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Spin exchange dominated by charge fluctuations of the Wigner lattice in the newly synthesized chain cuprate Na5Cu3O6

Na5Cu3O6, a new member of one dimensional charge ordered chain cuprates, was synthesized via the azide/nitrate route by reacting NaN3, NaNO3 and CuO. According to single crystal X-ray analysis, one dimensional CuO2 chains built up from planar, edge-sharing CuO4 squares are a dominant feature of the crystal structure. From the analysis of the Cu-O bond lengths we find that the system forms a Wigner lattice. The commensurate charge order allows to explicitly assign the valence states of either +2 or +3 to each copper atom resulting in a repetition according to Cu(2+)-Cu(3+)-Cu(2+)-Cu(2+)-Cu(3+)-Cu(2+). Following the theoretical analysis of the previously synthesized compounds Na3Cu2O4 and Na8Cu5O10, the magnetic susceptibility was expected to show a large dimer gap. Surprisingly, this is not the case. To resolve this puzzle, we show that the magnetic couplings in this compound are strongly affected by excitations across the Wigner charge gap. By including these contributions, which are distinct from conventional superexchange in Mott-insulators, we obtain a quantitative satisfying theoretical description of the magnetic susceptibility data.


I. Introduction
Multinary oxides constitute a remarkably versatile and prolific class of materials. They have continued to play a major role in the fields of high temperature superconductivity (HTSC) [1] and colossal magneto resistivity (CMR) [2], or, more recently, multiferroics and spintronics [3]. Although, during the past decades, much effort has gone into unraveling the phenomena of HTSC in cuprates [4] and of CMR in manganates, no fully consistent and conclusive microscopic explanation has become available yet. The theoretical difficulty is due to the high complexity of the problems resulting among others from strong electron correlations, and coupled charge, spin, orbital and lattice degrees of freedom in collective systems.
Furthermore, virtually all oxide materials showing HTSC or CMR include severe structural disorder, even decay into multiphase systems (phase separation, stripe formation) [5-8], a fact that has impeded theoretical analyses commonly relying upon translational invariance, and blurred experimental observations by inhomogeneous signal broadening effects. Thus, it would be highly desirable to employ fully periodic and chemically well defined materials as model systems for studying charge, spin and orbital ordering, either coupled or independent.
With the "azide/nitrate route" we have developed a rather efficient approach for the solid state synthesis of intrinsically doped multinary transition metal oxides [9, 10]. As a particular strength of this procedure, the oxygen content, and thus the valence state of the transition metal, can be precisely fixed by the alkali azide/nitrate ratio weighed in. At various illustrative examples, the "azide/nitrate route" has been proven to be rather versatile in providing highly defined materials showing interesting structural and physical properties.
Among these, in particular, a new family of quasi one-dimensional intrinsically doped sodium cuprates(II/III) [11 -16] has been found to be excellently suited for investigating the wealth of physical properties related. This includes the unique phenomenon of separation of spin and charge excitations [17], which constitutes potential for applications in non-linear photoelectric devices [18]. Furthermore, the close structural and electronic relationship of these materials to the high-temperature superconductors, and the known instabilities of the HTSC towards low-dimensional phenomena forms a background, which provides a strong motivation for close scrutiny of these materials [19 -21].
The sodium cuprates (II/III) realized thus far, Na 3 Cu 2 O 4 and Na 8 Cu 5 The temperature profile must strictly be followed to avoid any unpleasant circumstances. The obtained black powders, being very sensitive to humid air, were sealed in glass ampoules under argon atmosphere and all following manipulations with these substances were performed in inert atmospheres of purified argon. The X-ray investigation on powder samples was performed using a D8-Advance diffractometer (Bruker AXS, Karlsruhe, Germany) with Cu-Kα 1 radiation (λ = 1.54178 Å) at room temperature using a position-sensitive detector and a curved germanium monochromator.
Single crystals in the form of black needles can easily be singled out immediately after the reaction. However for better crystal quality the sample was post annealed at 873 K for 400 hours. Single crystal diffraction data were collected on a three circle diffractometer (Bruker AXS, Karlsruhe, Germany) equipped with a SMART-CCD (APEX I), at 293 K. The collection and reduction of data were carried out with the Bruker Suite software package [42]. Intensities were corrected for absorption effects applying a multi-scan method [43].  Table 1 shows the atomic parameters and equivalent isotropic displacement parameters.

B. Thermal analysis and magnetic measurements
The differential scanning calorimetry (DSC) measurements were carried out with a DSC device (DSC 404 C, Netzsch GmbH, Selb, Germany). The sample was heated at a rate of 30 The magnetic susceptibility χ(T) is measured in the temperature range from 2 K to 680 K in magnetic fields up to 7 T using a SQUID-Magnetometer (MPMS 5.5, Quantum Design). For measurements above 350 K the sample was contained in warily dried SUPRASIL ampoule (Ø = 3 mm) that was long enough to extend over the coils of the magnetometer inside the oven.
In order to nullify the contribution, whatsoever, from spurious ferro-and paramagnetic

A. Crystal Structure description
The crystal structure of Na 5 Cu 3 O 6 (Na 1,667 CuO 2 ), a new member of the one-dimensional commensurate composite crystal family Na 1+x CuO 2 [11,12], has been solved through single crystal X-ray diffractometry. Accordingly, the main structural characteristic is a onedimensional polyanionic 1 ∞ CuO 2 nchain, in which Cu is coordinated by oxygen in a square planar arrangement and these CuO 4 -squares knit together in linear chains, sharing edges in trans-position, with mean intrachain Cu-Cu distances of 2.80 Å ( Fig. 1 (b)).
Neighboring CuO  polyhedra, with Na ions forming a tubular honeycomb-like arrangement ( Fig. 1 (a)) with the channels occupied by cuprate ribbons.
The crystal structures of all members of chain cuprates belonging to the general family Na 1+x CuO 2 [11,12], differ in the Na/CuO 2 , and correspondingly in the Cu 2+ /Cu 3+ , ratios.
These two features determine the periodicity/modulation along b (chain direction). The resulting repetition unit for Na 5  In contrast to NaCuO 2 where all Na + ions are in the centers of the oxygen octahedra, in the title compound the sodium atoms are shifted off center, thereby giving freedom to accommodate more sodium atoms. This displacement in turn leads to two different oxygen environments for the Na atoms with Na-O bond length ranging from 2.270 to 2.767 Å.(c.f Table 2). The sodium atoms are shifted from the centers of the oxygen polyhedra in order to maximize the Na-Na distances. This leads in some cases to unusual thermal displacement parameters. This is true particularly for the position of Na5, which is thus better described applying a split position (Na5A and NA5B, Table 1).
Alternatively the structure of the title compound can also be interpreted within the 3+1D

B. Thermal analysis and magnetic characterization
As monitored by differential scanning calorimetry there is a sharp reversible thermal signal at T = 555 K for Na 5 Cu 3 O 6, which can be assigned to the WL melting [25] (Fig. 2) [60] as the only solid residues. The phase purity of the sample was monitored by X-ray powder analysis as can be seen in Fig. 3.
The specific heat for Na 5 Cu 3 O 6 was recorded in the temperature range of 2 -250 K. In the low temperature region one can see a λ-type anomaly at 23 K in the C p /T(T) curve, as shown in Fig. 4(a), which we assign to the onset of long-range AFM ordering. The ratio of T (χ max )/T(C max ) is closer to the S = 1/2 Heisenberg model than to the Ising value [61].
Consequently, the absolute values of C p are uncertain and a substantial lattice contribution cannot be ruled out. To probe the nature of the specific heat anomaly at T N in more detail, we also displayed C p /T 2 versus T in Fig. 4(a) and we plotted the temperature derivative of the quantity χ mol ×T ("Fisher's heat capacity", cf. ref. [62]) in Fig. 4(b) [63, 64]. Both show a lucid picture with the Neel temperature precisely determined to be 22 K.
The magnetic susceptibility data is fitted by a Curie Weiss law (c.f.

A. Long-range Coulomb interactions and charge order
In contrast to the corner-sharing geometry of copper-oxygen plaquettes as realized, for example, in the high-Tc cuprates, the edge-sharing geometry leads to strongly reduced hopping amplitudes. As a consequence, long-range Coulomb interactions within the chain become important. Here n j,σ counts the number of electrons with spin σ=↑,↓ and n j = n j,↑ + n j,↓ .
Here U is the local Coulomb repulsion and V d =V/d represent the long-range Coulomb interactions defined in terms of the nearest-neighbor matrix element V and the distance d=1,2,3, … . The 1/d Coulomb form of the interaction is appropriate here, as we are dealing with insulating systems. The value of V is, however, screened by the static dielectric constant of the core electrons.
As the Coulomb repulsion exceeds the kinetic energy, the valence electrons form a Wigner crystal on top of the underlying Cu lattice and thereby minimize their Coulomb interaction.
For the hole doping concentration x = 1/3, as realized in Na 5 Cu 3 O 6 , this leads to a charge localization with a unit cell (in a simplified picture considering only a single chain and only the copper atoms) comprising two Cu 2+ and one Cu 3+ ion as depicted in Fig. 1(b). The kinetic energy acts as a perturbation and the charge order is not frozen as the system will still undergo virtual charge excitations in order to take partial advantage of the kinetic energy, where c + j,σ and c j,σ are creation and annihilation operators for electrons at site j and spin σ, respectively, and the density operators are expressed as n j,σ = c + j,σ c j,σ . The resulting virtual transitions with hopping amplitudes t d lead to effective magnetic exchange interactions J d in a Heisenberg model with long-range interactions: The positions of the spin operators S j representing the spin-1/2 of Cu 2+ are determined by the charge ordering pattern. The relevant exchange constants for Na 5 Cu 3 O 6 are shown schematically in Fig. 6(a). Subsequently, in the effective spin model, Eq.(4), the sites with holes are unimportant and can be omitted, as seen in Fig. 6(b,c).
For Na 3 Cu 2 O 4 and Na 8 Cu 5 O 10 it was found that J 2 is antiferromagnetic and dominant [25]. Starting with the simplified magnetic structure as shown in Fig. 6(b) therefore suggests that the system consists of dimers which are coupled by weak ferromagnetic interactions. If this is the case, we can estimate J 2 by fitting the measured susceptibility at high temperatures to an independent dimer model. The result of an independent dimer fit is shown in Fig. 5. Whereby, we allow for a small constant contribution χ 0 amounting to -0.9 x 10 -4 emu/mol, well within the range of what is expected due to the diamagnetic and the van Vleck temperature independent contributions to the magnetic susceptibility.
While the independent dimer model with J 2 = 145 K does fit the data well for high which is expected to be of the order of the Neel temperature, J' ~ T N ~ 20 K. In this regard it is also important to note that the magnetic model shown in Fig. 6(b), which does take the coupling of the dimers by a nearest-neighbor exchange J 1 into account, smoothly connects for increasing ferromagnetic coupling J 1 a dimer model (for |J 1 | << J 2 ) with an effective antiferromagnetic S = 1 Heisenberg chain (for |J 1 | >> J 2 ). In the latter limit the model has a  Fig. 7(a), involving a doubly occupied site in the virtual state and consequently J d ~ 4t d 2 /U. Since t 2 is expected to be the dominant hopping amplitude due to the edge-sharing geometry, this approximation leads us to the dimer model shown in Fig. 6(b).

B. Virtual excitations across the Wigner charge gap
In contrast to a conventional Mott-Hubbard insulator, in a Wigner crystal there are also virtual excitations across the Wigner charge gap, as displayed in Fig. 7(b), which must be taken into account [26]. Careful analysis shows that these excitations also contribute to magnetism and can dramatically alter the exchange couplings J d as we will demonstrate in the following.
A particular exchange process is shown in Fig. 8(a). The sequence of processes 1-2-3 leads to an interchange of the 2 electrons involved. Thus the process in Fig. 8(a)  processes the hopping matrix elements may occur with odd powers, such that the character of the interaction depends on the signs of the involved hopping matrix elements. Thus 'Wignertype exchange' can give rise to antiferromagnetic but also to ferromagnetic couplings. The latter comes unexpected for a transition metal oxide without orbital degeneracy and in absence of Hund coupling. We also note that the intermediate states in Fig. 8(a) Fig. 9 we show the exchange constants as given in Eq. (5) as a function of D for U ~ 3.5 eV, t 1 ~ -0.08 eV, t 2 ~ -0.12 eV, and assuming t 3 ~ 0.05 eV.

C. Numerical results for the magnetic susceptibility
The exchange constants we have found for Na 5 Cu 3 O 6 imply that the magnetic structure of a single chain can be understood as a two-leg ladder with antiferromagnetic couplings along the legs and ferromagnetic coupling along the rungs as shown in Fig. 6(c). Such a system will also show an excitation gap, however, in this case the gap is only of order Δ ~ 0.2 |J 1 | ~ 20 K [67] and can easily be overcome by interchain couplings.
To calculate the susceptibility for the Heisenberg model as depicted in Fig. 6(c)