then fall abruptly starting at p*. This is crucially important because λ 0 − 2 is a truly ground-state property. One prominent hypothesis for the pseudogap is that it arises from incoherent pairing above T _c [4, 28]. But at T = 0, all thermal fluctuations are quenched so such a model would mean the pseudogap is absent at T = 0. But the superfluid density shows that the pseudogap is always present in the ground state for p < p*.

Finally, in Figure 9C, we show the upper critical field B _c2 obtained from our field-dependent analysis using Eq. 6, as well as the thermodynamic critical field B _c obtained from the values of ΔF _ns shown in Figure 9B. Like all other thermodynamic parameters, both collapse below p* as the pseudogap opens. The maximum value of B _c2 = 105 T is reasonable and consistent with other independent measures [29].

Though the topic here is pseudogap thermodynamics, applicable only to under- and optimally doped cuprates, it is worth briefly refocusing on the superfluid density in the over-doped region. It has long been realized that over-doped physics is probably just as puzzling and unconventional as under-doped physics, and this is no better highlighted than in the unusual behavior of λ(0)⁻² which has widely been reported to fall along with T _c as doping increases toward the end of the superconducting dome [30–32]. The fall-off in both T _c and λ(0)⁻² on both sides of the dome has come to be known as the “boomerang effect” and λ(0)⁻² was shown to reach a sharp maximum at p* [32]. Renewed interest in this long-established result came more recently with the highly detailed studies of Božović et al. [33] on high-quality thin films of La_2−x Sr _x CuO₄ which showed a linear decline in λ(0)⁻² with T _c across the entire over-doped region. However, the just-discussed field-dependent specific heat shows that for Bi2212, λ(0)⁻² remains largely constant in the over-doped region where p > p* (see Figure 9B), along with all the other properties plotted in Figure 9. Already, T _c has fallen from 92 K to 74 K with no significant fall-off in superfluid density. This is reassuring because, as shown previously, both Δγ ^mf/γ _n and U 0 / γ n T c mf  2 remain conventional (and close to their BCS ratios) in this over-doped region. It would surely be highly anomalous for λ 0 − 2 ( p ) to be exceptional in its collapse. Interestingly, Dordovic and Homes [34] recently suggested that the boomerang effect may not be present in bulk samples. However, we note that the early muon spin relaxation studies on λ 0 − 2 were in fact carried out on bulk samples. It is fair to say this remains an open question but it seems at least that in Bi2212, there is no observed suppression of superfluid density over nearly half of the over-doped region. Additionally, ac-susceptibility measurements on La_2−x Sr _x CuO₄ also showed no fall-off in λ 0 − 2 [35]. This remains a key point of investigation. It may well be that cuprates display a conventional over-doped behavior in all properties: λ 0 − 2 , Δγ, Δ 0 / k B T c mf , and U ₀. If confirmed, this would reflect a great simplification of cuprate-phase behavior.

Loram [38] has examined a simple BCS-like model with a temperature-independent competing gap, E*, using an approach similar to that of Bilbro and MacMillan [39] where the superconducting gap and pseudogap are combined quadratically. Thus, the order parameter Δ′ is given by Δ ′ = Δ 2 ( T ) − E * 2 , where Δ(T) is the spectral superconducting gap which must be distinguished in such a model from the order parameter. E*(p) is taken to be a linearly descending function of p. This model reproduces all of the elements discussed previously—the parallel suppression of entropy at a high temperature, combined with suppression of the jump, Δγ at T _c. This was done for s-wave gaps, and so superconductivity is suppressed when Δ(T) = E* and this then defines the reduction of T _c with E*. For d-wave superconductivity, because the pseudogap forms around the antinodes leaving ungapped Fermi arcs or pockets, it is possible for superconductivity to persist even for the values of E* well exceeding Δ₀. Loram’s s-wave calculations have been extended to d-wave by Williams et al. [40].

As already noted, Storey et al. [25] have used the rigid tight-binding model of Kaminski et al. [26] derived from ARPES measurements on Bi2212 to calculate S(T) and γ(T) in both the normal and superconducting states using a self-consistent solution of the gap equation in the presence of a pseudogap with Fermi arcs. The fitting parameters were E*, the height, Δ_vHs, of the Fermi level above the vHs and the length of the Fermi arc. The fits were extremely faithful in the superconducting and normal states, including the progression of the small local peak in γ(T) arising from the nearby vHs. They reproduce the parallel shift in entropy as the pseudogap widens. Of course, the pseudogap model was necessarily somewhat arbitrary but these calculations were extended [41] to the very specific Yang–Rice–Zhang (YRZ) model of Fermi surface reconstruction [42] and shown to accurately reproduce key observed thermodynamic features. Two particular features of the YRZ model include (i) a very asymmetric gap about E _F which nicely accounts for the rapidly growing thermopower, Σ(p, T), in the under-doped region [3, 43], and (ii) the appearance of AN electron pockets in a small doping range close to the reconstruction point. These were shown to account for a double undulation of Σ(p,T) at low temperature when p ≈ 0.175 [3].

Many authors have interpreted the distinctive phase behavior of the cuprates in terms of a crossover from a Bose–Einstein condensate (BEC) to a BCS condensate [28, 44, 45, 46, 47, 48]. In this way, the pseudogap state is viewed as a (partially) paired state which condenses to a coherent paired state at T _c [4]. This, of course, explains amongst other things the reduced specific heat jump in the under-doped region because it lacks the pairing contribution which has already reduced the entropy from well above T _c [48]. Many detailed theoretical studies have pursued this line of thought. As noted, our principal objection lies in the expectation that such models, in the ground state, lose all memory of their precursor pairing state, where at T = 0, all thermal fluctuation is quenched and all pairs are condensed. In contrast, the experimental observation is that the pseudogap is still present at T = 0. An equally compelling objection is found at high temperature where, depending on the pair-binding energy, pairs should dissociate thus recovering the “lost” entropy [4]. Such a recovery is not found in the experimental data. In the context of BEC to BCS crossover, it was recently shown that the phase diagram may exhibit multiple pseudogaps [49]. This is of course quite possible, but we note that the previous thermodynamic studies show no evidence of multiple pseudogaps. One would expect to see closure of the smaller pseudogap at some threshold temperature with associated partial entropy recovery, prior to closure of the larger pseudogap at higher temperature, thus completing the entropy recovery. Multiple pseudogaps, if present, should also be seen in the c-axis infrared conductivity, located at distinct frequencies. Only two gaps are observed [50, 51]: the superconducting gap that opens at T _c resulting in a downshift in spectral weight, and the pseudogap, already present at 300 K, resulting in a canonical upward shift in spectral weight.

In view of the continuing divergence of opinion regarding phenomenology, phase extent, and impact of the pseudogap, it is important to consider the thermodynamic data alongside other spectroscopies. Figure 10A shows our values of Δ₀(p) and E*(p) for Y_0.8Ca_0.2Ba₂Cu₃O_7−δ derived from specific heat measurements and compares these with those derived from (b) ARPES, (c) infrared optics, and (d) intrinsic tunneling. We now consider each of these spectroscopies in turn.

The effect of impurity scattering on superconductivity in the cuprates holds an important place in HTS physics because of the d-wave order parameter. Scattering mixes the phases of Cooper pairs and therefore mixes positive and negative quadrants of the order parameter. The result is a rapidly suppressed T _c and an even more rapidly suppressed superfluid density [35]. In the unitary scattering limit, the suppression of T _c and λ ⁻² is a purely thermodynamic process and the critical concentration of Zn, y _crit, for just suppressing superconductivity is a thermodynamic ground-state parameter. To justify this, we have previously shown the numerical agreement between S(T _c)/R and x _crit for Y_0.8Ca_0.2Ba₂Cu₃O_7−δ , Bi₂Sr₂CaCu₂O_8+δ , and La_2−x Sr _x CuO₄ [35] across the superconducting phase diagram. As we have shown, this implies that superconductivity is destroyed when the density of unitary scatterers equals the pair density, that is, each scatterer breaks one pair [35].

Figure 11A shows this rapid suppression of the T _c(p) phase curve with Zn substitution in Y_0.8Ca_0.2Ba₂Cu₃O_7−δ [59]. Significantly, the rate of suppression is much higher in the under-doped region, thus resulting in an asymmetric collapse of the phase curve around p* ≈ 0.19. The last vestige of superconductivity at critical doping occurs at p*, clearly a special point in the phase diagram of the cuprates. This enhanced suppression in the under-doped region is due to the presence of the pseudogap and the associated reduced DOS. We follow our treatment given earlier [59]. The T _c reduction for elastic scattering in weak-coupling d-wave superconductors follows the Abrikosov–Gorkov (A-G) equation [60]. Thus − ln T c / T c0 = Ψ 1 / 2 + Γ / 2 π k B T c − Ψ 1 / 2 , (8) where Ψ[x] is the digamma function; T _c0 = T _c (y = 0); and for unitary scattering, Γ = n _i/(πN ₀) is the pair-breaking scattering rate. Here, N ₀ is the DOS per spin at the Fermi level and n _i = αy _ab /abc is the density of impurity scatterers with y _ab being the “planar” concentration of Zn atoms in the CuO₂ plane; α being the number of CuO₂ planes per unit cell; and a, b, and c being the lattice parameters. For La214, α = 1 and y _ab = y, while for Y123, α = 2 and y _ab = 3y/2. For weak coupling, the critical scattering rate for suppression of superconductivity is Γ_c = 0.412Δ₀₀ = αy _crit/(abcπN ₀). Finally, the DOS is related to γ _n by the standard relation γ n = 2 / 3 π 2 k B 2 N 0 (9) This is, of course, true only if N(E) is not strongly energy dependent. However, the essentially constant γ(T) at high temperature, largely independent of doping, gives some credibility to this assumption.

The problem is then fully defined with no fitting parameters. The A-G equation can be linearized for small Γ to get the initial linear slope as follows T c / T c0 = 1 − 0.69 Γ / Γ c = 1 − 29 α / γ n Δ 00 y a b , (10) where Δ₀₀ is the T = 0 amplitude of the d-wave gap when y _ab = 0 and where γ _n is expressed in J mol⁻¹ K⁻². As the scattering rate is inversely proportional to N ₀, the role of the pseudogap is clear—as the AN gap opens, the DOS falls and the scattering rate increases. Observationally, γ _n is essentially constant, independent of doping, when p > 0.19 on the over-doped side, so the slope of T _c vs y _ab remains fixed on the over-doped side, as shown in Figure 11B. In contrast, it rises steeply on the under-doped side due to the presence of the pseudogap. Already at optimal doping, the curve has steepened showing that the pseudogap is present even there. The A-G equation is plotted by the curves shown in Figure 11B, where we have used for γ _n the quantity (S/T) _Tc which is the average value of γ _n between T = 0 and T _c. All curves for all dopings show a good fit to the data with no free fitting parameter. Similar results were found for Bi₂Sr₂CaCu₂O_8+δ [61] and La_2−x Sr _x CuO₄ [59].