Experimental and theoretical studies of vibrational density of states in ${\mathrm{Fe}}_3{\mathrm{O}}_4$ single-crystalline thin films

This paper presents experimental and theoretical studies of lattice vibrations in a single-crystalline ${\mathrm{Fe}}_3{\mathrm{O}}_4\left(001\right)$ thin film. The investigations were carried out in order to see how the lattice dynamics changes at the Verwey transition. Vibrational densities of states (DOS) were obtained from nuclear inelastic scattering (NIS) of synchrotron radiation in the temperature range 25 to 296 K, while theoretical DOS were calculated ab initio within density functional theory. Experimental phonon density of states shows good agreement with calculated DOS, reproducing both the general features of main line groups as well as the groups’ structure. This is also in qualitative accord with heat capacity data, provided that experimental DOS is augmented with that calculated for oxygen atoms. We have observed a gradual change in the NIS raw data as well as the relevant DOS while lowering the temperature. In particular, the main peak in the energy region 15–25 meV shows increasing splitting on cooling. The Lamb-Mössbauer factor calculated in the course of DOS evaluation shows a pronounced drop in the vicinity of the Verwey transition that may be partly connected to the observed abrupt lowering of the count rate at approximately 7 meV for $T<T_V$. Since this is an indication of lattice stiffening below $T_V$, we conclude that we have found experimental evidence for lattice participation in the mechanism leading to the Verwey transition.

Figure 1

Top: Variation of the (0 0 3) “forbidden” peak intensity while temperature rising for ${\mathrm{Fe}}_3{\mathrm{O}}_4$ single-crystalline thin film, where it is clear that the transition region extends from 125 K down to 120 K. Bottom: Time nuclear forward scattering (NFS) spectra taken for temperature range from 80 to 130 K. The arrow indicates the temperature region, 115 to 120 K, which corresponds to the Verwey transition.

Figure 2

Temperature dependence of the out-of-plane lattice parameter (solid triangles: cubic $c$ for $T>T_V$ and $c_m∕2$ for $T<T_V$) for ${\mathrm{Fe}}_3{\mathrm{O}}_4$ thin film obtained from the variation of the position of (0 0 4) and (0 0 8) reflections. The corresponding temperature dependence of the distorted cubic lattice parameters $a$ ($=a_m∕\sqrt{2}$, denoted by solid circles), $b$ ($=b_m∕\sqrt{2}$, solid diamonds), $c$ ($=c_m∕2$, open squares) for powdered ${\mathrm{Fe}}_3{\mathrm{O}}_4$ single crystal is also presented. The stars denote the relevant parameters obtained with the proper refinement from high-resolution XRD synchrotron measurements (Ref. 6). Note that the magnitude of the $c$ parameter distortion is comparable to that obtained from our simplified refinement, but the sign is opposite. Finally, ▷ symbols describe temperature dependence of doubled MgO lattice constant. The inset shows the expanded view of temperature variation of the cubic lattice parameter $c$ for the thin film and the powder. The vertical line allows one to compare the Verwey transition temperature and the width of the transition for the thin film and the bulk.

Figure 3

Temperature dependence of normalized (to the value at $T_V$) cubic lattice constants for ${\mathrm{Fe}}_3{\mathrm{O}}_4$ and MgO powders compared to the normalized out-of-plane lattice parameter $c$ for ${\mathrm{Fe}}_3{\mathrm{O}}_4$ thin film (same as presented in Fig. 2).

Figure 4

(a) Raw data of nuclear inelastic scattering for various temperatures (dots) and with subtracted central peak (solid line). (b) Density of states extracted from NIS.

Figure 5

Temperature dependence of Lamb-Mössbauer factor $f_{LM}$. Light-gray spots are the results of calculations from “long” scans and dark-gray spots from processed “short” scans. The dashed line is only to guide the eye.

Figure 6

Normalized (to the peak at 17 meV) raw data at a narrow energy range. Error bars, representative for all data points, are shown only for a few points for $T=140\mathrm{K}$ for clarity.

Figure 7

Low-energy reduced DOS [as $g\mathit{\left(}E\mathit{\right)}∕E^2$ vs $E$] calculated from long scans. Note that the usual quadratic density of states relation is approximately valid for high $T$ up to about 10 meV, while no leveling off for low temperatures is seen down to 3 meV. Error bars, representative for all data points, are shown only for a few points for $T=100\mathrm{K}$ for clarity.

Figure 8

Ab initio calculated phonon dispersion relation for ${\mathrm{Fe}}_3{\mathrm{O}}_4$.

Figure 9

Ab initio calculated phonon density of states for ${\mathrm{Fe}}_3{\mathrm{O}}_4$. The thin line shows the relevant DOS calculated ab initio within density functional theory in Ref. 22.

Figure 10

Comparison of experimental DOS at 25 and 296 K with those calculated for both crystallographic positions (A—tetrahedral; B—octahedral) of Fe. Note that the experimental results for 25 K are shifted by $0.01{\left(\mathrm{meV}\right)}^{-1}$.

Figure 11

Experimental DOS measured at 296 K in comparison with the results drawn from nuclear inelastic scattering at room temperature reported in Ref. 22, room temperature inelastic neutron studies (shaded), and literature data from optic studies (vertical bars; here, only peak positions are marked).

Figure 12

Lattice heat capacity calculated from experimental DOS supplemented with calculated DOS for oxygen atoms.