Oxygen vacancy-induced magnetic moment in edge-sharing CuO2 chains of Li2CuO2−δ

Li2CuO2 is a typical charge transfer insulator with CuO2 chains that are composed of edge-shared CuO4 plaquettes. The existence of oxygen vacancies for single crystals prepared under various oxygen partial pressures has been confirmed by the chemical and thermogravimetric analyses. The puzzling discovery of extra magnetic moment near the oxygen site by earlier neutron scattering studies has been verified by a thorough Curie–Weiss law analysis of spin susceptibilities, and resolved quantitatively with a molecular orbital model of edge-sharing CuO2 chains containing oxygen vacancies.

, the conventional coupling constants J i based on the Hamiltonian  = J S S i j ij i j å < · must have the J i = 2 J i * relationship for a consistent comparison. The J values shown in the 4th and 5th rows of table 6 in the paper by Shu et al should be doubled, i.e., J 1 for δ ∼ 0 and 0.16 are ∼130 K and 122 K, respectively, which is also consistent to the J 1 estimation as shown in the figure 2 of reply to the comment raised by Kuzian et al [3]. In addition, strictly speaking, it is not appropriate to position J¢/ J in the J 3 /J 5 columns, because the former is for the dipole-dipole inter-chain coupling for each FM chain as a unit, and the latter is for inter-chain individual spin exchange coupling.

Introduction
Li 2 CuO 2 is a charge-transfer insulator composed of weakly coupled CuO 2 chains with Li atoms in the interstitial sites, where each chain is formed with edge-shared CuO 4 plaquettes, as shown in figure 1. While the CuO 4 plaquette is the building block of the Cu-O plane as the common signature of high T c cuprate superconductors [1], Li 2 CuO 2 has been explored intensively in parallel, including its orthorhombic crystal structure [2], antiferromagnetic spin ordering of T N ∼9 K [2], field-induced ferromagnetic transition below ∼2.8 K [3], charge and orbital excitation [4], and Zhang-Rice type excitations [5]. In particular, neutron scattering studies suggest that oxygen has a magnetic moment of ∼0.11 μ B in Li 2 CuO 2 [6], which is puzzling and has been examined theoretically to suggest a strong moment transfer to the oxygen ions due to orbital hybridization [7].
While most of the studies of Li 2 CuO 2 assumed that the studied samples were stoichiometric without considering the Li or O defects, it was also found that the effective moments fitted from the paramagnetic regime of the spin susceptibility data varied from ∼1.8 to 2.07 μ B for samples prepared using different preparation conditions [3,[8][9][10]. Clearly, strong sample-dependent studies have been made without careful chemical composition analysis, especially on lithium and oxygen non-stoichiometry, which could invalidate the theoretical calculations and experimental interpretations made for the perfectly stoichiometric compound. In addition, the effective magnetic moments per Cu 2+ estimated from the Curie-Weiss law fitting were found to be consistently larger than the spin-only value for S=1/2 of 1.732 μ B (assuming g=2), which has often been attributed to the unquenched orbital contribution due to strong d-p hybridization, but a reasonable molecular orbital model remains to be established to illustrate the proposed hybridization. We believe that the role and stoichiometry of lithium and oxygen in Li 2 CuO 2 should be re-examined in detail, especially from a molecular orbital model approach, which allows a clear picture of localized spins that correspond to the unpaired electrons in hybridized orbitals of copper and oxygen.

Experimental details
Li 2 CuO 2−δ single crystals were grown using the optical floating-zone method under various pressure and oxygen partial pressures. The initial feed rod was prepared using a stoichiometric 2:1 ratio of dehydrated LiOH and CuO through mixing and grinding. The initial heating was performed at 470°C for 12 h, and the final annealing was performed at 550°C under oxygen flow. In the final annealing period, feed rods of ∼3-5 mm in diameter and ∼8-10 cm long were prepared with a hydraulic press for the optical floating-zone crystal growth. Li 2 CuO 2 melts congruently and the crystal can be pulled from the stoichiometric feed rod through stages from polycrystalline to single crystalline after the first ∼10 mm of pulling. Although lithium loss was expected for its high vapor pressure in the liquid state [11], to control the lithium and oxygen loss, mixed O 2 /(O 2 +N 2 ) gas (from 20% to 100%) of pressure up to 7 bar has been applied and reported.
Three chemical composition analysis methods have been applied to all crystal samples, including the inductively coupled plasma (ICP) mass spectrum, which focused on the Li and Cu content, the electron probe microanalysis (EPMA), which focused on the Cu and O ratio, and finally an independent oxygen content check, which was performed using the oxygen/nitrogen combustion analyzer (EMGA-920, Horiba). The O 2 /N 2 combustion technique provides an independent and direct method for oxygen content determination. We found that a high pressure atmosphere does help to reduce Li loss during growth, although Li loss can also be compensated effectively with approximately 10% Li excess in the feed rod for growth at ambient pressure. A series of crystal samples that grew under 7 bar gas pressure and have been confirmed to be without Li vacancy were investigated in this work. However, it was found that even if a high pressure (7 bar) of 100% pure oxygen was maintained in the floating-zone furnace growth chamber, oxygen vacancy was unavoidable to show δ as high as ∼0.16±0.01. Oxygen content can be tuned accurately using low-temperature post annealing with the thermogravimetric analysis method under oxygen flow. For example, Li 2 CuO 2.00±0.01 powder can be prepared through post-annealing at 400°C for two days using the as-prepared sample of Li 2 CuO 1.84±0.02 . Homogeneous oxygen vacancy-free single crystal sample was prepared with an additional ultrahigh oxygen pressure annealing, using a cubic anvil apparatus with mixed oxidizer of KClO 4 at 6 GPa and 800°C for 30 min.
The transmission electron microscope (TEM) samples were prepared by crushing Li 2 CuO 2−δ single crystal samples into small pieces, dropping them onto a carbon-coated Cu grid, and rapidly loading them into the TEM column to avoid possible oxidation in air. Electron energy loss spectroscopy (EELS) spectra of Li 2 CuO 2−δ were acquired at 200 kV with a JEOL-2100 field emission TEM equipped with a Gatan Tridiem 863 system.

Results and discussions
3.1. Phase purity and structure analysis The phase purity for the growth of Li 2 CuO 2−δ under various oxygen partial pressure (p O 2 ) has been examined using synchrotron x-ray, as shown in figure 2(a). Minor impurity phase of LiCu 2 O 2 can only be identified in the crystals grown under 100% Argon atmosphere at 7 bar pressure, which is consistent to that reported by Wizent et al on the optical floating-zone growth of Li 2 CuO 2 under 40 bar gas pressure [11]. The lattice parameters for Li 2 CuO 2 samples prepared under various oxygen partial pressures at 7 bar are summarized in table 1, where no significant oxygen partial pressure dependence is found and these refined values are in good agreement with those reported in the literature [12]. On the other hand, the structural refinement results indicate that a significant oxygen partial pressure dependence is observed on the average Cu-O bond length and O-Cu-O bond angle, as summarized in figure 2(b). The combined observations of insensitive lattice size change with significant change on the average bond length/angle strongly suggest the existence of randomly distributed oxygen vacancies, without breaking the bulk crystal symmetry for the emergence of an impurity phase. In order to examine this hypothesis, independent chemical analyses are required.
The bond angles shown in figure 2(b) revealed one more interesting phenomenon on bond angle also. For an ideal edge-sharing CuO 2 chain, it is expected that the O-Cu-O bond angle should be close to 90°considering the d-p hybridization. However, it is found that the O-Cu-O bond angle for sample with nearly zero oxygen vacancy still shows a bond angle approaching ∼91°. We believe such rectangular distortion is related to the Coulomb attraction between -O 2 and Li + arrays along the c-direction (see figure 1). In fact, similar rectangular distortion has also been found consistently in compounds containing edge-sharing CuO 2 chains, including Ca 2 Y 2 Cu 5 O 10 and La 6 Ca 8 Cu 24 O 41 [13].

Lithium and oxygen stoichiometry
Both lithium and oxygen contents of Li 2 CuO 2−δ have been cross examined using three different techniques, including ICP which focused on the ratio of Li:Cu, EPMA for the Cu:O ratio when Li is too light for an accurate determination, and the oxygen content which is determined directly and independently with an oxygen combustion analyzer, as summarized in table 2.
The Li content has been found to be constant near 2 per formula unit consistently within error, especially for crystals grown under 7 bar atmosphere or with feed rod containing 10% Li excess at the ambient pressure. The oxygen contents are significantly lower than 2 per unit when the EPMA is normalized relative to Cu, which is also verified by the oxygen combustion analysis independently with the expected 29.24% by weight for oxygen in the vacancy-free sample. Moreover, starting from a sample with a determined oxygen vacancy level, the oxygen content can also be tuned in situ and the final content could reach as high as 2.00(1) after the as-grown crystal sample is annealed in O 2 atmosphere at 400°C for two days, as demonstrated in figure 3. It is clear that the existence of oxygen vacancy defects is common and severe in Li 2 CuO 2−δ at the level of δ∼0.15-0.3, even for crystals grown under the high-pressure pure oxygen atmosphere up to 7 bar.

Magnetic susceptibilities
The magnetic susceptibilities for single crystal samples grown under various oxygen partial pressures at 7 bar atmosphere are confirmed to be paramagnetic (PM) at high temperature and with an antiferromagnetic (AF) ground state of T N ∼9 K, similar to those reported in the literature [2,14]. Since the powder data could average out the orbital anisotropy, and the Curie-Weiss law fitting is valid for > T J k B in the true PM state only, we have examined the susceptibility data using anisotropic single crystal measurement results first. The representative anisotropic measurement results of single crystal Li 2 CuO 2−δ (δ∼0.16) are displayed in figure 4. A complete Curie-Weiss law fitting of ( ) has been performed for magnetic field applied along the three major crystallographic orientations using data above ∼250 K, as summarized in table 3.
The powder-averaged homogeneous susceptibility (χ=M/H) of an orthorhombic system can be calculated from the anisotropic single crystal measurement results as For a perfect powder sample preparation, it is expected that Curie constant m = C N k 3 eff 2 B and g value obtained directly from the powder measurement should be close to the powder-average of the anisotropic measurement results following Table 1. The lattice parameters for Li 2 CuO 2−δ crystals grown under different oxygen partial pressures at 7 bar. Homogeneous oxygen vacancy-free crystal has been prepared with additional ultrahigh oxygen pressure (+HP) annealing.  crystals grown under different oxygen partial pressures at 7 bar and the vacancy-free crystal prepared with additional ultrahigh oxygen pressure (+HP) annealing, chemical composition analysis results from three methods are shown, including ICP (normalized to Cu), EPMA (normalized to Cu), and Oxygen Combustion Analysis.
respectively. The powder-averaged Curie constant C and the g-factor can be derived from equations (2) and (3) using the single crystal anisotropic measurement results, as shown in table 3. Alternatively, a direct Curie-Weiss law fitting can be applied to the powder averaged data of χ(T) which are derived from the single crystal measurement results with equation (1), as summarized in table 4. Comparing the C and g values derived from the single crystal anisotropic measurement results (table 3) and those obtained from the direct fitting using the powder averaged data of χ(T) (table 4), the satisfactory agreement suggests that the magnitude of the extracted Curie constant is accurate and reliable without potential contamination from the anisotropic orbital contribution due to preferred orientation and the temperature-independent terms. The temperature-independent term ( • c ) of χ(T) for Li 2 CuO 2 includes the core diamagnetic (c core ), Van Vleck paramagnetic (c VV ) , and Pauli paramagnetic (c Pauli ) contributions. Since c = 0 Pauli for the insulator,   [15], the c VV is deduced to be in the order of 10 −6 cm 3 mol -1 experimentally, which is found about one order smaller than those reported for some representative cuprate compounds, including CuBr 2 , LiCuVO 4 , and YBa 2 Cu 3 O 7 [16][17][18]. On the other hand, for the theoretical c VV derived via the second order perturbation of energy as c = ¶ ¶ E H 2 2 [19], such effective orbital Zeeman effect would contribute significantly only when the orbital splitting is small. Since the d 3 -orbital degeneracy of Cu 2+ in a square planar CEF is lifted into a four-fold splitting of b 1g -b 2g -a 1g -¢ e g [20], the sizable CEF gap is not expected to produce significant contribution to c VV . In fact, no matter how these nine electrons in 3d 9 are distributed within the four-fold splitted energy levels, no significant Zeeman energy gain is expected under the field.
For an ideal PM behavior which is best described by the Curie-Weiss law for isolated spins under the competing influence of temperature and field, it is expected that the thermal energy must overcome the nearest neighbor spin exchange coupling J. The exchange coupling constant reported in the literature for Li 2 CuO 2 has been shown in the range of -J k 100 228 K B [10]. The temperature range of 250-550 K used in the Curie-Weiss law fitting must safely satisfy the condition of > T J k B , as well as the Curie's law approximation condition of  in the Brillouin function [21]. The data of 1/χ are shown to deviate from the high temperature linear fitting below ∼135 K (figure 4), which suggests that the thermal energy becomes weaker than the spin coupling strength, and the onset of 1/χ linear deviation could be used as a rough estimate to the nearest neighbor spin exchange coupling. Following the mean field approximation of Weiss temperature Q µ å z J i i i [22], it is implied that the strongest spin exchange coupling J k B for Li 2 CuO 2 must be near ∼135 K, which is in good agreement with those estimated by the J 1 -J 2 -J c model and the molecular quantum chemical calculations by treating La 2 CuO 2 as an edge-sharing spin chain system [14,23].

Copper and oxygen valence
Based on the ionic model description, the existence of oxygen vacancy in Li 2 CuO 2 implies the emergence of Cu + following the requirement of neutrality as ( ) It is curious to examine this assumption by checking the actual copper valence change as a function of δ. The EELS spectra of the oxygen K-edge and copper Table 3. Curie-Weiss law fitting of ( ) ( ) for the anisotropic measurement of single crystals Li 2 CuO 2−δ of δ∼0.16 and 0. Powder averaged values of C and g are derived using equations (2) and (3), and under the assumption of the existence of single type of copper spin (S=1/2) in the fitted temperature range.

Single crystal
Fitting range 250-550 K δ∼0.16   Table 4. Curie-Weiss law analysis using powder averaged data of the anisotropic single crystal measurement results, i.e., fitting directly using the χ(T) data derived from the anisotropic measurement results with equation (1) between 250 and 550 K. g values are evaluated under the assumption of the existence of single type of copper spin (S=1/2) in the fitted temperature range. L-edge have been widely used as a typical technique for valence determination in transition metal oxides [24]. The EELS spectra of O-K and Cu-L 2,3 edges for both stoichiometric Li 2 CuO 2 and oxygen-deficient Li 2 CuO 1.84 samples are shown in figure 5(a), respectively, with the background subtracted. The Cu-L 2,3 -edge (figure 5(a)) displays two sharp spectral peaks at ∼935 and 955 eV corresponding to the L 3 , and L 2 ionization edge, respectively [25]. The spectral feature of the Cu-L 2,3 -edge is nearly identical for Li 2 CuO 2 and Li 2 CuO 1.84 , which implies that the copper valence does not change due to oxygen vacancy creation. The intensity ratio of L 2,3 peaks (L L 3 2 ) has also been used to analyze the valence change qualitatively; for example, Co 4+/3+ valence change has been supported by the Co L 3 /L 2 ratio [26]. The L L 3 2 ratios of CuO, a typical compound of Cu 2+ with 3d 9 configuration, and Cu 2 O, a typical compound of Cu + with 3d 10 configuration, have been shown to be close to ∼3.3 and ∼2.3, respectively [25,27]. Our results indicate that the L L 3 2 intensity ratios for both Li 2 CuO 2 and Li 2 CuO 1.84 are nearly identical (∼3.3±0.1) and very similar to that of CuO, which suggests that all Li 2 CuO 2−δ samples have an identical copper valence close to Cu 2+ , and that the copper valence does not change with the level of oxygen vacancy within the range of δ studied in this work.
In addition to the observation of copper L 2,3 -edge, two spectral peaks near ∼532 and 540 eV are assigned to the oxygen K-edge. Compared the O-K edge peaks, no significant peak shift is found to indicate an oxygen valence change, although the intensity of 532 eV peak is shown reduced significantly for Li 2 CuO 1.84 . Similar 532 eV peak reduction has also been observed in several other transition metal oxides [26,28,29], which could be closely related to the oxygen non-stoichiometry.

Magnetic moment near the oxygen vacancy site
For the stoichiometric Li 2 CuO 2 as a charge-transfer insulator [4], it is expected that the localized spin of Cu 3d 9 following the Curie-Weiss law should have a Curie constant equal to 0.374 cm K mol  with oxygen vacancy level, so that the derived m eff values are larger than the expected spin-only value of 1.732 μ B for d ¹ 0, as shown in figure 6 and tables 3, 4. Although it is convenient to assign the excess contribution of m eff to the unquenched orbital contribution, or to the possible large moment transfer between Cu-d and O-p orbitals qualitatively [7], a quantitative magnetic moment analysis is expected. Moreover, if we consider that all detected spins are coming only from copper, then an oxygen vacancy is expected to create the effect of an n-type electron doping to the system as ( ) under the ionic model assumption, which would lead to fewer total spins due to the increased amount of spinless Cu + (3d 10 ) at higher δ, as predicted by the equation of as shown in figure 6, where • N is the Avogadro number and k B is the Boltzmann constant. Unexpectedly, the ionic model prediction for ( ) d C has been shown failed to describe the experimental results correctly, as illustrated in figure 6. Since a molecular model using hybridized Cu-d and O-p orbitals is required to describe the physical properties of Li 2 CuO 2−δ correctly, as hinted by the spin-polarized local density approximation studies [7], we propose that based on a molecular orbital model of Li 2 CuO 2−δ with oxygen vacancy existence, each O-vacancy may donate at most two electrons (holes) to the system being either itinerant as free charge carriers or localized showing detectable spins. Since the Seebeck coefficients measured between 300 and 650 K ( figure 5(b)) indicate that the doped carrier for Li 2 CuO 2−δ (δ∼0.16 and 0.29) is p-type, the localized and unpaired electrons (holes in the valence band) must have energy below the Fermi level and act as localized spins to follow the Curie-Weiss law.
Starting from the inspiring observation that m eff shows the spin-only value of 1.732 μ B (i.e., C=0.374 cm K mol 3 1 for S=/1/2 with g=2) for δ≈0 only ( figure 6 and table 4), a modified Curie-Weiss law is proposed to correlate that every oxygen vacancy (δ) may generate two localized spins of ¢ = S 1 2, in addition to the existing Cu 2+ of S=1/2. As a result, the Curie constant should include both contributions proportional to the localized spins corresponding to Cu and oxygen vacancy, i.e., i.e., the C total versus δ plot is expected to have a linear relationship with an intercept of 0.374 cm K mol 3 1 and a slope of 0.374×2. For the five experimental data points of C total versus δ plotted in figure 6, a linear regression returns a slope close to 0.374×2, as predicted by equation (6) within experimental error. In particular, only sample of zero oxygen vacancy shows the Curie constant being close to 0.374 cm K mol 3 1 as the spin-only value for S=1/2 with g=2. In addition, based on the successful verification of the re-interpreted Curie-Weiss law analysis expressed by equation (6), earlier qualitative assumption of single type copper spin of S=1/2 with g>2 (see tables 3 and 4) has been reinterpreted satisfactorily by both the copper spins of S=1/2 with g=2 and the localized spins of ¢ = S 1 2 with g=2 generated by oxygen vacancies quantitatively. Figure 6. The Curie constant versus δ of Li 2 CuO 2−δ obtained from the Curie-Weiss law fitting of spin susceptibilities. The two straight lines correspond to the two models of ( ) d C described by equations (5) and (6), which clearly shows the validity of the proposed model described by equation (6).
Current results strongly support the fact that an additional magnetic moment can be generated from oxygen vacancies. These results are consistent with the proposals supported by the μSR and neutron scattering experiments [6,30], although early studies ignored the possible existence of oxygen vacancies completely. Based on the current study, the sample studied by Chung et al showing an oxygen moment of ∼0.11(1) μ B may correspond to an oxygen vacancy level of at least δ∼0.1 [6], which is consistent to the growth atmosphere studies reported in this work also.

Molecular orbital model of localized spins
The proposed molecular orbital model of Li 2 CuO 2 has also been examined using density functional theory (DFT) calculations, as shown in figure 7(a). DFT calculations, with the B3LYP functional and the 6-31G(d) basis set implanted in the Gaussian 09 package [31], have been performed to visualize the corresponding molecular orbitals and bonding characters of a single CuO 4 plaquette. The natural bond orbital analysis shows that Cu donates 4s, 4p x , 4p y , and d 4 z 2 atomic orbitals to form four hybridized Cu-O bonds, and the s, p, and d orbitals have contributions of ∼23%, 50%, and 27% to each bond, respectively, in reasonable agreement with the requirement of an sp 2 d orbital hybridization, as illustrated in figure 7(a). The σ bond formed with hybridized Cu-4sp 2 d orbitals and O-2p x/y orbitals in a CuO 4 square planar shape shows the HOMO state of a binding energy ∼5.5 eV. We find that the binding energy of the σ bond between Cu-O is consistent with the excitation of ∼5.4 eV obtained from the resonant inelastic x-ray scattering study conducted previously [4,5].
Based on the proposed molecular orbital model of an edge-sharing CuO 2 chain, we can examine the interesting consequence when one of the oxygen atom is removed from the chain, as illustrated in figure 7(b). Each oxygen vacancy is expected to create two dangling bonds (unpaired electrons) near the outer valence shell of Cu in proximity to the oxygen vacancy site. The oxygen vacancies can generate unpaired electrons at the excited state, but they are also trapped near the oxygen vacancy site, as reflected by the conducting property of a charge transfer insulator with detectable extra localized spins.
According to the proposed molecular orbital model of Li 2 CuO 2−δ with oxygen vacancy existence, it is also interesting to note that to avoid the destruction of an edge-sharing CuO 2 chain, at most one in six oxygen atoms can be missing from every two adjacent edge-shared CuO 4 plaquettes (see figure 7(b)), i.e., δ=0.33, which has been supported by the fact that the largest δ is found near ∼0.30 before decomposition. Moreover, no Li 2 CuO 2−δ crystal could be grown when the oxygen partial pressure is lower than ∼20%, which has also been verified by the failed attempt on floating-zone crystal growth conducted in Ar atmosphere up to 40 bar [11].

Emergent FM moment at low temperature
Although both T N and Θ of Li 2 CuO 2−δ show no significant δ-dependence (table 4), it is noted that emerging FM moment is observed for samples with d ¹ 0 below ∼3 K, as shown in figure 8. In contrast, no FM moment is detectable for sample with δ∼0, as verified by the non-hysteretic M(T) behavior and the zero spontaneous moment below ∼3 K (inset of figure 8). Similar FM moment has also been identified and confirmed with neutron scattering on the nominal Li 2 CuO 2 sample without oxygen content verification, where the observed FM moment was proposed due to spin canting [3,6].
Based on the confirmed 3D AF spin structure of antiferromagnetically coupled FM chains for Li 2 CuO 2 [2,6], as illustrated in figure 9(a), the magnetic moments near the randomly distributed oxygen vacancy sites must be sitting in the 1D (along the b-direction) and 2D (within ab-plane) FM environment constructed by the Cu 2+ spins. At high temperatures well above T N , both the Cu and oxygen moments show PM behavior, as described satisfactorily by the revisited Curie-Weiss law data analysis shown above. On the other hand, for temperatures well below T N , the localized spins due to oxygen vacancies are distributed randomly and at the excited state, which are expected to order at a temperature much lower than T N . In particular, the isolated spins near the oxygen vacancy sites are expected to be aligned by the local FM environment of the Cu 2+ spins in 1D and 2D (see figure 9), so that the total FM moment per ab-layer does not cancel out along the c-direction completely, similar to the ferrimagnetic ordering for the intermediate spins of Co 3+ which are built on the matrix of the A-type AF ordered Co 4+ spins below T N in Na 0.82 CoO 2 [32]. We have estimated that the spontaneous FM magnetization at 2 K for δ∼0.16 of ∼20 emu mol -1 (inset of figure 8) corresponds to about ∼0.1 μ B per oxygen, which agrees very well with the 0.11 μ B estimated from neutron diffraction study [6]. The observation that sample of δ∼0 does not show FM moment below T N is consistent with the proposed interpretation.

Exchange couplings estimated from χ(T)
The 3D AF spin structure of Li 2 CuO 2 has been solved by the neutron diffraction as antiferromagnetically coupled FM chains [2,6], as illustrated in figure 9(a) with the on-site spin anisotropy along the a-direction, which has also been confirmed by the anisotropic magnetic susceptibilities shown in figure 4. The spin structure can also be categorized as an A-type 3D AF spin ordering of antiferromagnetically coupled planes with FM intraplane (ab-plane) coupling. In the first order approximation for the magnet couplings among Cu 2+ spins, we may consider that the spin system is composed of antiferromagnetically coupled FM chains with three magnetic couplings, including the nearest neighbor intra-chain coupling J, the nearest-neighbor inter-chain coupling ¢ J along the a-direction, and the next nearest-neighbor inter-chain coupling  J along the a/2-c/2 diagonal direction, as illustrated in figure 9(b). Following the derived spin susceptibility χ(T) expression of an 1D isotropic Heisenberg ferromagnet in the mean field approximation [33,34], which implies that the inter-chain coupling strength is inversely proportional to the corresponding inter-chain distance. The implication that a coupling strength is inversely proportional to the distance is consistent to a magnetic dipole-dipole interaction type [35]. Although the spin structure does not distinguish the origin of the magnetic interaction type, it should be noted that the intra-chain coupling must be of electronic exchange type due to the Cu-O orbital hybridization via Cu-O-Cu superexchange route, but the inter-chain coupling lacks such route in Li 2 CuO 2 . The bonding structure of Li 2 CuO 2 are edge-sharing CuO 2 chains separated by the highly ionic interstitial arrays of Li (see figure 1), which supports the scenario of dipole-dipole type magnetic interaction among FM chains positively. In addition, based on the first-principles DFT electronic structure calculations by Xiang et al [36], it is suggested that the next-nearest-neighbor interchain interactions could be responsible for the absence of spiral magnetic order.
3.9. Exchange coupling constants J i and Θ The magnetism of edge-sharing CuO 2 spin chain system is the foundation to the understanding of high T c cuprate superconductors of edge-sharing Cu-O plane [37]. Comparing the representative samples having edgesharing CuO 2 chains, from (Ca,Y) 0.8 CuO 2 (i.e., Ca 4 Cu 5 O 10 or Ca 2 Y 2 Cu 5 O 10 ) with interstitial Ca/Y atomic arrays, to (La,Ca,Sr) 14 Cu 24 O 41 with sandwiched two-leg spin ladder layers [34,[38][39][40][41][42], Li 2 CuO 2 is the simplest as a prototype edge-sharing CuO 2 spin chain system, without complications from the effects of cation nonstoichiometry, hole doping, or spin ladder coupling on data interpretation. However, many calculated or fitted exchange coupling constants of Li 2 CuO 2 reported in the literature did not consider the possibility of oxygen non-stoichiometry, let alone its impact on the electronic structure and physical properties.
In general, the Weiss temperature Θ fitted from the Curie-Weiss law ( )