Investigation of the presence of charge order in magnetite by measurement of the spin wave spectrum

Inelastic neutron scattering results on magnetite $\left({\mathrm{Fe}}_3{\mathrm{O}}_4\right)$ show a large splitting in the acoustic spin wave branch, producing a $7\mathrm{meV}$ gap midway to the Brillouin zone boundary at $\mathbf{q}=(0,0,1∕2)$ and $\hslash \omega =43\mathrm{meV}$. The splitting occurs below the Verwey transition temperature, where a metal-insulator transition occurs simultaneously with a structural transformation, supposedly caused by the charge ordering on the iron sublattice. The wavevector $(0,0,1∕2)$ corresponds to the superlattice peak in the low symmetry structure. The dependence of the magnetic superexchange on changes in the crystal structure and ionic configurations that occur below the Verwey transition affect the spin wave dispersion. To better understand the origin of the observed splitting, several Heisenberg models intended to reproduce the pair-wise variation of the magnetic superexchange arising from both small crystalline distortions and charge ordering were studied. None of the models studied predicts the observed splitting, whose origin may arise from charge-density wave formation or magnetoelastic coupling.

Figure 1

Scattering plane in horizontal field configuration with a field of $1.1\mathrm{T}$ applied along the cubic [001] direction. Black (and gray) lines show the Brillouin zone boundaries of the cubic $(T>T_V)$ and orthorhombic $(T<T_V)$ lattices, respectively. Arrows indicate typical scattering vectors studied. The reduced wavevectors are equivalent in the cubic phase. In the orthorhombic phase, the cell doubling axis lies along the field and $\left[100\right]∕\left[001\right]$ directions are no longer equivalent.

Figure 2

Observed scattering from the acoustic spin wave at four cubically equivalent (0,0,1/2)-type reduced wavevectors (a) $(4,0,-1∕2)$, (b) (0,0,4.5), (c) (1/2,0,4), and (d) (4.5,0,0) at $T=130\mathrm{K}$ (cubic, empty circles) and $T=115\mathrm{K}$ (monoclinic, filled circles). Measurements were made in configuration A in a horizontal magnetic field along the cubic [001] direction. The (0,0,1/2) and (1/2,0,0) reduced wavevectors are inequivalent in the Verwey phase. The spin wave excitation is split below $T_V$ at the (0,0,1/2) position [(a) and (b)], but only weakly split along (1/2,0,0) [(c) and (d)] demonstrating that splitting occurs primarily along the cell doubling direction. Spin wave intensity is approximately twice as large for wavevectors along [001] [(b) and (c)] than along [100] [(a) and (d)] due to the neutron spin cross section.

Figure 3

Temperature dependence of (a) the (6,0,1/2) superlattice peak, and (b) the acoustic spin wave intensity at $\mathbf{Q}=(0,0,4.5)$ and $\hslash \omega =43\mathrm{meV}$ measured in configuration B. The appearance of the superlattice peak below $T_V$ occurs simultaneously with the splitting of the spin wave branch. Slanted arrows indicate cooling/warming.

Figure 4

(Color online) Constant-$\mathbf{Q}$ energy cuts made for several $\mathbf{Q}$-vectors along the [001] direction at (a) $300\mathrm{K}$ and (b) $115\mathrm{K}$, in configuration C. This instrument configuration has no applied magnetic field, thus the spin waves excitations below $T_V$ are averaged over all three orthorhombic directions. The gap is still clear despite this and is indicated by the slanted red line in (b) and the dotted line marks the gap energy at (0,0,4.5) and $T=115\mathrm{K}$. In each figure, gray circles mark the spin wave peaks. The dashed lines and arrow indicate phonon excitations.

Figure 5

(Color online) Constant energy contour plots of the scattering in the [h0l] plane near the (004) Brillouin zone above $T_V$ [$T=134\mathrm{K}$, left panels, (a), (b), and (c)] and below $T_V$ [$T=115\mathrm{K}$, right panels (d), (e), and (f)]. Mesh scans performed at constant energies of $39\mathrm{meV}$ [top row, (a) and (d)], $43\mathrm{meV}$ [middle row, (b) and (e)], and $46\mathrm{meV}$ [bottom row, (c) and (f)] were used to construct the contour plots. These energies correspond to the lower peak, gap, and upper peak of the low-temperature split spin wave excitation at (0,0,1/2). Measurements were performed in the D configuration with the HB1 spectrometer in zero-applied field.

Figure 6

Constant energy cuts along the $(h,0,4.5)$ direction above $T_V$ $(T=150\mathrm{K}$, filled circles) and below $T_V$ $(T=110\mathrm{K}$, empty circles) at (a) $\hslash \omega =39\mathrm{meV}$, (b) $\hslash \omega =43\mathrm{meV}$, and (c) $\hslash \omega =46\mathrm{meV}$. Measurements were performed with the B configuration on the HB3 spectrometer in zero-applied field.

Figure 7

Schematic drawing of the local electronic states for an ${\mathrm{Fe}}^{2+}$ ion on the B site in the cubic phase of magnetite. The free ion term is split by strong cubic crystal field and a relatively weaker trigonal field, resulting in an orbital singlet ground state. This orbital singlet is unsplit by weak spin-orbit interactions; however, the doublet excited state is split by this interaction. Subsequent symmetry lowering due to the Verwey transition cannot split the orbital singlet ground state. All states are subjected to Zeeman splitting by the molecular field. Given an approximate value of $350\mathrm{T}$, the Zeeman splittings can lead to low lying local excitations of $\sim 100\mathrm{meV}$. The lowest dipole excitation of local character that is observable by neutron scattering is $\sim 200\mathrm{meV}$. The local excitations are well above the energies considered here.

Figure 8

Spin wave dispersion of cubic magnetite along the principal symmetry directions as calculated from a Heisenberg model. (Heisenberg parameters were obtained from Ref. 10.)

Figure 9

(a) The acoustic spin wave dispersion of magnetite along [001] and above $T_V$ as calculated from the Heisenberg model of Ref. 10 and projections of the resolution ellipsoids for the various experimental configurations. (b)–(f) show various resolution convoluted cross sections obtained from the Heisenberg model as compared to the measured data for the experimental configurations annotated on the figures.

Figure 10

(Color online) Analytical curves for the acoustic spin wave dispersion along [001] are compared to the Heisenberg model (black). The red curve is calculated from the high-temperature formula given in Eq. (20). The blue curves are the two low-temperature branches with ${\omega }_L({\omega }_U$) the lower (upper) branch as calculated in Eq. (22). The continuation of these branches is presumed to be flat and is shown by dotted lines. The dispersive parts of ${\omega }_L$ and ${\omega }_U$ are primarily cubic modes whose strong intensity is governed by the fitting parameter $I_1$ (solid blue curve) while the weak and flat parts are governed by $I_2$ (dotted blue curve).

Figure 11

The acoustic spin wave dispersion of magnetite along [001] as obtained from fits to the resolution folded analytical dispersion relations in various experimental configurations (a) above $T_V$, and (b) below $T_V$. Empty circles are the fitted energies of the spin wave modes. The solid lines are the analytical dispersion relations for typical values of the fitting parameters. The dotted line in (b) indicates a possible continuation of the dispersion of the upper and lower branches.

Figure 12

Resolution folded fits to the spin wave excitation at (4,0,-1/2) as measured in experimental configuration A at (a) $T=130\mathrm{K}$ and (b) $T=115\mathrm{K}$. Open circles are the experimental data and solid lines are resolution folded fits. The fitting gives an intrinsic spin wave width of $\sim 1.5\mathrm{meV}$ both above and below $T_V$. Thus, the apparent narrowing of the spin wave modes below $T_V$ is a resolution effect due to the flattening of the dispersion near the gap.

Figure 13

Constant-$\mathbf{Q}$ energy cuts of the spin wave at $\mathbf{Q}$=(0,0,4.4) in configuration C at (a) $T=300\mathrm{K}$, and (b) $T=115\mathrm{K}$. Dots are the measured data and lines are the result of convolution of the analytical expressions for $S(\mathbf{Q},\omega )$ with the resolution function. Arrows point to the fitted spin wave energy for each model. While (a) contains only a single excitation, (b) appears to contain three excitations at $\mathbf{Q}$=(0,0,4.4). However, the resolution calculations were performed with the parameter $I_2=0$ in (b) meaning that only one branch exists in the model at (0,0,4.4). The two higher-energy peaks at $38\mathrm{meV}$ and $44\mathrm{meV}$ are the split modes at (0,0,4.5) which are picked up by the tail of the resolution function, as shown schematically in the inset. The dispersion of any weak extra branch is difficult to pick up due to the combination of steep dispersion and coarse energy resolution.

Figure 14

Low-temperature spin wave dispersion for magnetite along the original cubic [001] direction as calculated for a Heisenberg model with superexchange interactions modified by (a) atomic distortions in the Pmca structure, (b) charge ordering obeying the Anderson condition and, (c) charge ordering violating the Anderson condition. In each figure, the dashed line corresponds to the original high-temperature acoustic spin wave branch.

Figure 15

(Color online) Neutron intensities along [001] in the Pmca structure with locally varied superexchange due to atomic distortions. Despite the large number of branches which are folded into the smaller Pmca zone, only the main cubic branches show appreciable intensity due to the small atomic distortions in the Verwey phase.

Figure 16

Transverse [(a) and (b)] and longitudinal [(c) and (d)] constant-$\mathbf{Q}$ scans through the acoustic spin wave branch at $\mathbf{q}=(0,0,1∕2)$ [parallel to magnetic field direction, (b) and (c)] and (1/2,0,0) [perpendicular to field, (a) and (b)] above $T_V$. Measurements are made in configuration A with a magnetic field of 1 $T$ applied along the [001] direction. The dashed line is the fit to a spin wave and a phonon excitation using the full triple-axis resolution convolution. The solid line is the spin wave contribution and the dotted line is the phonon contribution to the cross section.

Figure 17

(a) Transverse and (b) longitudinal scans through the spin wave mode below $T_V$ in configuration A. Convolution fits are shown for the spin wave (solid), phonon (dotted) and total intensity (dashed). The lower branch of the spin wave moves down to the energy of the optical phonon below $T_V$.