Two-dimensional Quantum (S = 1/2) Heisenberg Antiferromagnets (2DQHAF) have been widely studied as systems where a phase transition from the renormalised classical to the quantum disordered regime can be induced by charge doping, or by a sizeable frustration. In particular, for a square lattice with a super-exchange coupling along the diagonals of the square, J2, about half of the one along the sides, J1, a cross-over to a spin-liquid ground state is expected [1]. For |J2/J1| < 0.35 Néel order occurs, whereas for |J2/J1| > 0.65 a collinear order should develop, with two degenerate ground states having spins ferromagnetically aligned along the x(y)-axis and antiferromagnetically along y(x).

Theoretical models predict that in 2-dimensions the quantum fluctuations affects the value of the order parameter, which will be reduced to m~0.6 mB/V4+. A further reduction of the magnetic moment is expected for a system close to the border with the non-magnetic ground state.

While the theoretical understanding of the J1-J2 phase diagram has progressed considerably, an experimental verification of the theoretical predictions was still missing because suitable model compounds were not available. However, two recently-synthesised vanadates have been considered as prototypes for frustrated 2DQHAF on a square lattice, Li2VOGeO4 and Li2VOSiO4. The latter compound has been the object of a number of theoretical and experimental investigations during the past few years [2].

In Li2VOSiO4, see Figure 113, the magnetically-active network of spin 1/2 V4+ ions is built up by layers of VO5 square pyramids sharing corners with SiO4 tetrahedra. Neutron powder diffraction experiments established that the system orders antiferromagnetically below 2.8 K, with propagation vector (1/2 1/2 0). However, several points required further clarification: namely whether the magnetic structure is 2 or 3-dimensional; which of the two possible spin-arrangements is correct; and a determination of the values or at least the ratio of the values of the exchange integrals.


Fig. 113: Perspective view of Li2VOSiO4 (left) showing the layered character of the structure. Notice the VO5 pyramids reversal. The small red circles are the Si4+ ions, whereas the V4+ ions are the larger green circles. On the right, the magnetic structure of Li2VOSiO4 projected along [001].



The first two of these ambiguities were solved by the results of Resonant X-ray Scattering (RXS) measurements performed at ID20 on a 3x2x0.1 mm3 single crystal. Determination of the exchange integrals will require much larger samples and a different experimental technique.

The results obtained have established that the ordered structure is three-dimensional with ferromagnetic coupling in the c-axis direction. The magnetic contribution to the scattering has a marked dependence on the magnetic structure and on the scattering channels (- or -). It was possible to demonstrate that only the magnetic structure shown on the right in Figure 113 is compatible with the polarisation dependence of the measured magnetic reflections. The absence of a measurable magnetic scattering at high energy (5.480-5.495 keV, see Figure 114) is intriguing when compared with other reported Vanadium compounds. This could be related to the coordination and the valence of the Vanadium in Li2VOSiO4, as our calculations show that this result is not related to a different magnetic symmetry at the p states, but it is consistent with the idea that only the d states carry the magnetisation.


Fig. 114: Photon Energy dependence around the V K-absorption edge of the intensity of the (1/2 1/2 3) superlattice magnetic reflection collected in the - channel. The inset shows the temperature dependence of the ordered magnetic moment as determined by neutron diffraction compared with the square root of the X-ray integrated intensity after normalisation at 1.7 K.



In conclusion, combining neutron powder diffraction with resonant X-ray scattering measurements on a single crystal has allowed us to establish the low temperature magnetic structure of Li2VOSiO4, which consists of collinear a-b antiferromagnetic layers stacked ferromagnetically along the c axis, confirming J2>J1. The neutron-measured ordered magnetic moment (0.6 mB/V4+) is larger than previously reported and points to a system that is far from the border of the non-magnetic ground state, in good agreement with the latest theoretical predictions. In this system a mechanism to relieve the degeneracy of the two ground states is expected but, surprisingly, no structural distortion was observed close to the AF transition.

[1] P. Chandra et al., Phys. Rev. Lett. 64, 88 (1990).
[2] R. Melzi et al., Phys. Rev. Lett. 85, 1318 (2000).

Principal Publications and Authors
A. Bombardi (a,g), J. Rodriguez-Carvajal (b), S. Di Matteo (c), F. de Bergevin (a), L. Paolasini (a), P. Carretta (d), P. Millet (e), and R. Caciuffo (f), Phys. Rev. Lett. 93, 27202 (2004).
(a) ESRF (France)
(b) LLB, CEA-Saclay (France)
(c) LNF-INFN (France)
(d) INFM (Italy)
(e) CNRS (France)
(f) Università Politecnica delle Marche (Italy)
(g) now at Diamond Light Source (UK)