Structure and magnetism of Cr${}_2$[BP${}_3$O${}_{12}$]: Towards the quantum-classical crossover in a spin-$\frac{3}{2}$ alternating chain

Magnetic properties of the spin-$\frac{3}{2}$ Heisenberg system Cr${}_2[$BP${}_3$O${}_{12}]$ are investigated by magnetic susceptibility $\chi \left(T\right)$ measurements, electron spin resonance, neutron diffraction, and density functional theory (DFT) calculations, as well as classical and quantum Monte Carlo (MC) simulations. The broad maximum of $\chi \left(T\right)$ at 85 K and the antiferromagnetic Weiss temperature of 139 K indicate low-dimensional magnetic behavior. Below $T_{\mathrm{N}}=28$ K, Cr${}_2[$BP${}_3$O${}_{12}]$ is antiferromagnetically ordered with the $\mathbf{k}=0$ propagation vector and an ordered moment of 2.5${\mu }_{\mathrm{B}}$/Cr. DFT calculations, including $\text{DFT}+U$ and hybrid functionals, yield a microscopic model of spin chains with alternating nearest-neighbor couplings $J_1$ and $J_1^{\prime }$. The chains are coupled by two nonequivalent interchain exchanges of similar strength ($\sim$1–2 K), but different sign (antiferromagnetic and ferromagnetic). The resulting spin lattice is quasi-one-dimensional and not frustrated. Quantum MC simulations show excellent agreement with the experimental data for the parameters $J_1\simeq 50$ K and $J_1^{\prime }/J_1\simeq 0.5$. Therefore, Cr${}_2[$BP${}_3$O${}_{12}]$ is close to the gapless critical point ($J_1^{\prime }/J_1=0.41$) of the spin-$\frac{3}{2}$ bond-alternating Heisenberg chain. The applicability limits of the classical approximation are addressed by quantum and classical MC simulations. Implications for a wide range of low-dimensional $S=3/2$ materials are discussed.

Figure 1

Crystal structure of Cr${}_2\left[{\mathrm{BP}}_3{\mathrm{O}}_{12}\right]$. (Top) View along $\left[0001\right]$. (Bottom left) View perpendicular to $\left[0001\right]$. The pathways of the leading interchain couplings $J_{\mathrm{ic}1}$ and $J_{\mathrm{ic}2}$ are shown with dark gray (dark blue) and light gray (red) lines, respectively. Thick arrows denote the experimental magnetic structure refined in ${\Gamma }_1$ (see Sec. 3b for details). (Bottom right) Dimers are connected via three PO${}_4$ tetrahedra and form chains running along $\left[0001\right]$. The intradimer coupling $J_1$ (double line) as well as the interdimer coupling $J_1^{\prime }$ (single line) are shown.

Figure 2

Cr${}_2[$BP${}_3$O${}_{12}]$. Temperature evolution of the magnetic 101 reflection (circles) and the fit with Eq. (2). The residual intensity above $T_N$ is due to the nuclear scattering and background. (Inset) Refinement of the subtracted ($I_{\text{4}\text{K}}-I_{\text{35}\text{K}}$) pattern with spins along $c$ (${\Gamma }_1$, dark solid line) and along $a$ (${\Gamma }_9$, light dashed line). Note that the temperature scan (main figure) and the angular scan (inset) are done on different instruments; hence, the respective intensities should not be compared.

Figure 3

(Top) Experimental magnetic susceptibility $\chi \left(T\right)$ of Cr${}_2[$BP${}_3$O${}_{12}]$ measured at 5 T (circles) and fit with the model of coupled bond-alternating $S=3/2$ chains (line). (Bottom left) Curie-Weiss fit to experimental $\chi \left(T\right)$. (Bottom right) Field dependence of $\chi \left(T\right)$. Note the long-range AF ordering transition at $T_N=28$ K.

Figure 4

Electronic structure of Cr${}_2[$BP${}_3$O${}_{12}]$. (Top) LDA density of states. The Fermi level ${\varepsilon }_{\mathrm{F}}$ is at zero energy. The contribution from Cr states is shown by shaded filling. (Bottom) LDA bands in the vicinity of the Fermi level and the fit with the tight-binding model based on Wannier functions.

Figure 5

Cr${}_2[$BP${}_3$O${}_{12}]$: Wannier functions for five Cr $3d$ orbitals (see subscripts). Cr, P, and O atoms are depicted by large (red), intermediate-size (green), and small (red) spheres, respectively. For each of the five WFs, a Cr${}_2$O${}_9$ dimer and three PO${}_4$ tetrahedra are shown. The projection is similar to the bottom right panel of Fig. 1.

Figure 6

Temperature dependence of the nearest-neighbor diagonal spin correlations $\langle S_i^z{S}_{i+1}^z\rangle$ in the $S=3/2$ alternating Heisenberg chain model ($J_1^{\prime }/J_1=0.5$). QMC results are plotted with squares ($J_1$) and circles ($J_1^{\prime }$); the classical MC results are depicted with pluses ($J_1$) and crosses ($J_1^{\prime }$). The dashed lines denote the ED results for the quantum model. The $\frac{1}{3}S(S+1)$ line marks the maximal diagonal correlation for $S=3/2$ spins. (Inset) At high temperatures ($T>2J_1$), the QMC and MC curves become almost indistinguishable.

Figure 7

(Top) Nearest-neighbor diagonal spin correlations $\langle S_i^z{S}_{i+1}^z\rangle$ in the $S=3/2$ alternating Heisenberg chain model as a function of the alternation ratio $J_1^{\prime }/J_1$ at $T=0.03J_1$. Dashed lines are guides for the eye. (Bottom) Root mean square of ${\sigma }_i\equiv (\langle S_i^z{S}_{i+1}^z\rangle -\frac{1}{3}S[S+1])$ ($i=1$, 2), reflecting the discrepancy between the quantum and the classical models.