Haldane topological spin-1 chains in a planar metal-organic framework

Abstract

Haldane topological materials contain unique antiferromagnetic chains with symmetry-protected energy gaps. Such materials have potential applications in spintronics and future quantum computers. Haldane topological solids typically consist of spin-1 chains embedded in extended three-dimensional (3D) crystal structures. Here, we demonstrate that [Ni(μ-4,4'-bipyridine)(μ-oxalate)]_n (NiBO) instead adopts a two-dimensional (2D) metal-organic framework (MOF) structure of Ni²⁺ spin-1 chains weakly linked by 4,4'-bipyridine. NiBO exhibits Haldane topological properties with a gap between the singlet ground state and the triplet excited state. The latter is split by weak axial and rhombic anisotropies. Several experimental probes, including single-crystal X-ray diffraction, variable-temperature powder neutron diffraction (VT-PND), VT inelastic neutron scattering (VT-INS), DC susceptibility and specific heat measurements, high-field electron spin resonance, and unbiased quantum Monte Carlo simulations, provide a detailed, comprehensive characterization of NiBO. Vibrational (also known as phonon) properties of NiBO have been probed by INS and density-functional theory (DFT) calculations, indicating the absence of phonons near magnetic excitations in NiBO, suppressing spin-phonon coupling. The work here demonstrates that NiBO is indeed a rare 2D-MOF Haldane topological material.

Fig. 1. Energy diagram of a Haldane spin-1 chain system and the Zeeman effect on the magnetic states.

Here, D_eff and E_eff are effective axial and rhombic (or transverse) ZFS parameters for the lowest spin excitations, respectively. The right panel illustrates the Zeeman effect on the states, leading to a magnetic phase transition at about 9 T for NiBO. The labels “+1” and “−1” represent M_S = +1 (or |1, + 1〉) and −1 (or |1,−1〉) states, respectively, of the excited triplet state. The energy splitting diagram is based on the z direction [i.e., along the Ni(ox) chains as shown in Fig. 2] aligned parallel to the magnetic field.

Fig. 2. Differences between a Haldane topological chain and an antiferromagnetic material.

a AKLT model representation of the Ni²⁺ spin-1 Haldane chain showing the antiferromagnetic coupling (gray oval) between two unpaired electrons (red and blue arrows) in the middle of the chain, leaving unpaired electrons (orange arrows) at ends of the chain. These two unpaired electrons may have antiparallel (shown) or parallel spins, giving singlet and triplet states, respectively, in Fig. 1. There is no magnetic unit cell in the Haldane system. b Schematic of the antiferromagnetic structure in NiO below its Néel temperature. The magnetic unit cell has twice the linear dimension of the crystal unit cell, as revealed by PND. In the crystal unit cell, Ni²⁺ ions form a face-centered cubic cell with ferromagnetically coupled sheets that are anti-parallel with adjacent sheets. O²⁻ ions are shown as small gray cycles.

Fig. 3. Schematic of NiBO and the crystal structure of NiBO-d₈.

a 2D schematic of NiBO showing the bonding around three Ni²⁺ ions within the chain. Blue arrows show intra- and inter-chain directions. b Structure of NiBO-d₈ viewed down the crystallographic a-axis. Green: Ni; Red: O; Purple: N; Gray: C. Selected bond lengths and angles: Ni-O = 2.0488(10)-2.0533(9) Å, Ni-N = 2.092(5) Å. cis-ligand angles: O-Ni-O 82.53(4)-97.47(4)°, O-Ni-N 89.68(11)-90.32(11)°; trans-ligand angles: O-Ni-O 179.4(2)-179.5(2)°, N-Ni-N 180.0°. The CIF (Crystallographic Information Framework) file of the crystal structure, which has been deposited in the Cambridge Structural Database (CCDC No. 2278974), is provided as a Source Data file.

Fig. 4. PND patterns of NiBO and the Rietveld refinement of the PND data.

a PND patterns of NiBO at 1.7 and 100 K. Patterns at 10 K, 20 K, and 100 K are given in Supplementary Fig. 7. b Rietveld refinement of the PND data at 1.7 K; R_{weighted profile} = 2.49%; GOF (Goodness of Fit) = 5.08. Insets: low d-spacing regions. Arb. unit = arbitrary unit. Source data are provided as a Source Data file.

Fig. 5. DC magnetic susceptibility data of NiBO.

a Temperature dependence of the DC magnetic susceptibility (χ) of NiBO at 0.1 T. b Plot of $\ln \left[\chi \left(\mathrm{T}\right)\text{–}\chi \left(0\right)\right]$ vs. 1/T for data below 11.5 K. c Temperature dependence of the DC susceptibility of NiBO at magnetic fields of 0.1-12 T. Source data are provided as a Source Data file.

Fig. 6. Quantum Monte Carlo results for the magnetic susceptibility of NiBO as a function of temperature.

The inset shows the magnetic susceptibility data in the low-temperature range. The QMC data fits well with the experimental data down to the low-temperature regime. Source data are provided as a Source Data file.

Fig. 7. Specific heat data and the temperature-magnetic field phase diagram of NiBO showing a phase transition.

a Magnetic field-dependent specific heats of NiBO from 1.92 to 6.00 K at 0-14 T. b Temperature-magnetic field phase diagram of NiBO from specific heat (C′) and magnetization (χ) including their respective error bars in the temperatures. The green dash line presents the fitting T_N vs (H′−H’_N)^φ. Inset: Plot of ln T_N vs ln (H′−H’_N) including error bars in ln T_N. T_N is the critical temperature. H’_N is the critical field. H′ is the magnetic field. φ is the crossover exponent. Source data are provided as a Source Data file.

Fig. 8. Energy diagram of NiBO showing three observed magnetic transitions.

Expected transitions A, B at 0 T and C inside magnetic fields for a single crystal of NiBO with the magnetic field aligned in the z-direction. The M_S = +1, −1, and 0 states are defined in Fig. 1.

Fig. 9. INS and HFESR spectra revealing the three observed magnetic transitions in NiBO.

a Temperature-dependent INS data for NiBO-d₈ at 5–20 K. The pink dashed are at the peaks from excitations A and B in Fig. 8 corresponding to the two spin gaps between levels 1 and 2/3 as well as between levels 1 and 4. b HFESR spectrum of NiBO at 4.5 K and 511 GHz. dΧ″ / dH′ is the first derivative of the absorption Χ″ vs magnetic field H’. Source data are provided as a Source Data file.