{'fraction_non_alphanumeric': 0.03584555340835891, 'fraction_numerical': 0.024932742222727445, 'mean_word_length': 4.3049246231155776, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Curie-like paramagnetism due to incomplete Zhang-Rice singlet formation in La2-xSrxCuO4

In an effort to elucidate the origin of the Curie-like paramagnetism that is generic for heavily-overdoped cuprates, we have performed high transverse-field muon spin rotation (TF-muSR) measurements of La2-xSrxCuO4 single crystals over the Sr content range 0.145<x<0.33. We show that the x-dependence of the previously observed field-induced broadening of the internal magnetic field distribution above the superconducting transition temperature Tc reflects the presence of two distinct contributions. One of these becomes less pronounced with increasing x and is attributed to diminishing antiferromagnetic correlations. The other grows with increasing x, but decreases above x ~ 0.30, and is associated with the Curie-like term in the bulk magnetic susceptibility. In contrast to the Curie-like term, however, this second contribution to the TF-muSR line width extends back into the underdoped regime. Our findings imply a coexistence of antiferromagnetically correlated and paramagnetic moments, with the latter becoming dominant beyond x ~ 0.185. This suggests that the doped holes do not neutralize all Cu spins via the formation of Zhang-Rice singlets. Moreover, the paramagnetic component of the TF-muSR line width is explained by holes progressively entering the Cu 3d_{x^2-y^2} orbital with doping.


I. INTRODUCTION
In the heavily-overdoped regime of hole-doped cuprates, T c decreases with increasing hole concentration p, the normal-state pseudogap vanishes, and a Curie-like temperature dependence of the normal-state bulk 1-10 and local [11][12][13][14][15] spin susceptibilities appears. In La 2−x Sr x CuO 4 (LSCO) where p = x, Oda et al. 3 reported a Curie-like term appearing in the bulk magnetic susceptibility χ at x ∼ 0.18, becoming more pronounced with increasing x, but weakening beyond x ∼ 0.30. They attributed the Curie-like contribution to localized moments that break Cooper pairs and drive down T c . However, the origin of such localized moments has never been established, and more recently this has prompted alternative explanations for the Curie-like behavior of χ. 16 While often attributed to magnetic impurities or oxygen-disorder induced defects, the Curie-like susceptibility appears to be a universal property of heavilyoverdoped cuprates, which has persisted through significant improvements in sample quality. In fact Takagi et al. 1 demonstrated early on that oxygen vacancies are not directly responsible for the Curie-like behavior. Nuclear magnetic resonance (NMR) studies of overdoped Tl 1−x Pb x Sr 2 CaCu 2 O 7 (Ref. 13) and Bi 2 Sr 2 CaCu 2 O 8+δ (Ref. 15) indicate that the source is paramagnetic moments localized in the CuO 2 planes. Wakimoto et al. 10 have proposed that beyond x = 0.22 where a Curie-like term appears in their measurements of χ, that 1/4 of the overdoped Sr ions may create the paramagnetic moments either via direct substitution of Sr on the Cu sites, or as a more plausible scenario, the holes they add enter the Cu 3d x 2 −y 2 orbitals. In either case, some fraction of the localized Cu magnetic moments are neutralized, breaking up the local AF order and liberating free Cu spins.
There is some recent evidence for holes predominantly entering the Cu 3d orbitals from Compton scattering measurements on a LSCO x = 0.30 sample. 17 This is a drastic departure from the widely accepted view that the doped holes mainly enter the O 2p band, creating Zhang-Rice singlets comprised of a Cu 3d x 2 −y 2 hole state hybridized with a coherent superposition of O 2p x,y orbitals from the four surrounding O atoms. 18 The formation of Zhang-Rice singlets justifies the use of an effective single-band Hubbard or t-J model for a satisfactory description of the low-energy physics of the cuprates, rather than a three-band model containing both the inplane Cu 3d x 2 −y 2 and O 2p x,y orbitals. Based on xray absorption spectroscopy measurements of LSCO and Tl 2 Ba 2 CuO 6+δ , Peets et al. 19 have proposed yet another scenario, whereby the doped holes in the O 2p x,y orbitals cease to hybridize with the Cu 3d x 2 −y 2 states beyond p ∼ 0.20. This would result in a loss of the oxygenmediated in-plane AF superexchange interaction between the localized Cu spins. However, this is incompatible with neutron scattering measurements on LSCO that show some remnants of AF correlations persisting to at least x = 0.25. 20 Hence it is likely that some degree of hybridization persists in LSCO beyond x ∼ 0.20.
Here we report high TF-µSR measurements on LSCO single crystals. Previous measurements of this kind revealed a T -dependent heterogeneous field response extending far above T c . [21][22][23][24][25] Through a systematic study of the width of the internal magnetic field distribution above T c as a function of Sr content x, MacDougall et al. 25 concluded that the most likely source of the heterogeneous line broadening are regions of staggered magnetization about the dopant Sr ions. In the present study we show that the T -dependent line broadening is associated with the Curie-like term in the bulk magnetic susceptibility. Consequently, a universal explanation is necessary to explain a similar p-dependent Curie term in χ for Tl 2 Ba 2 CuO 6+δ , 4 which is hole doped by increasing the oxygen content δ. In contrast to bulk magnetic susceptibility studies of LSCO, we make no assumptions about the T -dependent AF component to isolate the contribution of the localized paramagnetic moments. As a consequence, we find that the localized moments responsible for the Curie-like behavior of χ in heavily-overdoped samples of LSCO also exist at lower hole doping -gradually becoming more pronounced with increasing x. We attribute the lack of a Curie-like term in χ below x ∼ 0.18 to limitations of the assumed scaling function introduced by Johnston 26 to account for the AF correlations, and associate the paramagnetic moments with doped holes entering the Cu 3d x 2 −y 2 orbital.

A. Samples
The LSCO single crystals studied here were cut from travelling-solvent floating zone growth rods and subsequently annealed at elevated temperature and under oxygen partial pressure to minimize oxygen defects. The crystals were prepared by three different groups as follows: (i) Single LSCO crystals of Sr content x = 0.145, 0.15, 0.166, 0.176, 0.19, 0.196, 0.216, and 0.24 with T c = 37.3, 37.6, 37.3, 37.1, 31.4, 30.5, 28, and 17 K, respectively, were grown by S. Komiya. These crystals were annealed at 800 to 900 • C under low oxygen partial pressure (0.2 to 3.5 atm) for 200 hours. Measurements were also performed on an as-grown x = 0.19 single crystal prepared by S. Komiya, which has a substantially reduced and very broad superconducting transition temperature of T c = 23 ± 5 K. The x = 0.15, 0.166, 0.216, and 0.24 crystals were studied by zero-field (ZF) µSR in Refs. 27 and 28, and TF-µSR measurements of the x = 0.166 and 0.176 crystals were reported in Refs. 24 and 29. (ii) Six LSCO single crystals with x = 0.26 and T c = 12 ± 2 K, and two single crystals with x = 0.30 were synthesized by T. Adachi and Y. Tanabe. The x = 0.26 crystals were oxygen annealed at 1 atm and 900 • C for 50 hours, and subsequently slow cooled and kept at 500 • C for 50 hours. From chemical titration, the oxygen defect δ (defined as La 2−x Sr x CuO 4−δ ) is estimated to be 0.014 ± 0.01. The x = 0.30 crystals were oxygen annealed at 3 atm and 900 • C for 100 hours. Although we have not estimated δ for the x = 0.30 crystals, empirically we believe it is similar to the x = 0.26 sample. We note that bulk magnetic susceptibility measurements reveal superconducting dia-magnetism in a tiny volume fraction of the x = 0.30 crystals below T c ∼ 25 K. (iii) Two LSCO single crystals with x = 0.33 were supplied by N.E. Hussey. One of these was a 380 mg as-grown crystal. Bulk magnetic susceptibility measurements of this crystal down to 2 K show no signs of superconductivity. The other 33.7 mg single crystal was annealed for two weeks under an extreme oxygen partial pressure of 400 atm. Resistivity measurements show that this crystal exhibits no trace of superconductivity down to 0.1 K. Low-temperature ZF-µSR measurements have revealed a spin-freezing transition of unknown origin at 0.9 K. 27

B. Muon spin rotation
The TF-µSR experiments were carried out at TRIUMF in Vancouver, Canada, using the HiTime spectrometer, which consists of an ultra low-background sample holder contained inside a helium gas flow cryostat and a horizontal warm bore 7 T superconducting magnet. Measurements at temperatures above T ∼ 2 K were performed with an external magnetic field applied perpendicular to the direction of the initial muon spin polarization P(0), and parallel to the c-axis ( c) of the LSCO crystals. The TF-µSR spectra for each sample are well described by the sum of a small time-independent background component from muons that miss the sample and evade the background suppression system, and the following powerexponential depolarization function: where B is the magnitude of the average local magnetic field sensed by muons stopping in the sample, and φ i is the phase angle between P(0) and the axis of the i th positron detector (i = 1, 2, 3 and 4). The power exponential describes the relaxation of the TF-µSR signal, which occurs when there is a distribution of internal magnetic field. For static fields, a larger relaxation rate Λ signifies a broader field distribution, often referred to as inhomogeneous line broadening. If the local magnetic field sensed by each muon also fluctuates during its lifetime, motional narrowing of the line width occurs, such that Λ is reduced with increasing fluctuation rate. It is generally not possible to distinguish between static and dynamic depolarization of the TF-µSR signal.

III. RESULTS
A. Field-induced broadening above T ∼ 150 K [see Fig. 1(b)] indicates that the depolarization of the TF-µSR signal at these temperatures is dominated by the dipolar fields of the nuclear moments, whereas the fluctuation rate of the electronic moments becomes too fast to cause significant dynamic depolarization. Below T c there is an additional depolarization associated with the static inhomogeneous line broadening by vortices, which may or may not form a highly-ordered lattice. The vortex line broadening is proportional to λ −2 ab , where λ ab is the in-plane magnetic penetration depth. 30 The effect of the high-oxygen pressure post annealing on the temperature dependence of Λ at x = 0.33 is shown in Fig. 2. The difference between Λ in the as-grown and annealed crystals grows significantly with decreasing temperature. The higher values of Λ for the as-grown crystal is ascribed to a spatial distribution of hole doping caused by oxygen inhomogeneity, and in particular, electronic moments from more underdoped regions of the sample. Figure 3 shows the dependence of Λ on Sr concentration x at low temperatures. Below x ∼ 0.185 the dependence of Λ on x is opposite to the hole-doping dependence of the zero-temperature extrapolated quantity λ −2 ab (0). 31,32 This indicates that the width of the internal magnetic field distribution below x ∼ 0.185 is not dominated by static vortices. While the decrease of Λ with increasing x could be caused by motional narrowing of the line width due to rapid fluctuations of the vortices on the µSR time scale, Λ (and hence the line width) below T c increases with increasing magnetic field, 24,33 which is contrary to either a pure vortex-solid or vortexliquid state. Neutron scattering measurements on LSCO at H = 0 show that remnant AF spin correlations of the parent insulator La 2 CuO 4 vanish between x = 0.25 and x = 0.30. 20 At low temperatures the H = 7 T applied field appears to sufficiently stabilize the AF spin fluctutaions, such that the temporal fluctuations of the local magnetic field sensed by the muon do not average to zero. Here we note that while elastic neutron diffraction experiments indicate that a field of H ≥ 13 T is necessary to induce coexisting static AF order and superconductivity in x ≥ 0.145 samples, 34 non-uniform static magnetism is observed in the vortex state by TF-µSR at much lower fields. 29 The latter is explained by interlayer coupling that enables fluctuating magnetism to be stabilized by weakly interacting vortices. 35 With this said, the decrease of Λ with increasing x in Fig. 3 is explained by an increase in the Cu spin fluctuation rate, which reduces the dynamic depolarization of the TF-µSR signal.
Just above x ∼ 0.185, the near saturation of Λ in Fig. 3 indicates that the influence of the AF spin fluctuations on the TF-µSR line width becomes negligible. Since measurements of λ −2 ab (0) in 0.07 ≤ x ≤ 0.24 LSCO powders by Panagopoulos et al. 31 show a saturation of λ −2 (0) above x = 0.19, it appears that the TF-µSR linewidth immediately above x ∼ 0.185 is dominated by the static inhomogeneous field distribution of the vortex-solid phase. However, recent measurements of 0.06 ≤ x ≤ 0. 30  Above T c where the vortex contribution is absent, the component of Λ that increases with increasing x is more obvious and is seen to intensify up to x = 0.30 [see Fig. 4(b)]. At temperatures T > 60 K it becomes apparent that this component of Λ extends back to the lowest doping at x = 0.145. Hence it is clear that the source of this behavior is not restricted to heavily overdoped phase-separated samples, and does not onset at x ∼ 0.185. Instead x ∼ 0.185 appears to mark a lowtemperature crossover from a hole-doping range where the dominant electron spin contribution to Λ comes from remnant AF correlations of the parent compound, to a higher hole-doping range dominated by a different kind and/or arrangement of magnetic moments. The slight increase of Λ with increasing x at T = 200 K in Fig. 4(b) implies that the line broadening at this temperature is dominated by the nuclear dipole moments. This conclusion is also reached from inspecting the corresponding values of β in Fig. 4(a). A complete summary of our measurements of Λ as function of T and x is shown in Fig. 5.
The contribution χ 2D (x, T ) follows a simple universal scaling relation, 26 attributed to residual two-dimensional (2D) AF correlations in the CuO 2 planes. In particular, χ 2D (x, T )/χ 2D max = F (T /T max ), where χ 2D max and T max are x-dependent values corresponding to the maximum of χ 2D (x, T ), and F is a universal curve. The Tindependent term χ 0 (x) in Eq. (2) is the sum of the atomic core diamagnetism χ core , the anisotropic Van Vleck paramagnetism χ VV and the Pauli paramagnetism χ Pauli . As discussed later, the x dependence of χ 0 is caused by a variation of χ Pauli . The third term in Eq.
(2) has a Curie-like T -dependence that appears above x ∼ 0.18. 3,6,10 The Curie constant C(x) increases with increasing x above x ∼ 0.18, signifying the growing presence of localized paramagnetic moments, but decreases beyond x ∼ 0.30. The local spin susceptibility causes a muon Knight shift, which is defined as the fractional difference between the average magnetic field B at the muon site and the applied field H. Correcting for macroscopic contributions to B that are present in the external field, the Knight shift originating from microscopic contributions in the sample is where K dem,L = 4π(1/3 − N )ρ mol χ is a correction for demagnetization and Lorentz fields, N is the demagnetization factor for the sample, ρ mol is the molar density of the sample in units of mol/cm 3 , and χ is the bulk molar susceptibility displayed in Fig. 6(a). The applied field induces spin polarization of both the conduction electrons and localized electronic moments, such that The first term is a T -independent contribution from the Pauli spin paramagnetism of the conduction electrons that screen the µ + charge. The second term is Tdependent and arises from the polarization of localized Adip depend on the µ + site and the crystallographic structure, and are given by where the sum extends over all localized moments inside a Lorentz sphere, and r is the magnitude of the vector r = (x, y, z) connecting the muon site to the localized moment site in the crystal lattice. The hyperfine contact coupling, which is often considered as isotropic and Tindependent, results from further spin polarization of the conduction electrons by induced localized moments via the Rudermann-Kittel-Kasuya-Yoshida (RKKY) interaction. In cuprates, an RKKY interaction may occur between localized Cu spins and doped itinerant holes. Calculations show that the RKKY interaction is enhanced with increasing hole concentration, and may dominate over the antiferromagnetic superexchange coupling in the overdoped regime. [40][41][42] Figures 6(b) to 6(d) show the normal-state Knight shift (for H c) of the x = 0.15, x = 0.24 and x = 0.33 single crystals, corrected for demagnetization and Lorentz fields, and plotted versus χ . For a wide range of temperature above T c , K exhibits linear scaling with χ , indicating an equivalency of the bulk and local magnetic spin susceptibilities. The saturation of K at high temperatures for the x = 0.24 sample likely signifies the resolution limit of the measured TF-µSR frequency shift. Likewise, the Knight shift at x = 0.33 appears to be too small to measure accurately with the spectrometer used in our experiments.
The most notable feature of the K versus χ plots is the change in the slope from positive at x = 0.15 to negative at x = 0.24. A positive slope was previously reported for x = 0.07. 23 The slope is equal to the effective hyperfine coupling tensor element A eff for H c. Unfortunately, the present measurements for a single direction of applied magnetic field are insufficient for determining the dipolar and contact hyperfine couplings. The problem is compounded by the apparent presence of both AF correlated spins and local paramagnetic moments. Consequently, to explain the changes in slope we resort to a heuristic argument. Recently it has been determined that the muon stops near the non-axially symmetric site (0.120a, 0, 0.219c). 28 At this site and with H c, the calculated dipolar coupling constant for antiferromagnetically ordered Cu spins (canted by 45 • or lying in the a-b plane) is A dip = −0.33 kG/µ B . For parallel Cu spins aligned along the applied field direction, A dip = 0.97 kG/µ B . Hence the change in the sign of the slope between x = 0.15 and x = 0.24 can be explained by a change in the primary alignment of the Cu spins. However, the contact hyperfine coupling constant A c is also expected to vary with x, not only because of a change in the effective exchange interaction between the Cu moment and the conduction electrons, but also as discussed in the next section, because the density of electronic states at the Fermi energy is x-dependent.

IV. DISCUSSION
We have shown that the normal state of LSCO single crystals from three different sources is characterized by a broad field-induced distribution of internal magnetic field. The dependence of the TF-µSR relaxation rate Λ on x indicates two distinct T -dependent contributions. The first of these decreases with increasing x and is clearly associated with diminishing remnant AF correlations. While neutron scattering 20,43,44 and ZF-µSR 45 measurements on pure and Zn-doped LSCO in zero external magnetic field suggest that AF spin fluctuations likely persist out to the termination of superconductivity in the overdoped regime, the effect of AF correlations on the TF-µSR line width at H = 7 T is apparent only up to x ∼ 0.185. This is partly due to the diminishing effect of the applied magnetic field in stabilizing the increasingly faster AF fluctuations, but it is also a consequence of a second contribution to the TF-µSR line width that grows with increasing x. We find, however, that this second source of line broadening is greatly reduced when superconductivity completely disappears beyond x = 0.30.
Since the Knight shift K exhibits a linear scaling with χ for a wide temperature range above T c , it is instructive to consider the µSR results in the context of comprehensive studies of the normal-state bulk magnetization of LSCO. To begin with, the T -independent component of the bulk magnetic susceptibility χ 0 is strongly dependent on the hole concentration. Measurements on samples in the range 0.10 ≤ x ≤ 0.45 show that χ 0 increases with increased hole doping, plateaus above x ∼ 0.20 to 0.23, and decreases above x ∼ 0.30. 6,46,47 While the T -independent component of the total static magnetic susceptibility includes contributions from the core diamagnetism and Van Vleck paramagnetism, the doping dependence of χ 0 has been attributed to changes in the Pauli susceptibility χ Pauli . This is because the x dependences of χ 0 and the normal-state electronic specific heat coefficient γ are very similar below x ∼ 0.24, 46 and in a free-electron system χ Pauli and γ are each proportional to the density of states N (E F ) at the Fermi energy. The x dependences of χ Pauli and γ can be explained by the presence of a van Hove singularity (VHS) in the electronic density of states caused by the flat part of the conduction bands of LSCO being very close to E F . With increasing x, the (π, 0) flatband and VHS move to higher energy and cross E F at x ∼ 0.20, 48,49 where N (E F ), 50 and hence both χ 0 and γ reach their maximum values.
The increase of χ 0 with increasing x up to x ∼ 0.20 is accompanied by a reduction of the effective moment of the localized Cu spins, 6,46 due to the frustration effect of the doped holes in the O 2p x,y orbitals on the AF exchange interaction between Cu spins. As mentioned above, the suppression of the AF correlations of the localized Cu spins is the source of the initial decrease of Λ with increasing x in Figs. 3, 4(b) and 5. On the other hand, we attribute the additional T -dependent contribution to Λ that grows with increasing x to the same source of the Curie-like constant C(x) in the bulk magnetic susceptibility above x ∼ 0.18. To see this is the case, in Fig. 7 we compare C(x) from Ref. 3 to Λ(x) at T = 80 K, where the AF contribution is minor. Note as expected, the interpolated data of Ref. 3 falls between the two Λ values at x = 0.33, which correspond to the as-grown and high-oxygen pressure annealed extremes of sample preparation. While the Λ(x) data points at x = 0.33 in Fig. 7 seem to suggest that the decrease of C(x) beyond x = 0.30 is a sole consequence of annealing, the data at temperatures below T = 80 K in Fig. 4(b) clearly indicates that this is not the case. In fact the T -independent component χ 0 of the bulk magnetic susceptibility of the as-grown crystal at x = 0.33 is larger than that of the annealed crystal, such that the former resembles an annealed sample at a somewhat lower value of x.
In contrast to C(x), the second T -dependent contribution to Λ(x) extends below x = 0.18 into the underdoped regime. Strictly speaking, the Curie-like term observed in overdoped LSCO in Refs. 3 and 6 is the deviation of the bulk magnetic susceptibility from the assumed Tdependent form for χ 2D (x, T ) in Eq. (2), which is described by the universal scaling function F introduced by Johnston. 26 Consequently, we ascribe the separation of the data for C(x) and Λ(x) below x ∼ 0.21 in Fig. 7 to an inaccuracy in the assumed form for χ 2D (x, T ). In actuality, Eq. (2) is only valid if the doped holes in the O 2p x,y orbitals are weakly hybridized with the Cu 3d x 2 −y 2 states, 5 in which case χ 0 (x) and χ 2D (x, T ) can be treated as distinct terms. According to the x-ray absorption spectroscopy measurements by Peets et al. 19 , this may be the case only at x > 0.20, where we find that the xdependence of Λ closely follows C(x).
Based on the following observations we conclude that the paramagnetic moments are the result of doped holes that do not stimulate the formation of Zhang-Rice singlets: (i) NMR measurements indicate that the source of the Curie-like paramagnetism resides in the CuO 2 lay-ers, (ii) the Curie-like behavior is observed in cuprates hole-doped by either cation substitution or changes in oxygen content, (iii) we find good agreement between samples synthesized by different groups, and (iv) we have demonstrated that the localized electronic moments responsible also exist in the underdoped regime. Recently, Sordi et al. have shown that doublet formation (total spin S = 1/2) in a one-band Hubbard model grows upon doping the parent Mott insulator. 51 The same is true of the S = 1/2 three-spin polaron present in three-band models, 52,53 where the spin of a hole on the in-plane oxygen is antiparallel to the two nearest-neighbor copper spins. However, the hole in these and the triplet state (total spin S = 1) is mobile, with the effective magnetic moment presumably fluctuating too fast to cause dynamic depolarization of the TF-µSR signal.
The paramagnetic moments are more likely caused by a growing number of holes entering the Cu 3d orbitals, as evidenced by recent Compton scattering measurements of LSCO. 17 Localized spins are created by holes entering the out-of-plane Cu 3d z 2 −r 2 orbital, but this does not explain the reduction of C (and Λ) beyond x ∼ 0.30. Doped holes entering the half-filled in-plane Cu 3d x 2 −y 2 orbital neutralize existing Cu spins, hence disrupting local AF order and creating free Cu spins. With doping, an increasing number of free Cu spins are generated in this way until the disappearance of AF correlations in the overdoped regime. The additional holes introduced by further doping progressively neutralizes the free Cu 3d x 2 −y 2 spins, in agreement with the reduction of the Curie-like paramagnetism beyond x ∼ 0.30. This appears to be a universal property of the cuprates, and consequently is a restriction on the applicability of one-band models.
We thank G. A. Sawatzky and A.-M.S. Tremblay for informative discussions, and both the technical staff at the TRIUMF Centre for Molecular and Materials Science and N. Mangkorntong for technical assistance. This work was supported by the Natural Sciences and Engineering Research Council of Canada, and the Canadian Institute for Advanced Research. N.E. Hussey acknowledges the support of a Royal Society Wolfson Research Merit Award.