Field-dependent specific heat of the canonical underdoped cuprate superconductor YBa₂Cu₄O₈

The measured electronic specific heat coefficient, $\gamma(T)\equiv C_{P}(T)/T$ is shown in Fig. 1(a) for YBa₂Cu₄O₈ (Y124) with 0%, 2% and 4% planar Zn concentration at nine different applied fields, as annotated. As discussed later, the data extends to an unprecedented 400 K but we focus first on the transitions into the superconducting state. Because Zn substitutes only on the CuO₂ planes and not on the chains Williams1 these compositions correspond to YBa₂Cu_4-yZn_yO₈ with $y=$ 0, 0.04 and 0.08. It is immediately evident that $T_{c}$ is rapidly suppressed by Zn substitution (as observed previously in other cuprates Tallon_scattering ) and along with this a rapid reduction in the jump height at $T_{c}$, $\Delta\gamma_{\textrm{c}}$. At the same time there is a rapid suppression in $\Delta\gamma_{\textrm{c}}$ with applied field that becomes more extreme in the Zn-substituted samples. Further, at low $T$, $\gamma(T)$ fans out to higher values with applied field in the ‘pure’ sample but not appreciably in the doped samples. This is due to the Volovik effect Volovik – the field-induced pairbreaking at the nodes due to Doppler shift of quasiparticle energies, as discussed below.

The most notable feature, however, is the low-$T$ upturn due to impurity scattering. The inset to Fig. 1(a) shows $\gamma(T)$ plotted versus $\ln(T)$ and this reveals a common underlying energy scale given by the convergence of the dashed lines at 38 K. The dashed lines are subtracted from the raw data to give the $\gamma(T)$ versus $T$ plot in panel (b) and it is this that we proceed to analyze. (There is a small anomaly at 18 K, present in the Zn-doped samples but very weak in the pure; and another at 120 K, present only in the pure sample. The sample variability indicates unidentified impurities and these anomalies are ignored in the following).

Our first task is to identify the normal-state coefficient, $\gamma_{\textrm{n}}$, that would occur in the absence of superconductivity. This is very much constrained by the displayed data for $\gamma(T)$ because $\gamma_{\textrm{n}}$ must follow each of the three data sets above their respective $T_{\textrm{c}}$ values. This is the black dashed curve. It is further tightly constrained by the requirement for entropy balance. Because the area under a $\gamma(T)$ curve is entropy then integrating $\gamma(T)$ from $T=0$ to some $T_{0}>T_{\textrm{c}}$ must give the same result as integrating $\gamma_{\textrm{n}}(T)$ from $T=0$ to $T_{0}$. The fit function which satisfies these two requirements is

$T^{*}=156$ K is typical of values reported for Y124 from transport Bucher and NMR relaxation Raffa measurements but we emphasize this reflects an energy scale not a temperature entrant . An important implication of Fig. 1 is that there is no coupling between superconductivity and the pseudogap in the sense that the onset of superconductivity does not weaken the pseudogap. This is evident from the fact that a single $\gamma_{\textrm{n}}(T)$ curve fits all three samples i.e. $\gamma_{\textrm{n}}(T)$ is the same for 4% and 0% Zn even in the temperature range below $T_{\textrm{c}}$ for 0% Zn so that the onset of superconductivity in the latter case does not alter the underlying pseudogap energy scale, $E^{*}$. Close scrutiny of the $k$-dependent gap in Bi2212, as measured by angle-resolved photoelectron spectroscopy (ARPES) Vishik , (which allows separation of the antinodal pseudogap from the nodal superconducting gap on the Fermi arcs) confirms that the pseudogap amplitude does not alter on cooling below $T_{\textrm{c}}$.

We will now test this relationship in the present case of Y124. By integrating $\gamma_{\textrm{n}}(T)$ from $T=0$ to $T$ we obtain the normal-state entropy and this is plotted as $S_{\textrm{n}}(T)/T$ by the black dashed curve in Fig. 2. For fully-oxygenated Y123 $S_{\textrm{n}}(T)/T$ is essentially independent of temperature Loram3 , reflecting the fact that the pseudogap has closed at maximal doping ($p\approx 0.19$ holes/Cu). But for Y124 there is a large pseudogap present which suppresses $S/T$ at low $T$. This is also seen in the $T$-dependent NMR Knight shift which is linearly related to the spin susceptibility. To illustrate, we show in Fig. 2 the ¹⁷O Knight shift, referenced to the chemical shift, as reported by Tomeno et al. Tomeno . These authors also report the bulk susceptibility as a function of the Knight shift, thus enabling calibration of the spin susceptibility from the Knight shift. As a final step we multiply the spin susceptibility by the Wilson ratio, $a_{\textrm{W}}$, in order to express the $T$-dependent part of the Knight shift in entropy/$T$ units.

It can be seen in Fig. 2 that, not only the shape, but the absolute magnitude concurs remarkably well with the derived $S_{\textrm{n}}(T)/T$ suggesting, as already noted for the bulk susceptibility Loram , that the near-nodal states are consistent with a weakly interacting Fermionic system. Of especial interest is the fact that the O2 and O3 Knight shifts begin to diverge below 200 K. The difference in shift, ${}^{17}K_{2,c}-^{17}K_{3,c}$, expressed in entropy units, is also shown in the figure ($\times$10). This shows an abrupt onset in nematicity, consistent with that reported by the Matsuda group Sato using torque magnetometry. Its location at ($p$=0.13, $T$=200) is precisely consistent with the three data points in the ($p$,$T$) plane reported by the Matsuda group for Y123. (For the doping state of 0.13 see Materials). These are plotted in the inset to Fig. 2 by the red data points. Their susceptibility data was presented as evidence for a thermodynamic “phase transition at the onset of the pseudogap” however it is clear from Fig. 2 that the pseudogap is already open far above $T_{\textrm{nematic}}$, having already depleted half of the spin susceptibility. We observe no anomaly in $\gamma(T)$ at or near 200 K to suggest a phase transition. It must be very weak. We will see below that the pseudogap in fact extends at least to well above 400 K. Consequently this nematic phase transition occurs within a preexisting pseudogap state that extends far above and is not a transition into the pseudogap state, contrary to what has been claimed Sato .

We now consider the field-dependent low-$T$ behaviour of $\gamma(T)$ for the pure sample. Fig. 3 shows $\gamma(T,H)$ plotted as a function of $\sqrt{\mu_{0}H}$ for $T$ = 6, 8, 10, 12 and 14 K. Above 1 tesla the behaviour is linear, consistent with the expected phenomenology of the Volovik effect. Such behaviour has been seen in the specific heats of single-crystal Y123 Junod and La_2-xSr_xCuO₄ Wang , and in interlayer tunneling in Bi2212 Benseman . For $d$-wave symmetry, in the superconducting state the finite ground-state specific heat coefficient is Wang

in units of mJ/g.at.K², where $V_{\textrm{M}}$ is the molar volume, $d$ the unit cell length, $\phi_{0}$ the flux quantum, $\Delta_{0}$ is the antinodal amplitude of the $d$-wave gap, and $k_{\textrm{F}}$ the Fermi wave vector at the node along the ($\pi$,$\pi$) direction. For $\Delta_{0}=$ 30 meV the dash/dot line in Fig. 3 shows the expected slope. The data is very consistent with this expectation. The averaged slope of 0.0443 mJ/g.at.K²T^1/2 implies a gap amplitude of $\Delta_{0}=$ 34 meV. We have used the value $k_{\textrm{F}}=0.436\times 10^{10}$ m${}^{-1}\,$ Sebastian .

Fig. 4(a) shows the normal-state and superconducting state entropy below 100 K obtained by integrating $\gamma_{\textrm{n}}(T)$ and $\gamma_{\textrm{s}}(T)$, respectively (as displayed in Fig. 1(b)) from 0 to $T$. The curves shown are for pure Y124 in zero field and they are denoted $S_{\textrm{n}}(T)$ and $S_{\textrm{s}}(T)$, respectively. The difference is the condensation entropy $\Delta S_{\textrm{ns}}=S_{\textrm{n}}(T)-S_{\textrm{s}}(T)$. This is plotted underneath for fields of $\mu_{0}H$ = 0, 1, 3, 5, 7, 9, 11 and 13 T, colour-coded as in panel (b). Clearly the condensation entropy is rapidly suppressed in field. Another feature of note is the presence of fluctuations around $T_{\textrm{c}}$ which broadens the transition somewhat. This will be discussed later.

By integrating $\Delta S_{\textrm{ns}}(T)$ from a temperature $T_{0}$, sufficiently above the fluctuation regime that $\Delta S_{\textrm{ns}}(T_{0})=0$, down to a temperature, $T$, one obtains the condensation free energy $\Delta F_{\textrm{ns}}(T)$. However, a more efficient way of calculating the condensation free energy using just a single integration is given by:

As mentioned, Fig. 5(a) shows $\Delta F_{\textrm{ns}}(T)$ and its components $\Delta U_{\textrm{ns}}(T)$ and $-T\Delta S_{\textrm{ns}}(T)$. It is striking that fluctuations persist high above $T_{\textrm{c}}$ in both $\Delta U_{\textrm{ns}}(T)$ and $-T\Delta S_{\textrm{ns}}(T)$ while they are almost completely cancelled in $\Delta F_{\textrm{ns}}(T)$. The superconducting gap function $\Delta(T)$ may be calculated from these components of the free energy using Tallon2

It is highly instructive to compare the measured data for Y124 with that for Y123. This is shown for $\gamma(T)$ in Fig. 6(a) and for $S(T)$ in Fig. 6(b). The Y123 data is taken from ref. Loram_IRC which, in contrast to earlier reports Loram1 ; Cooper1 , includes a small correction for the background DOS in the undoped state. We also show the entropy-conserving normal-state functions as well. Two features are prominent. (i) the size of the specific heat jump for Y124 is much smaller than for Y123 due to the presence of the pseudogap in the former. (ii) while the $\gamma(T)$ curves converge above $T_{\textrm{c}}$ the $S(T)$ curves remain separated and parallel to the highest temperature. This is clear evidence that the pseudogap remains present in Y124 to the highest temperatures measured - here 400 K.

To see this, consider the Fermi window for the entropy given in Eq. 2. This is the single-peaked function shown in the inset to Fig. 6(b). Three different temperatures of 0.1, 0.3 and 1.0 $E^{*}/k_{\textrm{B}}$ are displayed, where $E^{*}$ is the magnitude of the triangular gap shown in the figure as a representation of the pseudogap. Provided that the pseudogap remains open, this Fermi window always sees the gap and at higher temperatures loses a fixed fraction of states so that $S(T)$ is displaced down in parallel fashion relative to that for Y123 - as evidenced in Fig. 6(b), and again in Fig. 7(b). In contrast, because $\gamma\equiv\partial S/\partial T$, the Fermi window for $\gamma(T)$ is the $T$-derivative of that for $S(T)$. This is a double-peaked function as shown in the inset to Fig 6(a). It can be seen that at high enough temperature this Fermi window falls outside of the gap. Therefore $\gamma(T)$ recovers its full ungapped magnitude at high $T$, despite the presence of the gap. The data in Fig. 6(a) and (b) are entirely consistent with this picture. The two systems have essentially the same background DOS. In the normal-state Y123 is ungapped while Y124 is gapped to the highest temperature as evidenced by the parallel suppression of $S(T)$. If the gap were to close with increasing $T$ then $S(T)$ for Y124 would recover to that for Y123. It does not. By extrapolating $S(T)$ back to the ordinate axis one can read off the normal-state gap magnitude. For a triangular gap, as in the insets to Fig. 6, but with a finite DOS of $N_{1}$ at $E_{\textrm{F}}$ and a constant DOS of $N_{0}$ above $E^{*}$ this negative intercept is $2\ln 2\,k_{\textrm{B}}(N_{0}-N_{1})V_{\textrm{M}}E^{*}$, where $V_{\textrm{M}}$ is the molar volume. The finite value of $\gamma_{\textrm{n}}$ at $T=0$ is given by $\gamma_{\textrm{n}}^{0}=4\ln 2\times 2.374k_{\textrm{B}}^{2}N_{1}V_{\textrm{M}}$, while at high $T$ we have $\gamma_{\textrm{n}}=(2/3)\pi^{2}k_{\textrm{B}}^{2}N_{0}V_{\textrm{M}}$. From these we obtain $E^{*}=19.1$ meV and $N_{0}=6.14\times 10^{-3}$ states/meV/cell.

We now extend this comparison to 400 K, a range never previously achieved in differential measurements. We combine this data with ⁸⁹Y Knight shift data, bulk susceptibility measurements and $1/^{63}T_{1}$ data in such a way that demonstrates, individually and collectively, the persistence of the pseudogap to 400 K and beyond. ⁸⁹K${}_{\textrm{s}}(T)$ probes the spin susceptibility of the CuO₂ planes that sandwich the Y atom in Y123 and Y124 Alloul . The ⁸⁹Y nucleus has the additional benefit of having no quadrupole moment so there is no quadrupole splitting of the resonance arising from electric field gradients. Further, the use of magic-angle spinning (MAS) enables extremely narrow line widths as will be used below Williams . The $1/^{63}T_{1}$ relaxation rate is a weighted sum over q of the imaginary part of the spin susceptibility, $\chi^{\prime\prime}(\textrm{\bf q},\,\omega)$, where, for the ⁶³Cu nucleus, the weighting form factor is strongly enhanced near the antiferromagnetic wave vector, $\textrm{\bf q}=(\pi,\pi)$ Crossover and hence $1/^{63}T_{1}$ is dominated by the antinodal pseudogap.

Fig. 7(a) shows the ⁸⁹Y Knight shift ⁸⁹K${}_{\textrm{s}}(T)$ for YBa₂Cu₃O_6+x as reported by Alloul et al. Alloul , with values of $x$ annotated. Also plotted is MAS ⁸⁹K${}_{\textrm{s}}$ for samples of Y124 from our laboratory Williams2 (red spheres), along with the $1/^{63}T_{1}$ data from Raffa et al. Raffa (up triangles) which we showed Williams2 scales precisely with ⁸⁹K${}_{\textrm{s}}$, as can be seen. (The conversion scale is ⁸⁹K${}_{\textrm{s}}=48.8\times 1/^{63}T_{1}-164.5$.) Notably, there is an excellent match over the entire temperature range with Alloul’s data for $x=0.75$ (green up-triangles) where the doping state (0.13) and $T_{\textrm{c}}$ for Y123 are much the same as those of Y124. As in Fig. 2, we convert ⁸⁹K${}_{\textrm{s}}$ to spin susceptibility using the calibration of Alloul et al. Alloul (see Methods for more detail) and multiply by $a_{\textrm{W}}$ to express in $S/T$ units. We find an excellent agreement between Alloul’s ⁸⁹K${}_{\textrm{s}}$ and the measured entropy for Y123 across a wide range of doping and temperature (see Fig. S4 in Supplementary Information, SI).

In Fig. 7(b) we assemble four distinct data sets ($S$, ⁸⁹K${}_{\textrm{s}}$, $1/^{63}T_{1}$, and $\chi_{\textrm{s}}$ from the bulk susceptibility) for Y123 at near full oxygenation ($x=0.97$), and for Y124. All are expressed in entropy units, in this case using the factor $a_{W}T$ to convert susceptibilities (including the $1/^{63}T_{1}$ data expressed as a susceptibility). The $\chi_{\textrm{s}}$ data is shown for Y123 with $x=0.97$ and $x=0.73$ (purple and green dashed curves, respectively, the latter as a proxy for Y124) and is taken from Loram et al. Loram_IRC . Evidently $S$ (blue solid curve), $a_{W}\,^{89}\chi_{\textrm{s}}T$ (purple squares) and $a_{W}\chi_{\textrm{s}}T$ (purple dashed curve) for Y123 at full oxygenation all track linearly to the origin indicating the absence of the pseudogap. For Y124 on the other hand, $S$ (red solid curve - this work), $a_{W}\,^{89}\chi_{\textrm{s}}T$ (red spheres) and $T/^{63}T_{1}$ (open black triangles), together with Alloul’s $a_{W}\,^{89}\chi_{\textrm{s}}T$ (solid green triangles) for Y123 with $x=0.75$ and $a_{W}\chi_{\textrm{s}}T$ (dashed green curve) for Y123 with $x=0.73$, both as proxies for Y124, all reveal a linear high-temperature behaviour that extrapolates to a negative intercept on the $y$-axis indicating the presence of a gap - the pseudogap. For Y124 the entropy curve almost completely overlays the bulk susceptibility green-dashed curve which is barely visible, so the figure is reproduced in the SI with the green-dashed curve overlaying the entropy. Evidently, the agreement over the full temperature range is excellent. Importantly, our entropy data extends to 400 K as does the $\chi_{\textrm{s}}$ data, and the ⁸⁹K${}_{\textrm{s}}$ data extends to 370 K. Fig. 7(b) represents the central result of this work. There is no indication, at any temperature, of the entropy recovering to the gap-less curve observed for fully-oxygenated Y123 that would signify the closing of the pseudogap at, or around, some $T^{*}$ value. The small upturn in $S(T)$ near 400 K simply represents the limitations of the present differential technique at such a high temperature and is not seen in the $\chi_{\textrm{s}}$ data. We conclude that the pseudogap does not close at some postulated $T^{*}$ in the range 150 to 200 K but remains open to the highest temperature investigated - 400 K. A similar conclusion has recently been drawn from $1/^{17}T_{1}$ planar oxygen NMR relaxation data for a number of cuprates Haase . We showed the same long ago Tstar for the in-plane resistivity and similarly for the $c$-axis resistivity entrant ; Bernhard1 .

Fig. 8 shows the as-measured field-induced change in specific heat coefficient, $\Delta\gamma(13,T)=\gamma(13,T)-\gamma(0,T)$. Recall that this difference contains no correction for the residual phonon contribution so is free of any imposed model. As well as the suppression of fluctuations around the specific heat anomaly near $T_{\textrm{c}}$ the low-temperature resonance is evident. We fit this using Eq. 8 with the parameters $A_{\textrm{res}}=0.61$ mJ/g.at.K² and $\varepsilon_{\textrm{r}}=-1.55$ meV, the latter value being nicely consistent with the STS result for Bi₂Sr₂CaCu₂O_8+δ Pan . This fit is the red curve in Fig. 8 and the difference is shown by the blue curve. The calculated resonance response is an excellent fit and shows the rapid decay at higher temperatures associated with the $T^{-3}$ tail. The difference is close to entropy conserving and requires a straightforward extrapolation below 2.8 K of its trend above 2.8 K to achieve exact entropy conservation. This rapid decay of the resonance in $\gamma(T)$ contrasts the predicted much slower $T^{-1}$ decay in the resonance part of the spin susceptibility. We have previously investigated the ⁸⁹Y NMR Knight shift in Zn-doped Y124 Williams1 . The Zn resonance contribution to the spin susceptibility is seen in a satellite peak which has a slowly-decaying Curie temperature dependence observable all the way up to 300 K, thus nicely confirming the behaviour predicted by Eq. 9.

In conclusion, we have measured the electronic specific heat of YBa₂Cu_4-xZn_xO₈ using a precision differential technique that allows separation of the electronic term from the lattice term up to an unprecedented 400 K. The pure sample reveals the expected Volovik effect which is fully suppressed in the Zn-doped samples. We show that the pseudogap, characteristic of underdoped cuprates, always remains open to above 400 K, far above the nominal pseudogap onset temperature usually proposed, $T^{*}\approx$ 150 - 200 K. Weak thermodynamic transitions reported at $T^{*}$, for example the onset of susceptibility nematicity, occur within the already fully established pseudogap and are not transitions into the pseudogap state as widely claimed. The spin susceptibility, derived from the ¹⁷O and ⁸⁹Y Knight shift and expressed in entropy units, is numerically the same as the electronic entropy divided by temperature, indicating that the near-nodal states are those of weakly-interacting Fermions. We derive the field-dependent condensation energy and superconducting energy gap. These expose the presence of strong pairing fluctuations extending well above $T_{\textrm{c}}$ as well as canonical impurity scattering behaviour, including the expected low-temperature resonance response which in the entropy channel decays rapidly as $T^{-3}$, but in the spin channel decays slowly as $T^{-1}$. The measurements and analysis reveal the remarkable utility of the differential specific heat technique in exposing the rich physics of strongly-correlated electronic materials.

Materials. The samples were prepared from stoichiometric proportions of high purity Y₂O₃, dried Ba(NO₃)₂, ZnO and CuO, pressed as pellets and reacted for 16 hours at 935^∘C under an oxygen pressure of 60 bar. The samples were ground finely and the process repeated three more times. X-ray diffraction revealed single-phase YBa₂Cu₄O₈ (see Fig. S1) with impurity less than 2% (only CuO identified). Lattice parameters were found to be $a=0.3842$ nm, $b=0.3870$ nm, and $c=2.7235$ nm under Ammm symmetry with a preferred alignment of the $c$-axis normal to the plane of the pellets. The $T_{\textrm{c}}$ values, determined from sharp diamagnetic onset were 81.2 K (0% Zn), 51.8 K (2% Zn) and 29.2 K (4% Zn). The quoted Zn concentrations are those referred to the CuO₂ plane as essentially no Zn resides on the chains. Thus 2% Zn refers to the composition YBa₂Cu_3.96Zn_0.04O₈ and 4% refers to YBa₂Cu_3.92Zn_0.08O₈. Thermoelectric power measurements at 290 K give 6.95 $\mu$V/K (0% Zn), 6.89 $\mu$V/K (2% Zn) and 6.93 $\mu$V/K (4% Zn). From the correlation of thermoelectric power with doping Obertelli this amounts to essentially identical doping states of 0.130 holes/Cu for each.

The suppression of $T_{\textrm{c}}$ with Zn substitution is very much in line with that for underdoped Y123 at the same doping state. Fig. S2 shows $T_{\textrm{c}}$ as a function of doping for 0, 2, 4 and 6% planar Zn substitution for Y_0.8Ca_0.2Ba₂Cu₃O_7-δ while the red stars show the data for Zn-substituted Y124. They are very consistent. The rapid suppression of $T_{\textrm{c}}$ for underdoped samples (open symbols) compared with overdoped (filled symbols) is a signature of the pseudogap which lowers the DOS at the Fermi level and hence raises the scattering rate Tallon_scattering . Fig. S3 shows $T_{\textrm{c}}$ as a function of planar Zn concentration for Y_0.8Ca_0.2Ba₂Cu₃O_7-δ at various doping states while red stars show the same data for Y124. Again they are very consistent. Typically, for higher Zn concentrations, the $T_{\textrm{c}}$ value sits higher than that expected from Abrikosov-Gorkov pairbreaking due to the statistical overlap of nearest neighbours to the Zn substituent Tallon_scattering .

Spin susceptibility and Knight shift. The spin susceptibility, $\chi_{\textrm{s}}$, is related to the bulk magnetic susceptibility, $\chi_{\textrm{m}}$, by $\chi_{\textrm{m}}=\chi_{\textrm{s}}+\chi_{0}$ where the constant $\chi_{0}$ comprises diamagnetic and van Vleck terms and is evaluated for Y123 by Alloul et al. Alloul . The measured Knight shift, ⁸⁹K${}_{\textrm{s}}$, is given by ⁸⁹K${}_{\textrm{s}}$ = $\sigma_{0}(x)+a(x)\,^{89}\chi_{\textrm{s}}$, where $\sigma_{0}(x)$ is the $T$-independent chemical shift and $a(x)$ is the relevant hyperfine coupling constant and, as indicated, both change only with oxygen content, $x$. For each $x$ therefore, $\chi_{\textrm{s}}$, $\chi_{\textrm{m}}$ and ⁸⁹K${}_{\textrm{s}}$ are linearly related to within an additive constant. We used values of $a(x)$ reported by Alloul Alloul , while for each $x$ the additive constant was determined by matching the ${}^{89}\chi_{\textrm{s}}$ data to our bulk susceptibility data, $\chi_{\textrm{s}}$ Loram_IRC . This fixed the value of the constant $\sigma_{0}(x)$ which differed somewhat from those of Alloul but other literature values also reflect those differences Williams2 . The overall $T$-dependence (independent of $\sigma_{0}(x)$) was an excellent match.

We are grateful to Dr. J. R. Cooper for helpful comments on the Knight shift and spin susceptibility. We also thank Dr Martin Ryan for assistance with the x-ray diffraction analysis.

JLT synthesized and characterized the samples, JWL carried out the specific heat measurements and the initial analysis to extract the electronic specific heat. JLT analyzed the data and wrote the paper.

^† [email protected]

^‡ deceased November 2017.