Extension of the spin-$\frac{1}{2}$ frustrated square lattice model: The case of layered vanadium phosphates

We study the influence of the spin lattice distortion on the properties of frustrated magnetic systems and consider the applicability of the spin-1/2 frustrated square lattice model to materials lacking tetragonal symmetry. We focus on the case of layered vanadium phosphates ${\text{AA}}^{\prime }\text{VO}{\left({\text{PO}}_4\right)}_2$ (${\text{AA}}^{\prime }={\text{Pb}}_2$, SrZn, BaZn, and BaCd). To provide a proper microscopic description of these compounds, we use extensive band structure calculations for real materials and model structures and supplement this analysis with simulations of thermodynamic properties, thus facilitating a direct comparison with the experimental data. Due to the reduced symmetry, the realistic spin model of layered vanadium phosphates ${\text{AA}}^{\prime }\text{VO}{\left({\text{PO}}_4\right)}_2$ includes four inequivalent exchange couplings: $J_1$ and $J_1^{\prime }$ between nearest-neighbors and $J_2$ and $J_2^{\prime }$ between next-nearest-neighbors. The estimates of individual exchange couplings suggest different regimes, from $J_1^{\prime }/J_1$ and $J_2^{\prime }/J_2$ close to 1 in $\text{BaCdVO}{\left({\text{PO}}_4\right)}_2$, a nearly regular frustrated square lattice, to $J_1^{\prime }/J_1\simeq 0.7$ and $J_2^{\prime }/J_2\simeq 0.4$ in $\text{SrZnVO}{\left({\text{PO}}_4\right)}_2$, a frustrated square lattice with sizable distortion. The underlying structural differences are analyzed, and the key factors causing the distortion of the spin lattice in layered vanadium compounds are discussed. We propose possible routes for finding new frustrated square lattice materials among complex vanadium oxides. Full diagonalization simulations of thermodynamic properties indicate the similarity of the extended model to the regular one with averaged couplings. In case of moderate frustration and moderate distortion, valid for all the ${\text{AA}}^{\prime }\text{VO}{\left({\text{PO}}_4\right)}_2$ compounds reported so far, the distorted spin lattice can be considered as a regular square lattice with the couplings $\left(J_1+J_1^{\prime }\right)/2$ between nearest-neighbors and $\left(J_2+J_2^{\prime }\right)/2$ between next-nearest-neighbors.

Figure 1

(Color online) Phase diagram of the FSL model[9] and the respective model compounds (see text for references). The inset shows the regular FSL with the NN $\left(J_1\right)$ and NNN $\left(J_2\right)$ couplings denoted by solid and dashed lines, respectively.

Figure 2

(Color online) Crystal structure of the ${\text{AA}}^{\prime }\text{VO}{\left({\text{PO}}_4\right)}_2$ compounds and the underlying spin model. The left panel shows a single $\left[{\text{VOPO}}_4\right]$ layer. The middle panel presents the stacking of the $\left[{\text{VOPO}}_4\right]$ layers and the $\left[{\text{AA}}^{\prime }{\text{PO}}_4\right]$ blocks as well as the angle $\phi$ measuring the layer buckling: larger and smaller spheres denote the A and ${\text{A}}^{\prime }$ cations, respectively. The right panel shows the magnetic interactions in the $\left[{\text{VOPO}}_4\right]$ layers (compare with the regular model in the inset of Fig. 1): solid, dash-dotted, dashed, and dotted lines indicate $J_1,J_1^{\prime },J_2$, and $J_2^{\prime }$, respectively.

Figure 3

(Color online) LDA density of states for $\text{BaCdVO}{\left({\text{PO}}_4\right)}_2$ (the contribution of barium is not shown, because it is negligible in the whole energy range). The Fermi level is at zero energy. The inset shows the orbital resolved DOS for vanadium.

Figure 4

(Color online) LDA band structure (thin light lines) and the fit of the tight-binding model (thick green lines) for $\text{BaCdVO}{\left({\text{PO}}_4\right)}_2$ (left panel) and $\text{SrZnVO}{\left({\text{PO}}_4\right)}_2$ (right panel). The Fermi level is at zero energy. The notation of the $k$ points is as follows: $\Gamma \left(0,0,0\right)$, $X\left(0.5,0,0\right)$, $M\left(0.5,0.5,0\right)$, $Y\left(0,0.5,0\right)$, $Z\left(0,0,0.5\right)$, and $T\left(0.5,0,0.5\right)$ (the coordinates are given along $k_x$, $k_y$, and $k_z$ in units of the respective reciprocal lattice parameters).

Figure 5

(Color online) Exchange couplings in ${\text{Li}}_2{\text{VOSiO}}_4$ (left panel) and ${\text{Li}}_2{\text{VOGeO}}_4$ (middle panel) calculated for different values of Coulomb repulsion parameter $U_d$ and the resulting frustration ratios $J_2/J_1$ for both the compounds (right panel). Shaded stripes show experimental estimates from Refs. 18, 19, 20, 25.

Figure 6

(Color online) Crystal structures of $\text{BaCdVO}{\left({\text{PO}}_4\right)}_2$ (left panels) and $\text{SrZnVO}{\left({\text{PO}}_4\right)}_2$ (right panels): the upper panels show the interlayer $\left[{\text{AA}}^{\prime }{\text{PO}}_4\right]$ blocks, while the bottom panels present the buckling of the $\left[{\text{VOPO}}_4\right]$ layers. The change of Ba and Cd for Sr and Zn leads to the reduction of the coordination numbers and the resulting reorganization of the structure: the layer buckling and the stretching in the $ab$ plane (see text for details).

Figure 7

(Color online) Distortion of the $\left[{\text{VOPO}}_4\right]$ layer as implemented in the model structures. Arrows show displacements of the vanadium atoms away from their ideal positions in the centers of the ${\text{VO}}_5$ square pyramids. The displacements yield two different V–O distances ($d$ and $d^{\prime }$) and two inequivalent NNN couplings ($J_2$ and $J_2^{\prime }$). Small spheres with solid and hatched filling denote different types of oxygen atoms with shorter (white bonds) and longer (shaded bonds) V–O distances, respectively. Dashed and dotted lines indicate the interactions $J_2$ and $J_2^{\prime }$.

Figure 8

(Color online) Spin lattice distortion $\left(J_2^{\prime }/J_2\right)$ for the model structures with the layer buckling (empty circles) and the layer stretching (filled circles) and for the real ${\text{AA}}^{\prime }\text{VO}{\left({\text{PO}}_4\right)}_2$ compounds (filled diamonds). The layer buckling is quantified by the buckling angle $\phi$ (Fig. 2). The layer stretching is imposed by shifting vanadium atoms and quantified by $\Delta d=d^{\prime }-d$ (see Fig. 7). For the real compounds, $d=\left(d_{\text{V}\text{−}\text{O}\left(2\right)}+d_{\text{V}\text{−}\text{O}\left(8\right)}\right)/2$ and $d^{\prime }=\left(d_{\text{V}\text{−}\text{O}\left(6\right)}+d_{\text{V}\text{−}\text{O}\left(9\right)}\right)/2$ (see Table ).

Figure 9

(Color online) Full diagonalization results for the distorted FSL model: magnetic susceptibility (primary figures) and specific heat (insets). The simulations are performed at fixed averaged couplings ${\overline{J}}_1=\left(J_1+J_1^{\prime }\right)/2$ and ${\overline{J}}_2=\left(J_2+J_2^{\prime }\right)/2$ and the fixed frustration ratio ${\overline{J}}_2/{\overline{J}}_1=-2$. In the left and right panels, $J_2$ and $J_2^{\prime }$ or $J_1$ and $J_1^{\prime }$ are varied, respectively. The arrows show the changes upon increasing the distortion. The thermodynamic energy scale $J_c$ is defined as $J_c=\sqrt{\left(J_1+J_1^{\prime 2}+J_2+J_2^{\prime 2}\right)/2}$.