Introduction

The understanding of the complex properties of the cuprates, and what causes their high critical temperature of superconductivity (Tc), is one of the greatest challenges in condensed matter physics. From it one expects clues that make the synthesis of cuprates with much higher Tc possible or of how to improve other fundamental properties. In particular, it is still not well understood what sets the very different, maximum Tc’s for different families of materials. An early experimental observation in this regard is the famous Uemura plot1. It shows that the maximum Tc is correlated with the muon spin relaxation rate σ0 (extrapolated to T=0 K) that is proportional to the superfluid density divided by the effective mass (σ0∝ns/m*). This relation holds for the underdoped materials and orders different cuprate families with respect to their maximum Tc. The Uemura relation and subsequent scaling laws have remained stimulating, up to now, and some were shown to be valid for other superconductors as well2,3,4,5,6,7,8,9. While there are various attempts at a theoretical explanation of the Uemura relation4,10,11,12, a connection to other experimentally probed properties of cuprates is still lacking, in particular to material chemistry parameters.

The various cuprate families have in common, see Fig. 1a, a CuO2 plane and charge reservoir (CR) layers that separate the planes from each other. While the nearly square CuO2 plane, defined by the Cu orbital bonding to four O 2pσ orbitals, is very similar for all systems, the CR chemistry can vary significantly. The nominal hole at Cu (3d9 configuration) is responsible for strong magnetic correlations that make parent materials antiferromagnetic. Holes or electrons can be added to the CuO2 plane by alteration of CR layers. As a result, static magnetism vanishes and new electronic phenomena emerge and the systems become conducting or superconducting. There are many similarities between different cuprate families, and one typically differentiates only between the hole and electron-doped phase diagrams, depicted in Fig. 1b, that appear to show a distinct asymmetry. However, the extent in temperature and doping of the different phases and observed phenomena varies substantially between different families, also for the much more thoroughly investigated hole doped materials. In addition, the comparability between different families is somewhat obstructed with regard to doping. For example, while for La2−xSrxCuO4 the doping level can be varied over a large range quite reliably by stoichiometry, interstitial doping with oxygen (Oδ) as, for example, in HgBa2CuO4+δ leaves uncertainties with regard to the actual doping level. In addition, ionic migration may cause phase separation or other ordering phenomena as in the YBa2Cu3O6+y systems. Given the very different Tc for various cuprate families, it was questioned whether the average doping level of the CuO2 plane is the appropriate chemical parameter for discussing all aspects of the complex physical properties of the cuprates, or whether other parameters should be considered as well, for example, distances within the plane, buckling, disorder, the role of the apical oxygen, or interlayer coupling. Nevertheless, from the understanding of the electronic properties we expect, in particular, clues as how to raise Tc.

Figure 1: General features of the cuprates.
figure 1

(a) The cuprates′ layered structure consists of CR layers and CuO2 planes with the bonding orbitals Cu and O 2pσ, which share the nominal 3d hole of the Cu2+ ion. Columns indicate occupation of Cu (2−nd) and O 2pσ (2−np), with hole contents nd and np measurable with NMR. (b) Schematic representation of electronic phase diagram of the cuprates for electron and hole doping x: AF phase below Néel temperature (TN), SC below critical temperature (Tc), and pseudogap regime below pseudogap temperature (T*). Doped electrons (e−) go to the Cu orbital almost exclusively, while doped holes (h+) predominantly go to the O 2pσ orbital, arrows next to columns indicate changes of nd and np caused by doping. (c) Solid red: Uemura plot1, that is, Tc versus muon spin relaxation rate (σ0, upper abscissa); black symbols: Tc versus planar oxygen quadrupole splitting 17νQ (lower abscissa). For list of abbreviations see Table 1. For triple layer Tl-2223 and Hg,Tl-1223 the pairs connected with a dotted line belong to the same sample and correspond to planar O sites of inner and outer layer (smaller splitting corresponds to underdoped inner CuO2 layer). (d) Tc versus planar O hole density np calculated from 17νQ for all available data (see text). Black dotted lines are guides to the eye and connect different doping levels for one family. Solid grey line indicates increase of the maximum Tc (that is, for the optimal doping level) as a function of np. AF, antiferromagnetic; SC, superconducting phase.

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Nuclear magnetic resonance (NMR), as a local probe of the magnetic spin susceptibility, focusses mostly on measurements of shift and relaxation caused by the interaction of the nuclear magnetic dipole moment with electronic magnetic moments. However, the nuclear electric quadrupole moment for nuclei with spin I>1/2 (I=3/2 for 63,65Cu, I=5/2 for 17O) interacts with the local electric field gradient (EFG) causing a quadrupole splitting (νQ) of the NMR lines in high magnetic fields. The EFG at the nuclear site is very sensitive to the local charge symmetry, and has been useful for assigning NMR signals to the various lattice positions or detecting inhomogeneous charge distributions in the CuO2 plane13,14. Since the quadrupole splittings of planar Cu and O depend on doping, models have been put forward that attempted to understand these changes in terms of the hole content of certain orbitals, see for example, refs 15, 16, 17, 18 and works cited therein. While trends relating the local charge distribution with Tc could be established, the uncertainty with regard to the EFG contributions from a variable CR chemistry, limited quantitative agreement with charge contents expected from stoichiometry, as well as insufficient experimental data hampered advances of such analyses.

Here we show that if we plot Tc versus the planar 17O NMR quadrupole splitting, a functional dependence very similar to that of the Uemura plot emerges. This documents that the superfluid density is a function of the EFG at planar O. Based on very recent progress in the understanding of NMR quadrupole splittings in terms of the charge distribution in the CuO2 plane19 we show that the maximum Tc increases with the hole content of the planar O 2pσ orbital, at the expense of that at Cu . Thus, we identify material chemistry parameters, the hole contents at planar Cu and O, that are largely temperature independent, yet determine the superfluid density at low temperatures. This finding stimulates the use of these orbital hole contents, calculated from NMR literature data16,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49, to draw a three-dimensional cuprate phase diagram that encompasses all cuprate families and has the superconducting domes ordered according to the maximum Tc. We argue that such a phase diagram might be very useful in discussing the complex properties of the cuprates.

Results

Superfluid density and charge densities in the CuO2 plane

We plot in Fig. 1c, together with the original Uemura plot (in red), Tc versus 17νQ for similar materials and doping, and find a striking correspondence. This shows that the muon spin relaxation rate deep inside the superconducting state must be tied to the almost temperature independent EFG at the planar O nucleus, which determines the 17O NMR splitting measured far above Tc, a rather unanticipated result.

It was confirmed recently, based on NMR data on the electron-doped and parent compounds, that NMR quadrupole splittings provide a quantitative measure of the charge distribution in the CuO2 plane of apparently all cuprates19. A list of materials with abbreviations is given in Table 1. It was shown that the hole densities in the Cu orbital (nd) and the O 2pσ orbital (np) in the CuO2 plane are related to the experimentally measured splittings 63νQ at 63Cu and 17νQ at 17O as follows18,19:

Table 1 List of abbreviations.
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The planar oxygen splitting in equation (1) is only dependent on the hole content np of the onsite bonding orbital 2pσ with the prefactor 2.45 MHz derived from the electric hyperfine interaction experimentally determined with atomic spectroscopy of the O 2p5 state18. The term of 0.39 MHz is due to the charge symmetry at planar oxygen in the ubiquitous CuO2 plane, and this term is found to be rather independent on doping and similar for all families19. Therefore, one can easily convert the experimentally measured 17νQ into a reliable hole content np. The situation is somewhat more complicated for the Cu splitting in equation (2), which depends on the hole densities of both bonding orbitals. The first term is from the onsite hole content nd of with the prefactor 94.3 MHz, again derived from atomic spectroscopy of Cu 3d9 state18. The second term in equation (2) accounts for the EFG at the Cu nucleus caused by the charge in the bonding orbitals of the four surrounding planar O atoms, with the prefactor derived from the orbital overlap of O 2pσ with the empty Cu 4p and the electric hyperfine interaction of the latter18.

First, we use equation (1) and convert all the planar oxygen splittings from the literature to np. The result is plotted in Fig. 1d, that is, we plot Tc versus np for the different materials. Of course, this plot is very similar to Fig. 1c, but it includes non-superconducting underdoped and parent materials with Tc=0 K. We see that different cuprate families have rather different np, which results in the sorting of the families as in the Uemura plot. We also recognize that a large np is a prerequisite for a high maximum Tc, that is, at optimal doping. In Fig. 1d one can also notice a parabolic-like dependence of Tc on the oxygen charge np, which resembles the typical phase diagram that shows a dome-like dependence of Tc on the average doping level. The correlation between σ0 and 17νQ is lost in the overdoped regime where σ0 decreases with increasing doping50,51, which was attributed to a decrease of ns (ref. 52).

In Fig. 1d, we also included recent results for the electron-doped materials19. For Nd1.85Ce0.15CuO4 the superfluid density was reported to be very similar to that of hole doped YBa2Cu3O6+y, albeit measured optically and not by muon spin relaxation3,53,54. We find that these results are also in agreement with 17νQ splittings (see Supplementary Fig. 1) and corresponding hole contents for those two families, cf. Fig. 1d. Electron doping appears to be less efficient in providing a high Tc, but the rather high oxygen hole contents of the parent materials Pr2CuO4 and Nd2CuO4 suggest that hole doping should result in much higher Tc. The so-called infinite layer cuprate Sr1−xLaxCuO2, for which there are no reports of 17O splittings, has the highest Tc among electron-doped families and a very high muon spin relaxation rate (σ0≈4.5 μs−1) (ref. 53), and we expect a high np. Indeed, a rather high Tc of more than 100 K was reported in the infinite layer system upon hole doping55,56.

Clearly, a large np is a prerequisite for a high Tc, but is not sufficient, as expected for such a material chemistry parameter. If this empirical relation (max Tc∝np) remains valid for higher oxygen hole content, the Tc of the cuprates might be raised substantially by the proper chemistry (we estimate 300–400 K per oxygen hole from the straight line in Fig. 1d).

The splittings of the 63Cu NMR lines can only be converted into nd if there are also 17O NMR data available, cf. equation (2). However, there are much less 17O splittings reported since the materials have to be enriched with 17O (the naturally abundant 16O nucleus has spin I=0) and therefore only part of 63Cu splittings can be converted. (In Supplementary Fig. 1, we plot Tc versus experimentally measured splittings).

Note that the simple analysis using equations (1) and (2) gives hole densities that are in astonishingly good quantitative agreement with the total charge in the CuO2 plane expected from the stoichiometry of the materials19, that is,

where the factor of 2 accounts for the two O atoms per CuO2. This means that the sum of the hole contents nd and np as determined with NMR (r.h.s.) equals the inherent Cu 3d9 hole content plus the hole content added by doping x (l.h.s.). This agreement was shown to apply for electron, as well as hole doping, and different parent materials differ only in terms of the charge transfer between Cu and O19. One can therefore also infer from Fig. 1d that compounds with the highest maximum Tc favour a smaller Cu hole content, and we conclude that it is the transfer of hole density to the O sites that is important for the highest Tc. With this result, one may ask if other properties of the cuprates should be discussed in terms of np and nd, as well? This leads us to propose a cuprate phase diagram based on NMR.

Phase diagram of the cuprates based on NMR

In Fig. 2 we plot Tc as a function of nd and 2np for all cuprates for which we could find both, Cu and O quadrupole splittings in the literature (see Supplementary Tables 1 and 2) with nd and np calculated from equations (1) and (2). All materials appear in four separate groups, marked by colour: (1) La2−xSrxCuO4; (2) YBa2Cu3O6+y, and other cuprates of that structure, for example, (CaxLa1−x)(Ba1.75−xLa0.25+x)Cu3O6+y as well as YBa2Cu4O8; (3) Bi, Tl and Hg based families; and finally, (4) the two electron-doped systems Pr2−xCexCuO4 and Nd2−xCexCuO4. The parent line, that is, the line that is given by nd+2np=1 (bold dashed line) separates hole doped and electron-doped systems. Note that the lines parallel to the parent line are given by nd+2np=1+x, and represent constant hole (x=+0.1, +0.2) or electron (x=−0.1, −0.2) doping. While there may be material-specific uncertainties, for example, La2CuO4 is not located exactly on the parent line, our straightforward analysis uncovers simple systematic trends concerning all cuprates, and we discuss some salient features now.

Figure 2: Cuprate phase diagram from NMR.
figure 2

Tc as a function of oxygen (2np) and copper (nd) hole content for hole doped La-214, Y-123, Y-124, CLBLCO_x and Bi-, Hg-, Tl-based compounds, as well as electron (e−) doped Pr-214 and Nd-214. For list of abbreviations see Table 1. The parent line (dashed bold black) indicates expectation for the undoped case (nd+2np=1 from x=0), parallel lines (thin black) correspond to expectation for doping x=nd+2np−1 changing with a step of 0.1. The commonly used phase diagram (T versus x) appears as a projection (upper left).

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While nd and np change significantly between different parent compounds along the line nd+2np=1, antiferromagnetism persists as long as there is one hole per CuO2. Such a large range of variation in the charge transfer in different parent compounds, with 2np ranging from 0.15 to 0.45, is perhaps quite surprising, and its further increase, if possible, could raise Tc substantially.

Doping holes means entering the right upper half of the (2np, nd)-plane. While nd and np increase with doping, the ratio of the respective changes (Δnd/2Δnp) appears to be a family property, that is, parent materials with low np (for example, La2CuO4) add more holes to O than those with high np. With electron doping we enter the lower left half of the (2np, nd)-plane. Here, predominantly Cu holes disappear while the (large) O hole content changes only slightly. It is not apparent from the phase diagram why most parent materials can only be doped with one type of carrier. As a function of doping, Tc increases with a slope that depends on the position on the parent line, as well, and hole doping seems to be more effective in raising Tc. Another important observation concerns optimal doping, that is, the doping level for which one finds the highest Tc for a given family. According to our analysis it is related to x=nd+2np−1 and not particular values of nd and np. However, we do observe a slight increase of the optimal x with increasing np (decreasing nd). Note that the doping level x follows from our analysis in terms of nd and np inserted into (3) and is not deduced from material chemistry. Our analysis agrees with expectations also for materials doped by interstitial Oδ where doping level x is often derived from the Tc dome19. Interestingly, the latter materials we find located in the same group, despite significant structural differences between Hg-, Tl- and Bi-based cuprates. Also the number of close CuO2 layers in multi-layer systems does not result in significant differences in the charge distribution.

Discussion

In Fig. 2 we plotted only Tc in the (2np,nd)-phase diagram, but it might be of great interest to investigate whether other cuprate properties are better presented as a function of the local charge distribution, instead of the average doping level. While further analysis is beyond the scope of our paper, we shortly discuss some other cuprate properties with regard to our phase diagram.

The Néel temperature depends on the interlayer coupling and therefore is not expected to be dominated by the charge distribution in the CuO2 plane. For example, YBa2Cu3O6 has a higher TN than Pr2CuO4 and La2CuO4. It would be interesting, however, to find out how the exchange coupling (J) changes along the parent line. Recently, there have been contradicting reports regarding J in the cuprates57,58,59. Mallet et al.58 found no correlation between J and Tc,max in R(Ba, Sr)2Cu3Oy, while Wulferding et al.57 claimed that J is correlated with Tc,max in (CaxLa1−x)(Ba1.75−xLa0.25+x)Cu3O6+y, which was later questioned by Tallon59.

Structural parameters of the CuO2 plane such as distances, buckling, or disorder appear to show no clear trend with respect to nd and np. However, the apical oxygen distance from the CuO2 plane increases as one follows the parent line beginning from low np, similar to the maximum possible Tc. This behaviour and the concomitant change in density of states of Cu 4s was noted before60.

Pressure applied to underdoped cuprates usually increases Tc, while the structural changes to even hydrostatic pressure can be complicated61. For example, specifically strained HgBa2CuO4+δ can have almost identical CuO6 octahedra as La2−xSrxCuO4, however, the large difference in their Tc values remains62. This might be related to the different values of nd and np for these families. As a result of recent progress in anvil cell NMR63,64, it is now possible to study cuprates at high pressures also with NMR65, and it was found that the Cu splitting increases with pressure indicating changes in the planar hole contents66. However, single-crystal studies are necessary and ongoing efforts by our group aim at providing a quantitative measure of the local charge distribution as a function of pressure.

Another important issue concerns the heterogeneity of the cuprates. We know from NMR that the static charge and spin density can vary drastically within the CuO2 plane, in particular between different cuprate families67. For example, the charge density in terms of the total doping x may easily vary by Δx≈0.05 (refs 32, 68, 69). Since Tc is not in a simple relation to this static inhomogeneity, only the average nd and np appear to matter. From this, one would conclude that inhomogeneity is either not important for the maximum Tc, or it is ubiquitous and dynamically averaged for NMR, depending on the chemical environment.

To conclude, NMR measures the charge distribution in the bonding orbitals in the CuO2 plane quantitatively, and since it reproduces the Uemura plot, that is, it finds the same ordering of families with respect to their maximum Tc, we now have material chemistry parameters that are responsible for setting the highest Tc and superfluid density. These findings inspired a different perspective on the cuprate phase diagram and it is likely that the complex cuprate properties might be better understood when discussed in the context of the charge distribution in the CuO2 plane.

Additional information

How to cite this article: Rybicki, D. et al. Perspective on the phase diagram of cuprate high-temperature superconductors. Nat. Commun. 7:11413 doi: 10.1038/ncomms11413 (2016).