Observation of field-dependent magnetic parameters in the magnetic molecule $\left\{{\mathrm{Ni}}_4{\mathrm{Mo}}_{12}\right\}$

We investigate the bulk magnetic, electron paramagnetic resonance, and magneto-optical properties of $\left\{{\mathrm{Ni}}_4{\mathrm{Mo}}_{12}\right\}$, a magnetic molecule with antiferromagnetically coupled tetrahedral ${\mathrm{Ni}}^{\mathrm{II}}$ in a diamagnetic molybdenum matrix. The low-temperature magnetization exhibits steps at irregular field intervals, a result that cannot be explained using a Heisenberg model even if it is augmented by magnetic anisotropy and biquadratic terms. Allowing the exchange and anisotropy parameters to depend on the magnetic field provides the best fit to our data, suggesting that the molecular structure (and thus the interactions between spins) may be changing with applied magnetic field.

Figure 1

(Color online) Ball-and-stick representation of the $\left\{{\mathrm{Ni}}_4{\mathrm{Mo}}_{12}\right\}$ molecule (Ni, numbered big spheres; Mo, medium size spheres; O, small spheres; H, not shown) emphasizing a slightly stretched ${\mathrm{Ni}}_4^{\mathrm{II}}$ pyramid with a near-equilateral triangle base (Ni ions labeled 2, 3, and 4) and an elevated apex (Ni ion labeled 1). $\mathrm{Ni}\text{−}\mathrm{Ni}$ distances: $d_{12}=6.700\left(5\right)\mathrm{\AA }$, $d_{13}=d_{14}=6.689\left(1\right)\mathrm{\AA }$, $d_{23}=d_{24}=6.616\left(1\right)\mathrm{\AA }$, $d_{34}=6.604\left(1\right)\mathrm{\AA }$ (Ref. 19).

Figure 2

Simplified structure of ${\mathrm{Ni}}_4$: the superexchange interactions $J^{\prime }$ and $J$ are represented by dashed and solid lines, respectively.

Figure 3

(Color online) Low-field susceptibility: Experimental data are given by black squares whereas the solid curve shows the result assuming a simple Heisenberg Hamiltonian (1) with $J∕k_B=-3.4\mathrm{K}$ and an isotropic spectroscopic splitting factor $g=2.25$. The inset shows a close-up view of the low-temperature region for different magnetic fields.

Figure 4

(Color online) Magnetization: Experimental data are given by squares (dark symbols, NHMFL, $T=0.44\mathrm{K}$; light symbols, OHMFL, $T=0.40\mathrm{K}$). The theoretical magnetization assuming a simple Heisenberg Hamiltonian (1) with $J∕k_B=-3.4\mathrm{K}$ and an isotropic spectroscopic splitting factor $g=2.25$ is given by a solid curve for $T=0.44\mathrm{K}$. The inset shows the experimental differential magnetization $d\mathcal{M}∕dB$ as data points as well as the theoretical $d\mathcal{M}∕dB$ (solid curve) using the same parameters as above.

Figure 5

(Color online) The figure shows the dependence of the observed EPR resonance frequencies P1 (circles), P2 (triangles), and P3 (squares) on the magnetic field. The lines provide linear fits to the data. The data were taken at $T=4.2\mathrm{K}$.

Figure 6

(Color online) The figure shows schematically the Zeeman level splittings in a pure Heisenberg model together with the assignments of allowed and forbidden transitions.

Figure 7

(Color online) Optical conductivity of a $\left\{{\mathrm{Ni}}_4{\mathrm{Mo}}_{12}\right\}$ pellet at $4.2\mathrm{K}$ (solid curve) and $300\mathrm{K}$ (dashed curve), calculated from reflectance measurements (inset) by Kramers-Kronig analysis. The energy range of our magneto-optical work is indicated by the arrow.

Figure 8

(Color online) (a) The normalized magneto-optical response $R\left(B\right)∕R(B=0\mathrm{T})$ of $\left\{{\mathrm{Ni}}_4{\mathrm{Mo}}_{12}\right\}$ pressed powder in an applied magnetic field from 0 to $32\mathrm{T}$ at $4.2\mathrm{K}$. $1\mathrm{T}$ steps are shown. No hysteresis is observed. (b) Absolute value of the integrated area of the magneto-optical reflectance ratio feature as a function of applied magnetic field (solid symbol). Results from the integrated optical conductivity data are similar. The inset shows the change in the optical conductivity of $\left\{{\mathrm{Ni}}_4{\mathrm{Mo}}_{12}\right\}$ with magnetic field, ${\sigma }_1(B=30\mathrm{T})∕{\sigma }_1(B=0\mathrm{T})$. The $B=0\mathrm{T}$ curve ratios data taken before and after the field sweep, giving an indication of overall spectral reproducibility.

Figure 9

(Color online) Low-field susceptibility: The measured data are given by symbols. The fits for $B=0.5\mathrm{T}$ are obtained using $J∕k_B=-3.2\mathrm{K}$, $J^{\prime }∕k_B=-3.1\mathrm{K}$, $j∕k_B=1.6\mathrm{K}$, $j^{\prime }∕k_B=0$, $D∕k_B=-1.0\mathrm{K}$. The data for field values of $B=3.5\mathrm{T}$ up to $5\mathrm{T}$ are approximated using $J∕K=-3.2\mathrm{K}$, $J^{\prime }∕k_B=-3.2\mathrm{K}$, $j∕k_B=1.5\mathrm{K}$, $j^{\prime }∕k_B=0$, and $D∕k_B=-3.2\mathrm{K}$. $g=2.195$ in all cases. The estimated impurity concentrations are 0.216 and 0.146 individual ${\mathrm{Ni}}^{\mathrm{II}}$ ions per $\left\{{\mathrm{Ni}}_4{\mathrm{Mo}}_{12}\right\}$ molecule for samples 1 and 2, respectively. The simulated data are averaged over 100 orientations.

Figure 10

(Color online) Magnetization of sample 2. Experimental data (NHMFL) are given by squares. The solid curve provides the best fit using a Hamiltonian with an exponentially field-dependent coupling. The simulated data are averaged over 100 orientations. The inset shows the experimental differential magnetization $d\mathcal{M}∕dB$ as data points as well as the theoretical $d\mathcal{M}∕dB$ (solid curve) using the same parameters as above.

Figure 11

(Color online) The figure shows the Zeeman level splittings for the realistic Heisenberg model (3) (parametrization 2) for several orientations of the field relative to the molecules. The various curves, which differ especially at avoided level crossings, represent relative angles of $10°$.

Figure 12

Relative EPR transmission observed at $\nu =190\mathrm{GHz}$, which corresponds to an energy level difference of $9.12\mathrm{K}$. The dominant (allowed) transition P3 is rather broad, whereas the weaker (forbidden) transition P1 results in a much sharper peak. The other weak (forbidden) transition P2, which is masked by P3, is broader than P1.