Evidence of magnetic mechanism for cuprate superconductivity

A proper understanding of the mechanism for cuprate superconductivity can emerge only by comparing materials in which physical parameters vary one at a time. Here we present a variety of bulk, resonance, and scattering measurements on the (Ca_xLa_{1-x})(Ba_{1.75-x}La_{0.25+x})Cu_3O_y high temperature superconductors, in which this can be done. We determine the superconducting, Neel, glass, and pseudopage critical temperatures. In addition, we clarify which physical parameter varies, and, equally important, which does not, with each chemical modification. This allows us to demonstrate that a single energy scale, set by the superexchange interaction J, controls all the critical temperatures of the system. J, in-turn, is determined by the in plane Cu-O-Cu buckling angle.


Introduction
The critical temperature for superconductivity T c in the metallic superconductors Hg, Sn, and Tl as a function of M −1/2 , where M is the atomic mass, is presented in Fig. 1 on a full scale including the origin [1]. In the case of Sn a clear isotope effect is observed resulting in a 4% variation of T c upon isotope substitution. For Hg and Tl the variations are observed only by zooming in on the data. In all cases a linear fit goes through the data points and the origin quite satisfactorily, namely, T c is proportional to M −1/2 . This observation, known as the isotope effect, plays a key role in exposing the mechanism for superconductivity in metallic superconductors. However, had nature not provided us with isotopes, and we had to draw conclusions only by comparing the different materials in Fig. 1, we would conclude that T c has nothing to do with the atomic mass. Thus, Fig. 1 demonstrates that it is dangerous to compare materials where several quantities vary simultaneously. The isotope experiment overcomes this problem and reveals the origin of metallic superconductivity.
The mechanism for high temperature superconductivity (HTSC) in the cuprate is still elusive, but is believed to be of magnetic origin [2,3]. Verifying this belief would require an experiment similar to the isotope effect, namely, a measurement of T c versus the magnetic interaction strength J, with no other structural changes in the compounds under investigation. Unfortunately, varying J experimentally can only be done by chemical variation, usually leading to very different materials. For example, YBa 2 Cu 3 O y (YBCO) has a maximum T c of 96 K, La 2−x Sr x CuO 4 has a maximum T c of 38 K, and both have roughly the same J [4,5,6]. This fact has been used to contradict the magnetic mechanism although these materials are different in crystal perfection, number of layers, symmetry, and more. Clearly they are uncomparable exactly as Hg, Sn and Tl.
In the present manuscript we describe a set of experiments designed to overcome this problem and to perform a magnetic analog of the isotope experiment by making very small and subtle chemical changes, which modify J but keep all other parameters intact. This is achieved by investigating a system of HTSC with the chemical formula (Ca x La 1−x )(Ba 1.75−x La 0.25+x )Cu 3 O y and acronym CLBLCO. Each value of x = 0.1 . . . 0.4 is a family of superconductors. All families have the YBCO structure with negligible structural differences; all compounds are tetragonal, and there is no oxygen chain ordering as in YBCO. Within a family, y can be varied from zero doping to over doping.
We present measurements of the critical temperature of superconductivity T c using resistivity [7], the spin glass temperature T g [8] and the Néel temperature T N [9] of the parent antiferromagnet (AFM) using zero field muon spin relaxation (µSR), the level of doping and the level of impurities by Nuclear Quadruple Resonance (NQR) [10], the superconducting carrier density using transverse field muon spin rotation [11], the lattice parameters, including the oxygen buckling angle, with neutron scattering [12], and the pseudogap (PG) temperature T * with susceptibility [13]. This allowed us to The superconducting critical temperature T c of three metallic superconductors as a function if the inverse root of their mass [1]. The solid lines are strait. The insets are zoom on the data. generate one of the most complete phase diagrams of any HTSC system, to demonstrate a proportionality between T c and J, and to draw more conclusions.
The paper consists of two main sections in addition to this introduction: in Sec. 2 the main experimental results and parameters extracted from the raw data are presented. In Sec. 3 the conclusions are summarized. Experimental details, raw data, and analysis description are given in appendices.

Main Results
The phase diagram of CLBLCO including T N , T g , T c , and T * versus oxygen level y is shown in Fig. 2. Details of the magnetic measurements are given in Appendix A and pseudogap measurements in Appendix B. In the doping region up to y = 6.5, the T N curves of the different families are nearly parallel. The maximum T N (T max N ) of the x = 0.1 family is the lowest, and of the x = 0.4 family the highest. Upon further doping T N decreases rapidly and differently, leading to a crossing point after which the x = 0.4 family has the lowest value of T N , and the x = 0.1 family the highest. By further doping, the long-range order is replaced by a spin glass phase, where islands of spins freeze. The spin glass phase penetrates into the superconducting phase which exists for y = 6.9 to y = 7.25. This phase starts earlier as x increases. The superconducting domes are nearly concentric with maximum T c (T max c ) decreasing with decreasing x. T max c varies from 80 K at x = 0.4 to 56 K at x = 0.1, a nearly 30% variation. This is much stronger varation than the strongest isotope effect in nature. As for the pseudogap temperatures T * , it seems that the x = 0.1 family has the highest T * and the x = 0.4 the lowest.
Perhaps the clearest feature of this phase diagram is the correlation between T max N and T max c . The family with the highest T max N has the highest T max c . However, T N is not a clean energy scale. It is well established that a pure 2D AFM orders magnetically only at T = 0, and that T N is finite only for 3D AFM. Intermediate cases are described by more complicated interactions where J is the isotropic intralayer Heisenberg interaction, and α ef f J represent interlayer and anisotropic coupling [5]. In order to extract J from T N , α ef f must be determined. One method of extracting α ef f is from the magnetic order parameter M versus temperature T [5,16]. For small α ef f the reduction of the magnetic order parameter M with increasing T is fast so that at α ef f = 0 the 2D limit is recovered. On the other hand, in the three dimensional case, where α ef f = 1, we expect a weak temperature dependence of M at T → 0 due to lack of antiferromagnetic magnons states at low frequencies. A plot of the normalized order parameter σ = M/M 0 , where M 0 is the order parameter at T → 0, versus T /T N should connect (1,0) to (0,1) as depicted in Fig. 3 [9]. The differences between curves are set only by α ef f and they can be fitted to experimental data. Given α ef f and T N , J can be extracted. This is not a very accurate method of α ef f determination, but J depends only on ln(α ef f ) so high accuracy is not required [5].
Using zero field muon spin rotation, we determined the muon angular rotation frequencies ω in the different compounds [9]. The normalized order parameter is given by σ(T ) = ω(T )/ω(0). The order parameter extracted from the high angular frequency, around a few tens of MRad/Sec (ω ∼ 27 MRad/Sec in our case), is known to agree with neutron scattering determination of σ.
In Fig. 3 we also present σ for two different underdoped CLBLCO samples with x = 0.1 and 0.3. Clearly the reduction of the magnetization with increasing temperatures is not the same for these two samples, and therefore their anisotropies are different. Since σ is less sensitive to increasing T in the x = 0.1 family than in the x = 0.3 family, the α ef f of x = 0.1 must be larger. Using the muon spin rotation frequency versus T and calculations based on the Swinger boson mean field (SBMF) theory [9,5], we determined α ef f for all samples [9]. Knowing T N and α ef f for all samples we extract a corrected T N (T cor N ). For the very underdoped samples this T cor N = J. For doped samples the situation is more complicated since the samples are described more accurately by the t-J model. We will present T cor N shortly after discussing the doping. Doping in CLBLCO is done by controlling the oxygen level in a chain layer as in YBCO. This leaves some ambiguity concerning the doping of the CuO 2 planes. One possibility to determine the amount of charge present in this plane is to measure the in-plane Cu NQR frequency ν Q . Assuming the lattice parameters variations within a family can be ignored (an assumption tested with neutrons in Appendix E) the NQR frequency is proportional to the level of doping in the plane. The in-plane Cu NQR frequencies, discussed further in Appendix C, are shown in Fig. 4(a). It is clear that ν Q grows linearly with doping in the underdoped side of the phase diagram. The most interesting finding is that, within the experimental error, the slope of ν Q (x, y) in the underdoped side is x-independent, as demonstrated by the parallel solid lines. This means that the rate at which holes p are introduced into the CuO 2 planes, ∂p/∂y, is a constant independent of x or y in the underdoped region. Using further the ubiquitous assumption that the optimal hole density, at optimal oxygenation, y opt , is universal, we conclude that the in-plane hole density is a function only of ∆y = y − y opt . The same conclusion was reached by X-ray absorption spectroscopy (XAS) experiments [17].
In contrast, the CLBLCO family obeys the Uemura relations in the entire doping region, namely, T c is proportional to n s /m * where n s is the superconducting carrier density and m * the effective mass [11]. This is determined with transverse field muon spin relaxation measurements where the Gaussian relaxation rate R µ is proportional to n s /m * as explained in Appendix D [18,19]. The experimental results are depicted in Fig. 5 for all families. This experiment seems to contradict the NQR results for the following reason. If all holes had turned superconducting (p = n s ), then samples of different x but identical ∆y should have identical ∆p and identical ∆n s . In addition, if m * is universal, samples with a common ∆y should have the same T c , in contrast to the phase diagram of Fig. 2. Something must be wrong in the hole counting. A similar conclusion was reached in the investigation of . This problem can be solved by assuming that not all holes are mobile and turn superconducting. Therefore, we should define the mobile hole concentration ∆p m by multiplying ∆y by a different constant per family K(x), namely, ∆p m = K(x)∆y. The superconducting carrier density variation ∆n s is now proportional to ∆p m with a universal factor. The Ks are chosen so that the superconducting critical temperature T c domes, normalized by T max c of each family, collapse onto each other. This is shown in  order of unity. This is the main finding of this paper. Theoretical indications for the importance of J and n s in setting up T c could be found since the early days of HTSC [3].
Having established a proportionality between T c and J, it is important to understand the origin of the J variations between families in CLBLCO. As we show in appendix Appendix E using lattice parameters measurements with neutron diffraction, the Cu-O-Cu buckling angle is responsible for these J variations since it is the only lattice parameter that shows strong differences between the families; there is an about 30% change from the x = 0.1 family to x = 0.4 [12]. This change is expected since as x increases, a positive charge is moving from the Y to the Ba site of the YBCO structure, pulling the oxygen toward the plane and flattening the Cu-O-Cu bond.
From the lattice parameters it is possible to construct the hopping integral t and super-exchange J of the t-J model, assuming that the Hubbard U and the charge transfer energy ∆ are family-independent. The basic quantity is the hopping integral t pd between a Cu 3d x 2 −y 2 and O 2p orbitals [21]. This hopping integral is proportional to bond length a to the power -3.5 [22]. The hopping from the O 2p to the next Cu 3d x 2 −y 2 involves again the bond length and cosine of the buckling angle θ. Thus, the Cu to Cu hopping depends on a and θ as t dd ∝ cosθ/a −7 and J is proportional to t dd 2 , hence, J ∝ cos 2 θ/a 14 . Estimates of the t dd and J, normalized to the averaged values of the x = 0.1 family, t dd 0.1 and J 0.1 are presented in Fig. 7(a) and (b). Although there is a variation in t and J within each family, the variation is much larger between the families. J increases with increasing x, in qualitative agreement with experimental determination of J. There is a 10% increase in J which is not big enough to explain the variations in T cor N , but the magnitude and direction are satisfactory. More accurate calculations are on their way [23]. It is also important to notice that there is an about 5% difference in the t/J ratio between the two extreme families. Equally important is to understand what is not changing between families. For example, we would like to check whether the crystal quality is the same for all families. Figure 4(b) shows the width of the NQR lines discussed in Appendix C [10]. This width is minimal at optimal doping and is identical within experimental resolution for all families. More recent experiments with oxygen 17 lines, and better resolution, provide the same conclusions [24]. A third evidence for the similarity of crystal quality comes from XAS [25]. Thus, the variations in T c for the different families cannot be explained by disorder or impurities.
The unified phase diagram in Fig. 6(a) reveals more information about the CLBLCO family than just Eq. 1. In particular, the fact that the Néel order is destroyed for all families at the same critical ∆p m is very significant. The disappearance of the longrange Néel order and its replacement with a glassy ground state could be studied more clearly by following the order parameter as a function of doping. Naturally, ω(T → 0) disappears (drops to zero) only when the Néel order is replaced by the spin glass phase as seen in Fig. 6(b). We found that the order parameter is universal for all families, and in particular the critical doping is family-independent [12]. To demonstrate this point we show, using the two arrows in Fig. 6(b), what should have been the difference in the critical doping had it changed between the x = 0.4 and x = 0.1 by 5% of the doping from zero, namely, 5% of ∆p m = 0.3. Our data indicates that M 0 (∆p m ) is familyindependent to better than 5%. This is a surprising result considering the fact that t/J varies between families by more than 5% and that the critical doping is expected to depend on t/J [26]. A possible explanation is that the destruction of the AFM order parameter should be described by a hopping of boson pairs where t is absorbed into the creation of tightly bound bosons leaving a prominent energy scale J [27]. The proximity of the magnetic critical doping to superconductivity makes this possibility appealing.
Finally we discuss the scaling of the pseudogap temperature. T * determined by magnetization measurements (See Appendix B) behaves like the well-known PG or the spin gap measured by other techniques on a variety of superconductors samples [28]. More importantly, a small but clear family dependence of T * is seen. At first glance in However, this conclusion is reversed if instead of plotting the T * as a function of oxygen level, it is normalized by T max N , and plotted as a function of mobile hole variation ∆p m [13]. This is demonstrated in Fig. 6 [13]. We conclude that a PG does exist in CLBLCO and that it scales with the maximum Néel temperature of each family. Therefore the PG is a 3D phenomenon involving both in-and out of-plane coupling. A similar conclusion was reached by resistivity analysis [14] and theoretical considerations [15].

Conclusions
In this work four families of cuprate superconductors with a maximum T c variation of 30% are investigated. It is demonstrated experimentally that these families are nearly identical in their crystal structure and crystal quality. The only detectable property that varies considerably between them is the Cu-O-Cu buckling angle. This angle is expected to impact the holes hopping rate t, hence, the magnetic super-exchange J between Cu spins. J in turn sets the scale for the Néel temperature where long range antiferromagnetic order is taking place. Independent measurements of J show that indeed J varies between families and that T c grows when J increases. A linear transformation from oxygen concentration to mobile hole concentration can generate a unified phase diagram in which T c is in fact proportional to J for all doping. Since T c is also proportional to the superconducting carrier density it obeys Eq. 1.
Surprisingly, the critical density, where the Néel order is destroyed at zero temperature upon doping, is identical for all families. The critical doping is expected to depend on t/J. This result has two implications. On the one hand it supports the validity of the linear doping transformation; on the other it suggests that this transformation has eliminated t from the low temperature effective Hamiltonian.
Finally, it is found that the pseudogap temperature T * , as measured by susceptibility, scales better with the Néel temperature than with J (or T c ). This suggests that T * is determined by both in-and out-of-plane coupling, and should be viewed as a temperature where the system attempts unsuccessfully to order magnetically.

Acknowledgements
The author acknowledges very helpful discussions with A. Kanigel, R. Ofer, E. Amit, A. Auerbach, Y. J. Uemura, and H. Alloul. Financial support from the Israel Science Foundation is also acknowledged.

Appendix A. Magnetic critical temperatures
The Néel and spin glass temperatures presented in Figs. 2, 3 and 6 are obtained by zero field µSR. In these experiments we determine the time-dependent spin polarization P z (t) of a muon injected into the sample at different temperatures. z represents the initial muon spin direction. Figure A1  moments. In this case the time-dependence of the polarization exhibits a Gaussian decay. As the temperature is lowered the sample enters a magnetic frozen phase and the polarization relaxes much more rapidly. While the transition from the paramagnetic to the frozen state looks identical for all samples, the behavior at very low T is different and indicates the nature of the ground state. Figure A1(a) is an example of an antiferromagnetic ground state. When the temperature decreases, long range magnetic order is established at ∼ 377 K reflected by spontaneous oscillations of P (t). Figure A1(c) is an example of a spin glass (SG) transition at ∼ 17 K. In this case the ground state consists of magnetic islands with randomly frozen electronic moments [8], and consequently the polarization shows only rapid relaxation. When the transition is to a Néel or spin glass state, the critical temperatures are named T N and T g , respectively. Figure A1(b) presents an intermediate case where the sample appears to have two transitions. The first one starts below 240 K, where the fast decay in the polarization appears. Between 160 K and 40 K there is hardly any change in the polarization decay, and at 30 K there is another transition manifested in a faster decaying polarization. This behavior was observed in all the samples on the border between antiferromagnet and spin-glass in the phase diagram.
In order to determine the magnetic critical transition temperatures, the data were fitted to a sum of two functions: a Gaussian with amplitude A n representing the normal fraction of the sample, and a rapidly relaxing and oscillating function, with amplitude A m and angular frequency ω, representing the magnetic fraction and describes the magnetic field due to frozen electronic moments [9,12]. In this function the sum  Figure A2. The magnetic amplitude A m fraction extracted from the muon depolarization as a function of temperature for the 3 samples shown in Fig. A1.
A m + A n = 1 is constant at all temperatures. Figure A2 shows A m as a function of temperature, for the three samples in Fig. A1. Above the transition, where only nuclear moments contribute, A m is close to zero. As the temperature decreases, the frozen magnetic part increases and so does A m , at the expense of A n . For the pure AFM and SG phases, the transition temperature was determined as the temperature at which A m is half of the saturation value. For the samples with two transitions, two temperatures were determined using the same principle. The magnetic order parameter is extracted from the muon rotation frequency in zero field as seen in Fig. A1(a). These rotations allow us to determine ω(T, x, y). The temperature dependence of ω is used to determine the effective interlayer and anisotropic interaction α ef f in Fig. 3. ω(0, x, y) is used to determine the family-dependent critical doping in Fig. 6(b).

Appendix B. Pseudogap
We determine T * using temperature-dependent magnetization measurements. In Fig. B1(a) we present raw data from four samples of the x = 0.2 family with different doping levels. At first glance the data contain only two features: A Curie-Weiss (CW) type increase of χ at low temperatures, and a non-zero baseline at high temperature (∼ 300). This base line increases with increasing y. The CW term is very interesting but will not be discussed further here. The baseline shift could be a consequence of variations in the core and Van Vleck electron contribution or an increasing density of states at the Fermi level.
A zoom-in on the high temperature region, marked by the ellipse, reveals a third feature in the data: a minimum point of χ. To present this minimum clearly we subtracted from the raw data the minimal value of the susceptibility χ min for each sample, and plotted the result on a tighter scale in Fig. B1(b). The χ minimum is a result of decreasing susceptibility upon cooling from room temperature, followed by an increase in the susceptibility due to the CW term at low T . This phenomenon was previously noticed by Johnston in YBCO [29], and Johnston [30] and Nakano [31] et al. in La 2−x Sr x CuO 4 (LSCO). The minimum point moves to higher temperatures with decreasing oxygen level as expected from T * . There are three possible reasons for this decreasing susceptibility: (I) increasing AFM correlations upon cooling, (II) opening of a SG where excitations move from q = 0 to the AFM wave vector [32], or (III) disappearing density of states at the Fermi level as parts of the Fermi arc are being gapped out when the PG opens as T /T * decreases [33].
In order to determine the T * we fit the data to a three-component function χ = C 1 /(T + θ) + C 2 tanh(T ⋆ /T ) + C 3 . The fits are presented by solid lines in Fig. B1. The values of T ⋆ are shown in Fig. 2 and Fig. 6.

Appendix C. Doping and Impurities
The Cu-NQR experiment is done on powder samples fully enriched with 63 Cu. We measured between five and seven different samples for each x in the normal state at 100 K. The most overdoped sample is a non superconducting x = 0.1 compound. The NMR measurements were done by sweeping the field in a constant applied frequency f app =77.95 MHz, using a π/2-π echo sequence. The echo signal was averaged 100,000 times and its area evaluated as a function of field. The data are presented in Fig. C1. The full spectrum of the optimally doped x = 0.4 sample (y = 7.156) is shown in the inset of Fig. C1. The main planes emphasize the important parts of the spectrum using three axis breakers.
The evolution of the main peaks as x increases is highlighted by the dotted lines. It is clear that as x decreases the peaks move away from each other. This means that ν Q at optimal doping is a decreasing function of x. A more interesting observation is the fact that there is no change in the width of the peaks, at least not one that can easily be spotted by the naked eye. This means that the distribution of ν Q is x-independent and that there is no difference in the disorder between the optimally doped samples of the different families. Thus, as mentioned in Sec. 2, disorder is not relevant to the variation of T max c between the different families. This conclusion is supported by more rigorous analysis [10]. ν Q and ∆ν Q presented in Fig. 4 are obtained by fitting NQR line shape to this data as demonstrated by the solid lines.  Figure D1. Demonstrating the real and imaginary polarization in a TF-µSR experiment as the temperautre is lowered below T c . The data are presented in a rotating reference frame.
average spin polarization. This situation is demonstrated in Fig. D1 where we present P ⊥ (0) in two different perpendicular directions (called real and imaginary) in a rotating reference frame. At temperatures above T c the field is homogeneous and all muons experience the same field, and therefore no relaxation is observed. Well below T c (of 77 K in this case) there are strong field variations and therefore different muons precess with different frequencies, and the average polarization quickly decays to zero. At intermediate temperatures the field variation are not severe and the relaxation is moderate. It was shown that in powder samples of HTSC the muon polarization P ⊥ (t) is well described by P ⊥ (t) = exp(−R 2 µ t 2 /2)cos(ωt) where ω = γµH is the precession frequency of the muon, and R µ is the relaxation rate [18]. The solid line in this figure is the fit result. The fact that the whole asymmetry relaxes indicates that CLBLCO is a bulk superconductor. The fit results for R µ are shown in Fig. 5. As can be seen, the dependence of T c on R µ is linear in the under-doped region and universal for all CLBLCO families, as expected from the Uemura relations. However, there is a new aspect in this plot. There is no "boomerang" effect, namely, overdoped and underdoped samples with equal T c have the same R µ , with only slight deviations for the x = 0.1. Therefore, in CLBLCO there is one to one correspondence between T c and R µ , and therefore n s /m * , over the whole doping range.

Appendix E. Lattice parameters
Neutron powder diffraction experiments were performed at the Special Environment Powder Diffractometer at Argonne's Intense Pulsed Neutron Source (see Ref. [34] for more details). Figure E1 shows a summary of the lattice parameters. The empty symbols represent data taken from Ref. [34]. All the parameters are family-dependent, but not to the same extent. The lattice parameters a and c, depicted in Fig. E1(a) and (b), change by up to about 0.5% between the two extreme families (x = 0.1 and x = 0.4). The in-plane Cu-O-Cu buckling angle is shown in Fig. E1(c). This angle is non-zero since the oxygen is slightly out of the Cu plane and closer to the Y site of the YBCO structure. As mentioned in Sec. 2 it changes by 30% between families.