Large exchange anisotropy in quasi-one-dimensional spin-$\frac{1}{2}$ fluoride antiferromagnets with a $d{\left(z^2\right)}^1$ ground state

Hybrid density functional calculations are performed for a variety of systems containing $d^9$ ions $(\mathrm{C}{\mathrm{u}}^{2+}$ and $\mathrm{A}{\mathrm{g}}^{2+})$ and exhibiting quasi-one-dimensional magnetic properties. In particular, we study fluorides containing these ions in a rarely encountered compressed octahedral coordination that forces the unpaired electron into the local $d\left(z^2\right)$ orbital. We predict that such systems should exhibit exchange anisotropies surpassing that of $\mathrm{S}{\mathrm{r}}_2\mathrm{Cu}{\mathrm{O}}_3$, one of the best realizations of a one-dimensional system known to date. In particular, we predict that the interchain coupling in the $\mathrm{A}{\mathrm{g}}^{2+}$-containing $\left[\mathrm{AgF}\right]\left[\mathrm{B}{\mathrm{F}}_4\right]$ system should be nearly four orders of magnitude smaller than the intrachain interaction. Our results indicate that quasi-one-dimensional spin-$\frac{1}{2}$ systems containing chains with spin sites in the $d{\left(z^2\right)}^1$ local ground state could constitute a versatile model for testing modern theories of quantum many-body physics in the solid state.

Figure 1

(a) Arrangement of elongated $M{X}_6$ octahedral leading to formation of chains featuring half-occupation of the $B_{1g}$ orbital of the $M^{2+}$ cation. (b) Arrangement of compressed $M{X}_6$ octahedral leading to formation of chains featuring with half-occupied $A_{1g}$ cation orbitals. Blue/red atoms balls depict $M/X$ atoms; long/short $M\text{−}X$ bonds are depicted with light blue/dark blue color.

Figure 2

(a) Structure of $a\text{−}\mathrm{KCu}{\mathrm{F}}_3$; K atoms were omitted for clarity. (b) Schematic description of the connectivity of the $J_{1\mathrm{D}}$ (black) and $J_{\perp }$ (red) superexchange paths.

Figure 3

(a) Structure of $M_2^{\prime }\mathrm{Cu}{\mathrm{O}}_3$ $\left(M^{\prime }=\mathrm{Ca},\mathrm{Sr}\right)$; $M^{\prime }$ atoms were omitted for clarity. (b) Schematic description of the connectivity of the $J_{1\mathrm{D}}$ (black line), $J_{\perp }^1$ (red line), and $J_{\perp }^2$ (green line) superexchange paths. (c) The $1\times 2\times 2$ supercell used for the calculation of the coupling constants.

Figure 4

(a) Structure of $\left[\mathrm{CuF}\right]\left[\mathrm{Au}{\mathrm{F}}_4\right]$ with gold atoms marked as orange balls. (b) The $2\times 2\times 2$ supercell used for calculations of magnetic coupling constants, together with the depiction of the superexchange paths.

Figure 5

(a) Structure of $\left[\mathrm{AgF}\right]\left[\mathrm{B}{\mathrm{F}}_4\right]$ with $\mathrm{B}{{\mathrm{F}}_4}^{-}$ anions marked as green tetrahedra. (b) The $\sqrt{2}\times \sqrt{2}\times 2$ supercell used for calculations of magnetic coupling constants, together with the depiction of the superexchange paths.

Figure 6

Depiction of the magnetic states used for calculations of $J_{1\mathrm{D}}^{\mathrm{NNN}}$. Open/closed balls depict spin-up/spin-down sites.

Figure 7

Comparison of experimental and theoretical values of the $T_N/J_{1\mathrm{D}}$ ratios for compounds that feature the unpaired electron in the $d(x^2-y^2)$ orbital of a metal. The red line marks a linear regression passing through (0,0) with a slope of $1.453\pm 0.004$. The values for $\mathrm{AgCu}{\mathrm{F}}_3$ and $\mathrm{NaCu}{\mathrm{F}}_3$, not included in the fit, are shown by open symbols.

Figure 8

Calculated dependence of the Néel temperature scaled by $J_{1\mathrm{D}}(T_N/J_{1\mathrm{D}})$ as a function of $|J_{\perp }^{\mathrm{eff}}|/J_{1\mathrm{D}}$. Systems with $A_{1g}$ chains are marked in black, those exhibiting $B_{1g}$ chains in red. The dotted line represents the dependence given by Eq. (3).