Alternating spin-orbital order in tetragonal Sr${}_2$VO${}_4$

Considering spin-orbit coupling, the tetragonal crystal field, and all relevant superexchange processes including quantum interference, we derive expressions for the energy levels of the vanadium ions in tetragonal Sr${}_2$VO${}_4$. The used parameters of the model Hamiltonian allow us to describe well the excitation spectra observed in neutron scattering and optical experiments at low temperatures. The free energy exhibits a minimum which corresponds to a novel alternating spin-orbital order with strong thermal fluctuation of the orbital mixing parameter.

Figure 1

Left: Unit cell of tetragonal Sr${}_2$VO${}_4$ with symmetry I4/mmm (Ref. 15) highlighting the VO${}_2$ planes and the octahedral coordination of the V ions. Right: Splitting of the V${}^{4+}$ $t_{2g}$ levels due the tetragonal crystal field and spin-orbit coupling.

Figure 2

(a) Calculated energy levels as a function of the orbital mixing angle $\vartheta$ using Eqs. (13)–(15) and parameters ${\lambda }_c=-30$ meV, ${\lambda }_{ab}=-28$ meV, $D=-33$ meV, $J_a=-15$ meV, $J_f=-9.3$ meV, $J_f^{\prime }=-0.7$ meV, and $J_{\mathrm{int}}=7.5$ meV. Excitation energies observed by neutron scattering (Ref. 13) and optical spectroscopy (Ref. 19) are shown as dashed and dash-dotted lines, respectively. (b) Free energy as a function of $\vartheta$ calculated with the same parameters 10 K and 100 K.

Figure 3

Sketch of the proposed alternating spin-orbital order in the $ab$ plane of tetragonal Sr${}_2$VO${}_4$. Short and long arrows correspond to spin and orbital moments of the V ions, respectively.