Temperature independent cuprate pseudogap from planar oxygen NMR

Planar oxygen nuclear magnetic resonance (NMR) relaxation and shift data from all cuprate superconductors available in the literature are analyzed. They reveal a temperature independent pseudogap at the Fermi surface, which increases with decreasing doping in family specific ways, i.e., for some materials the pseudogap is substantial at optimal doping while for others it is nearly closed at optimal doping. The states above the pseudogap, or in its absence are similar for all cuprates and doping levels, and Fermi liquid-like. If the pseudogap is assumed exponential it can be as large as about 1500 K for the most underdoped systems, relating it to the exchange coupling. The pseudogap can vary substantially throughout a material, being the cause of cuprate inhomogeneity in terms of charge and spin, and consequences for the NMR analyses are discussed. This pseudogap appears to be in agreement with the specific heat data measured for the YBaCuO family of materials, long ago. Nuclear relaxation and shift show deviations from this scenario near $T_{\mathrm{c}}$, possibly due to other in-gap states.


INTRODUCTION
was developed [16][17][18]. It fostered the understanding of charge sharing in electron and hole doped cuprates, as it was found that 1+x = n Cu +2n O , i.e., the charges measured with NMR in the planar Cu (n Cu ) and O (n O ) bonding orbitals add up to the total charge, inherent plus doped hole (x > 0) or electron (x < 0) content. An astonishing correlation appeared in this context, as the maximum T c of a curpate system (T c,max ) is nearly proportional to n O [18,19]. This explains the differences in T c,max between the various families that differ in charge sharing considerably, and it calls into question the usefulness of what one calls the cuprate phase diagram, rather, a phase diagram in terms of n Cu and n O appears advantageous [20].
These findings suggested that some cuprate properties might be family dependent, and that a broader look at NMR data might be useful, as well. Since planar O NMR requires the exchange of 16 O by 17 O, which is not easily performed for single crystals and can have consequences for the actual doping and its spatial distribution, the focus was on planar Cu data that appeared more abundant and more reliable.
Immediately, the overview of the Cu shifts across all families [21] demands different shift and hyperfine scenarios, as the changes in the shifts are not proportional to each other (similar to what was found with special NMR experiments before [12,14,15]). Likely, it involves two spin components, one that has a negative uniform response and is located at planar Cu, coupled to a second component (presumably on planar O) with the usual positive response. In a next step, all planar Cu relaxation data were gathered [22,23], and from the associated plots it became obvious that, surprisingly, the Cu relaxation is quite ubiquitous, very different from what was concluded early on. It turns out that the relaxation rate measured with the magnetic field in the plane (1/T 1⊥ ) does neither change significantly between families, nor as a function of doping, with 1/T 1⊥ T c ≈ 21/Ks. Only the relaxation anisotropy changes by about a factor of three across all cuprates. Thus, no enhanced, special spin fluctuations are present in the underdoped systems. This leaves as an explanation for the failure of the Korringa relaxation (discovered early on [3]) only a suppression of the NMR shifts [22]. This also means that there is no pseudogap effect in planar Cu relaxation, while the Cu shifts do have a temperature dependence above T c presumably from pseudogap effects. Finally, it was shown that the planar Cu relaxation can be understood in terms of two spin components, as well [24], where a doping dependent correlation of the Cu spin with that of O explains the relaxation anisotropy. Furthermore, the unusual planar Cu shift component that is a function of doping and not necessarily temperature was found to be present in the planar O high temperature data [25], where it causes the hallmark asymmetry of the total quadrupole lineshape, observed long ago [26][27][28], but not understood.
Here, we present all temperature dependent shift and relaxation data of planar 17 O collected in an intensive literature search (data points from about 80 publications were taken).
The main conclusion from the data will be that planar O relaxation, different from Cu, is affected by the pseudogap that also dominates the planar O shifts. Here, the pseudogap represents itself as a loss in density of states close to the lowest energies (at the Fermi surface) for the underdoped materials, and this gap is temperature independent, but set by doping, different from what is often assumed [29,30]. This scenario is in agreement with early specific heat data [31] that also discussed such a pseudogap in YBa 2 Cu 3 O 7−δ . The largest found pseudogap is in agreement with a node-less suppression of states of the size of the exchange coupling, 1500 K. It rapidly decreases with increasing doping, e.g., it is closed for YBa 2 Cu 3 O 7−δ at optimal doping, but not for optimally doped La 2−x Sr x CuO 4 .
Nuclear relaxation of planar oxygen shows strikingly simple behavior in these most studied materials, and we will find the conclusions to be generic to the cuprates.

Planar Oxygen Relaxation
In Fig. 1, next to a sketch of expected behavior for a Fermi liquid (A) we plot the relaxation rate (1/T 1 ) vs. temperature (T ). It is apparent that optimally and overdoped (B) are Femi liquid-like: above T c , an increase (decrease) in temperature adds (subtracts) additional states for nuclear scattering and even the density of states (DOS) seems to be rather constant up to about 250 K (above that temperature the relaxation appears to begin to lag behind the expected value [32]).
It is important to note that at high temperatures, changes in temperature (∆T ) lead to proportional changes in relaxation (∆(1/T 1 )) with a slope of 0.36 /Ks that intersects the origin. With other words, the proportionality of the rate to temperature is only disturbed by the opening of the superconducting gap at T c , below which relaxation drops more rapidly as pairing sets in (no Hebel-Slichter peak is observed). Thus, planar O relaxation of optimally and overdoped YBa 2 Cu 3 O 7−δ appears determined by Fermi liquid-like electrons, turning into a spin singlet superconductor.
The underdoped materials behave distinctively different, Fig. 1B. Here we observe a rapid change of relaxation with doping at given temperature, but we find nearly the same high-temperature slope of about 0.36 /Ks, i.e., increasing the temperature adds states at the same rate as for optimally or overdoped systems. However, the shifted slope signals an offset in temperature below which relaxation must disappear. This means, even at much larger temperatures one is aware of the lost, low temperature states. This is exactly what one expects if a temperature independent, low-energy gap in the DOS develops with doping  Note, the relaxation ceases completely at the lowest temperatures for all materials. While electric contributions (electric quadrupole interaction) to the relaxation have been shown to exist and contribute at lower temperatures [33,34] their contribution vanishes, as well.
The true magnetic relaxation dependences might be systematically shifted to lower rates at lower temperatures compared to what is seen in Fig. 1. Therefore, the apparent increase in relaxation could signal quadrupolar relaxation, as well. A thorough study of these effects might be in order.

Planar Oxygen Shifts
For planar O the orbital shift is almost negligible [26], making the spin shifts rather reliable with uncertainties arising only from the diamagnetic response below T c . Shift ref- erencing is simple, as well, as ordinary tap water can be used for 17 O NMR referencing (there is significant confusion about Cu shift referencing in the literature [21]). Nevertheless, there appear to be deviations between the shifts measured on similar samples, even for stoichiometric YBa 2 Cu 4 O 8 [35], and it is not always clear if shifts were corrected for the diamagnetic response. We will show the bare shifts without correction, in order to avoid introducing systematic errors. For example, it is possible that the uniform spin response from Cu 2+ is negative [21,22] leading to a negative term for planar O at low temperatures.
Note that the diamagnetic response of the cuprates was experimentally determined with 89 Y NMR, early on [11], by assuming that this nucleus' spin shift is negligible at low temperatures (4.2K). A value of about 0.05% was derived [11]. This value appears to be rather large [36], and as experiments with 199 Hg NMR of HgBa 2 CuO 4+δ showed [15], the diamagnetic response measured at 199 Hg is probably less than 0.01% (note that 199 Hg is located far from the plane and should not suffer from large spin shifts, different from 89 Y that might be affected by a negative term, as well).
For a Fermi liquid with a fixed DOS near the Fermi surface one expects a temperature independent spin shift (K) above T c , since an increase in temperature adds new states from an opening Fermi function, but the occupation decreases at the same rate, cf. Fig. 2A.
Now, in view of the planar O relaxation, a temperature independent gap at the Fermi surface should be assumed. Then, qualitatively, we expect a behavior shown in Fig. 2A: at the highest temperatures, far above the gap, low temperature states will still be missing, leading to a lower spin shift. As the temperature is lowered, the effect of the gap will be more severe. This is in agreement with data in Fig. 2B and C. Below T c , we note that there is no sudden loss of states as for optimally or overdoped materials, which one might naively expect if the same superconducting gap opens on the states still available. Quite to the opposite, a less rapid decrease of the shifts below T c is observed (we noted a different low temperature behavior for relaxation, as well).
Note that the Korringa relation is given by , and with S 0 = 1.4 · 10 −5 Ks one estimates a spin shift of about K = 0.23% from the relaxation slope

Numerical Analysis
The planar O relaxation data point to a pseudogap that is simply caused by missing low energy states. This gap is not temperature dependent, but rapidly increases with decreasing doping. In a very simple picture, we use the Fermi function with fixed DOS and calculate the relaxation as being proportional to the sum of the product of occupied states times empty states (the nuclear energy change is negligible for the electrons), i.e.
As a result one finds the Heitler-Teller dependence [6], 1/T 1 ∝ T , cf. Fig. 3. Now, one can remove manually states near the Fermi surface with a width ∆E given in temperature as defined by, by assuming a U-or V-shaped gap in the DOS, respectively [31]. For the U-shaped gap all states within ∆E are removed (exponential decrease), for a V-shaped gap a linear decrease in DOS is assumed, vanishing at E = µ. This simple scenario leads to the found behavior, i.e. we obtain nearly parallel high-temperature lines for different sizes of this pseudogap, cf. Fig. 3A. For a given offset, the cutoff temperature is different for both gaps, cf. Fig. 3B.
With such an approach we find for YBa 2 Cu 4 O 8 a gap of about T U PG ≈ 300 K (T V PG ≈ 650 K). Obviously, one cannot decide on the shape of the gap. Note that the BCS gap is not included in the fit and that there are uncertainties from quadrupolar relaxation at lower temperatures.
Since the action of the gap is to cause a near parallel shift of the high-temperature dependence, any spatial inhomogeneity of the gap will lead to similar lines, as well, very different from how it affects the shifts that we will discuss now.
One can estimate what such a pseudogap will do for the NMR shifts (by assuming a slightly different µ for spin up and down). Examples are shown in Fig. 4 for various T U PG (A), and T V PG (B). Clearly, for small gap sizes the shift will approach the Fermi liquid value (normalized to 1). The V-shaped gap has more total DOS and the action of the gap is weaker.
Above T c , one should be able to fit the experimental shifts, and by comparing Figs. 4 and 2 one finds qualitative agreement. However, a more quantitative determination of the gap appears difficult since (i) there is a large spread in shifts already for similar samples, and (ii) at lower temperatures the shifts for the underdoped systems appear larger, cf. An important feature of this pseudogap is a high temperature shift offset. It arises from the fact that even far above the pseudogap energy one still misses the low temperature states. Even if the shifts are temperature independent, they can carry a doping dependence (as the pseudogap depends on doping), i.e. two variables are needed to describe the shifts (K(x, T )).  The simple mean of the shifts is indicated by a dashed line, emphasizing that a gap inhomogeneity can cause a different temperature dependence of the apparent magnetic shift. Also, the magnetic linewidths will behave differently (the linewidths will grow as the temperature decreases, before it finally decreases).

PLANAR OXYGEN RELAXATION IN OTHER CUPRATES
In Fig. 5 we plot relaxation data from the literature for all other cuprates. Note that only the temperature axis is different (up to 600 K) compared to that in Fig. 1B, C. with T U PG ≈ 1450 K, the size of the exchange coupling in the cuprates. A V-shaped gap appears to fit better the low temperature behavior. It could be the states near the gap edge that are special (coherence peaks), also in-gap states could play a role in enhancing the relaxation at low temperature. Again, the loss of parts of the inhomogeneous sample with a large gap favors states from lower gap areas with increased relaxation. Quadrupolar relaxation plays some role, as well. Thus, the shape of the gap cannot be deduced from the low-temperature behavior. The gap rapidly closes with doping, as widely assumed.
Note that the high temperature behavior is similar for all materials, which does support the idea of a temperature independent gap set by doping, and, importantly, very similar high-temperature Fermi liquid-like states.
To conclude, planar O NMR relaxation appears ubiquitous to the cuprates, and it measures the pseudogap in a rather simple way.

PLANAR OXYGEN SHIFTS IN OTHER CUPRATES
Shift data from all other materials are presented in Fig. 6

DISCUSSION AND CONCLUSIONS
Planar O relaxation and spin shift data were collected and simple plots reveal that they demand a temperature independent pseudogap at the Fermi surface with a size set by doping.
The pseudogap rapidly opens, coming from the overdoped side by decreasing doping, and it approaches the size of the exchange coupling, J, for strongly underdoped systems.  planar Cu spin shifts [22,24], as well. Thereafter, it was shown that this doping dependent planar Cu spin shift explains the conundrum of the correlation of high-temperature spin shifts with the local charge [25], resulting in the hallmark asymmetric total planar O lineshapes (that include the quadrupolar satellites) of the cuprates [25,28].
Here, we argue that it is the doping dependence of the pseudogap that plays the dominant role for these effects. Then it follows that it is the pseudogap that can be spatially very inhomogeneous [25]. This distinction could not be made earlier [28], but it is in agreement with STM data [37]. With a large distribution of the pseudogap, shift and relaxation can be affected. An inhomogeneous broadening changes the apparent temperature dependence of the shift, cf. Fig. 4, as small pseudogap areas contribute more to the shift at lower temperatures than those with large pseudogaps. For relaxation, the faster relaxing regions, i.e. those with a smaller pseudogap, may dominate throughout the whole temperature range, if spin diffusion in possible. Thus, one has to be very careful in analyzing shift and relaxation quantitatively [38].
Not only changes an inhomogeneous pseudogap the temperature dependence of the average shift, also the NMR linewidths are affected. In view of Fig. 4 one concludes that in such a case the NMR linewidth should grow towards lower temperatures before it finally disappears, while the shift is decreasing monotonously. This is exactly what is known from experiment [25,28].
The relation of this pseudogap to the intra-unit cell charge variation that was first proposed from NMR data [39] and very recently shown to exist in the bulk of the material [40] is not clear. However, the response of the local charge symmetry to an external magnetic field and pressure found with NMR [40,41], must be similar to the discussed charge ordering phenomena and special susceptibilities [29,30], which are associated with the pseudogap.
The total charge involved in the ordering is small (1 to 2 % of the total planar O hole content) and may come from states in the pseudogap.
Note that the superconducting transition temperature T c appears to be not affected by this inhomogeneity, as it is nearly proportional to the average planar oxygen hole density of the parent compounds [18,19]. Then, with the size and distribution of the pseudogap set by doping, there appears no simple relation to maximum T c .
The pseudogap behavior was first reported with measurements above T c for 89 Y NMR of [42], and these data show a high-temperature offset in the shifts, as well. So we believe that 89 Y NMR data are in agreement with what we found for planar O here.
A U-shaped gap in our simulation means that all states contributing to planar O relaxation vanish suddenly within the gap. With such an assumption the largest pseudogap appears to be set by the exchange coupling. Then, effectively, doping decreases the energy gap that needs to be overcome for electrons to flip the nuclear spin for relaxation. Of course, the true shape of the gap and the nature of the states within the gap are not known.
If the above scenario describes the essential electronic states involved in cuprate conductivity and superconductivity, it should leave its typical signature in electronic specific heat.
Indeed, the YBa 2 Cu 3 O 7−δ family of materials appears to fit the specific heat data by Loram et al. [31] rather well [43]. Loram et al. [31] argue similarly in their specific heat investigations, as the specific heat is linear in temperature in the pseudogap range. Additional states are added by temperature at the same rate as for overdoped systems where there is no gap. and O NMR will be investigated in a forthcoming publication.
Unfortunately, we feel that it is difficult to conclude on the superconducting gap from the planar O data. An inhomogeneous pseudogap dominates the shifts and the relaxation may be partly electric [44] in the vicinity of T c . The latter clearly points to the involvement of charge fluctuations [45,46], very different from the relaxation of planar Cu [23], which is also rather ubiquitous at low temperatures in the cuprates, when normalized by T c [23]. Naively, one might assume that the states not already lost to the pseudogap disappear rapidly below T c , further slowing down relaxation, but the opposite behavior is found, i.e., the rate appears to increase at lower temperature before it finally decreases. This could be due to additional quadrupolar relaxation, alternatively, the magnetic relaxation could show a special increase, but perhaps the inhomogeneity of the pseudogap is most important as regions with fast relaxation (small pseudogap) will dominate. Details of the spin shift, including the behavior below T c , are difficult to evaluate, as well, not only due to the inhomogeneity, but also because of the uncertainty of the low-temperature data (loss of signal etc.). A small negative spin shift appears to be observed for a number of materials, which would be expected from the suggested shift scenario [21,22].  Here we list the references for other cuprates, about 44 publications with relevant data.
If a data set appeared in multiple papers, typically from the same group, we only show the last published account.