Hindered magnetic order from mixed dimensionalities in ${\text{CuP}}_2{\text{O}}_6$

We present a combined experimental and theoretical study of the spin-$\frac{1}{2}$ compound CuP${}_2$O${}_6$ that features a network of two-dimensional (2D) antiferromagnetic (AFM) square planes, interconnected via one-dimensional (1D) AFM spin chains. Magnetic susceptibility, high-field magnetization, and electron spin resonance (ESR) data, as well as microscopic density-functional band-structure calculations and subsequent quantum Monte Carlo simulations, show that the coupling $J_{2\mathrm{D}}\simeq 40$ K in the layers is an order of magnitude larger than $J_{1\mathrm{D}}\simeq 3$ K in the chains. Below $T_N\simeq 8$ K, CuP${}_2$O${}_6$ develops long-range order, as evidenced by a weak net moment on the 2D planes induced by anisotropic magnetic interactions of Dzyaloshinsky-Moriya type. A striking feature of this 3D ordering transition is that the 1D moments grow significantly slower than the ones on the 2D units, which is evidenced by the persistent paramagnetic ESR signal below $T_N$. Compared to typical quasi-2D magnets, the ordering temperature of CuP${}_2$O${}_6$ $T_N/J_{2\mathrm{D}}\simeq 0.2$ is unusually low, showing that weakly coupled spins sandwiched between 2D magnetic units effectively decouple these units and impede the long-range ordering.

Figure 1

Crystal structure of CuP${}_2$O${}_6$ (left) and the relevant magnetic model (right). Note two different Cu positions that form planar 2D units in the $bc$ plane (Cu1, green), and buckled chains stretched along the $c$ direction (Cu2, brown). For details of the Cu1 and Cu2 sublattices see Figs. 9 and 8, respectively. Weak couplings $J_{i1}$ (thin solid line) and $J_{i2}$ (dotted line) are leading interactions between the sublattices. Crystal structures are visualized using the vesta software (Ref. [13]).

Figure 2

Left panel: Magnetic susceptibility ($\chi$) of CuP${}_2$O${}_6$ measured in the applied fields of 0.1 T, 0.5 T, and 2 T. The inset shows the magnetization curve at 2 K, with a tiny hysteresis and the remnant magnetization $M_r\simeq 1.5\times {10}^{-3}$ ${\mu }_B$/f.u. Dotted lines are guides for the eye. Right panel: Inverse susceptibility ($1/\chi$). The inset shows the field-cooled and zero-field-cooled susceptibility measured at 10 mT. In both panels, the solid line is the QMC fit with the two-sublattice model, and the dashed line (right panel) is the Curie-Weiss fit. See text for details.

Figure 3

Magnetization isotherm of CuP${}_2$O${}_6$ measured at $T\simeq 1.8$ K in static and pulsed fields. Solid line is the fit with the two-sublattice model, as described in the text, whereas short-dashed (brown) and long-dashed (green) lines denote individual contributions of the 1D and 2D sublattices, respectively. Dotted line shows the extrapolation of the linear region to zero field.

Figure 4

ESR spectra measured at 5 K and 293 K and fits with powder-averaged Lorentzian lines assuming the easy-plane anisotropy. Note that at low temperatures the shape of the spectral line becomes more complex than at high temperatures, and additional terms beyond the single powder-averaged line may be required for a good fit.

Figure 5

Temperature dependence of the ESR spectral parameters: linewidth ($\Delta B$, top panel), $g$-values (middle panel), and inverse line intensity ($I^{-1}$, bottom panel). Circles and diamonds denote the components perpendicular and parallel to the field, respectively.

Figure 6

LDA density of states (DOS) for CuP${}_2$O${}_6$. The Fermi level ($E_F$) is at zero energy. Note well-defined crystal-field levels of Cu${}^{2+}$ in the planar CuO${}_4$ local environment.

Figure 7

LDA band structure of CuP${}_2$O${}_6$ in the vicinity of the Fermi level. Green (dark) and gray (light) circles show the contributions of Cu1 and Cu2 $d_{x^2-y^2}$ orbitals, respectively. Note the different bandwidths that indicate different energy scales within the Cu1 and Cu2 sublattices. The $k$ path is defined as follows: $\Gamma (0,0,0)$, $X(0.5,0,0)$, $M(0.5,0.5,0)$, $Y(0,0.5,0)$, $Z(0,0,0.5)$, $T(0.5,0,0.5)$, $R(0.5,0.5,0.5)$, $A(0,0.5,0.5)$, where the coordinates are given in units of the respective reciprocal lattice parameters.

Figure 8

Comparison of the chain sublattices in CuP${}_2$O${}_6$, Na${}_2$CuP${}_2$O${}_7$, and Sr${}_2$CuP${}_2$O${}_8$. Note the flat and linear chain in Sr${}_2$CuP${}_2$O${}_8$ (large exchange $J\simeq 140$ K), buckled linear chain in Na${}_2$CuP${}_2$O${}_7$ (moderate exchange $J\simeq 27$ K), and buckled zigzag chain in CuP${}_2$O${}_6$ (weak exchange $J_{1\mathrm{D}}\simeq 4$ K).

Figure 9

Comparison of the 2D sublattices in CuP${}_2$O${}_6$ (left) and Ba${}_2$CuGe${}_2$O${}_7$ (right). Large arrows show the spatial arrangement of the DM vectors (only in-plane components), which are uniform in Ba${}_2$CuGe${}_2$O${}_7$ and alternate in CuP${}_2$O${}_6$. Dotted lines denote the rotation of the neighboring CuO${}_4$ plaquettes by an angle $\phi \simeq {40}^{\circ }$. In CuP${}_2$O${}_6$, this rotation is very weak ($\phi \simeq 0^{\circ }$); hence the Cu–O bonds are nearly co-aligned, and the isotropic coupling $J_{2\mathrm{D}}\simeq 40$ K is much larger than $J\simeq 11$ K in Ba${}_2$CuGe${}_2$O${}_7$. In CuP${}_2$O${}_6$, the crystallographic axes $b$ and $c$ match, respectively, the directions $y$ and $z$ for the DM vectors.

Figure 10

Simulated magnetic specific heat ($C_{\mathrm{mag}}/R$) per lattice site for the spin lattice of CuP${}_2$O${}_6$ (circles) and individual contributions of the 1D and 2D sublattices shown by short-dashed and long-dashed lines, respectively.

Figure 11

A Cu2 spin chain sandwiched between two successive Cu1 square-plane antiferromagnets.

Figure 12

Staggered susceptibilities of the uniform spin chain (1D), square lattice (2D), and their combination, as observed in CuP${}_2$O${}_6$, where ${\overline{\chi }}^s=\sqrt{{\chi }_{1\mathrm{D}}^s{\chi }_{2\mathrm{D}}^s}$. Dashed line shows ${(z{J}_{i1}/J_{2\mathrm{D}})}^{-1}=50$ according to $J_{i1}/J_{2\mathrm{D}}=0.01$ and $z=2$ in CuP${}_2$O${}_6$. The susceptibility of the uniform chain is multiplied by a factor of 2 to account for $z=4$ in a quasi-1D system. The truncation of the data at $T/J_{2\mathrm{D}}=0.21$ marks the lowest temperature where the convergence with respect to the lattice size could be obtained for the 2D system.