Quantum spin ladder with ferromagnetic rungs in Bi$_2$CuO$_3$(SO$_4$)

Abstract

We introduce Bi$_2$CuO$_3$(SO$_4$) as a rare example of a spin-ladder magnet with ferromagnetic interactions on the rungs. Its magnetic response is studied through measurements of heat capacity, temperature-dependent magnetic susceptibility, and field-dependent magnetization, as well as electron spin resonance spectroscopy. These experiments are complemented by density-functional-theory calculations combined with the construction of maximally localized Wannier functions and an analysis of the relevant superexchange pathways. Quantum Monte Carlo simulations are employed to model thermodynamic properties and to quantitatively determine the magnetic exchange parameters. Our combined approach identifies Bi$_2$CuO$_3$(SO$_4$) as a two-leg spin-ladder system with ferromagnetic rungs ($J'$ $\approx -208$ K) and antiferromagnetic legs ($J$ $\approx 258$ K). These interactions of similar magnitude arise from remarkably different superexchange pathways, with the Cu--Cu distance along the leg being almost twice as long than the respective distance along the rung. The antiferromagnetic leg coupling represents the strongest oxygen-mediated long-range superexchange in a Cu$^{2+}$ compound reported to date and sets the benchmark for the role of complex superexchange pathways in quantum magnets.

Figures (6)

• Figure 1: General projections of the crystal structure and spin lattice of Bi$_2$CuO$_3$(SO$_4$) along $c$ (a,b) and within the $ac$ plane (c,d) (Cu - blue balls, Bi - purple balls, CuO$_4$ plaquettes are shown in blue, (SO$_4$)$^{2-}$ tetrahedra are yellow). The spin lattice comprises the leg $J$ (cyan), rung $J'$ (pink), diagonal $J"$ (green), $J_{\parallel}$ (red) and $J_{\perp}$ (black) couplings. $J_{\perp}$ interactions are illustrated with different line styles: the 5.68 Å pathway is shown as a black dotted line, whereas the 6.09 Å pathway is shown as a black dashed line.
• Figure 2: XRD pattern refinement of Bi$_2$CuO$_3$(SO$_4$) in the $C2/c$ space group by Rietveld method from the synchrotron powder data at 80 K (ID22, ESRF). Experimental data are shown as green circles (Y$_{obs}$), the calculated profile is shown by the red solid line (Y$_{calc}$) and the difference curve (Y$_{obs}$ - Y$_{calc}$) is displayed as a blue line at the bottom. Vertical tick marks indicate the Bragg reflection positions of the main phase Bi$_2$CuO$_3$(SO$_4$) (green) and the impurity phase Bi$_{28}$O$_{32}$(SO$_4$)$_{10}$ (gray).
• Figure 3: (a) Temperature-dependent magnetic susceptibility of Bi$_2$CuO$_3$(SO$_4$) measured in the applied field of 1 T with the QMC fit described in the text. The inset shows the low-temperature region of $\chi(T)$ at 0.1 T, 1 T, 5 T and highlights the anomaly at 16 K. (b) Field-dependent magnetization of Bi$_2$CuO$_3$(SO$_4$) measured at 2 K, 10 K, 50 K, 100 K, and 200 K with the mean-field fit described in the text. (c) Temperature-dependent specific heat $C_p$(T) of Bi$_2$CuO$_3$(SO$_4$) measured at 0 T, 1 T, 2.5 T, and 9 T. The colored circles show the raw data, the blue line represents the estimation of phonon contribution using the model with two Debye temperatures, as explained in the text.
• Figure 4: (a) Exemplary ESR lines at different temperatures at $180$ GHz. Shading highlights the area under the curve. The sharp spikes in the spectra reflect the ESR signal of the DPPH marker. (b) Frequency-field diagram at $2$ K. Solid line corresponds to $g=2.12$. (c) ESR line intensity at $180$ GHz. Symbols represent the data, solid line is a Curie--Weiss fit. Dashed line is the bulk magnetic susceptibility scaled to match the ESR line intensity at low temperatures.
• Figure 5: Calculated electronic density of states for Bi$_2$CuO$_3$(SO$_4$) at the PBE level. Colored areas represent the element-resolved contributions of Bi, Cu, S, and O states, while the dashed line shows the total DOS. The Fermi level is set to $E = 0$ eV. The states in the vicinity of the Fermi level are dominated by Cu 3$d$ contributions. Inset: Cu-projected DOS highlighting the dominant contribution of the 3$d_{x^{2}-y^{2}}$ orbital.