Nuclear inelastic scattering studies of lattice dynamics in magnetite with a first- and second-order Verwey transition

Nuclear inelastic-scattering studies were performed to infer temperature evolution of iron atom dynamics in magnetite samples exhibiting Verwey transition of first- and second-order (type-I and type-II materials). The possible difference in this evolution could rationalize the distinct properties of these classes of materials observed in heat capacity and diffuse scattering below the Verwey transition temperature $T_V$ and could explain the change in transition order triggered by a minute (below 0.3$\%$) altering of the iron sublattice. Although we have found the apparent stiffening of the phonon iron spectrum in the low-temperature phase, at the same time, we have shown that these spectra are rather similar for type-I and type-II materials, rendering the lattice vibration-based explanation of the distinct behavior of heat capacities very improbable. The calculation of phonon spectra, aimed at tracing the origin of various features in the phonon density of states (DOS), has shown that the local Coulomb interaction $U$ may have a large effect on phonon DOS. However, the change in the $U$ parameter cannot explain the difference in heat-capacity results for both classes of materials. Thus, an additional factor that differentiates these materials and possibly is responsible for the discontinuous character of the Verwey transition in stoichiometric magnetite still must be found.

Figure 1

Resistivity vs inverse temperature for magnetite Fe${}_{3(1-}$${}_{\delta }$${}_{)}$O${}_4$ with different stoichiometry parameters (based on Ref. 2) (a) and the universal $T_V$ vs 3δ = $x$ relation (based on Ref. 2 with subsequent results added). Note the change in transition order manifested by the transition spreading (a) and the corresponding two slopes in (b). The composition of the samples used for the present studies are marked by large bullets.

Figure 2

Experimental characteristics that differentiate between the Verwey transition of first and second orders. (a) Temperature dependence of the diffuse scattering at (8 0 $\frac{3}{4}$) for δ = 0 (type-I) and δ = 0.006 (type -II) materials (after Ref. 8). (b) The temperature dependence of heat capacity in Fe${}_{3-x}$Zn${}_x$O${}_4$. The baseline for $x$ = 0.028 (second-order transition) at $T$ < $T_V$ is larger than for first-order samples as better seen in the $T$ dependence of Debye ${\theta }_D$ (the inset; the peak is removed for clarity). For $T$ > $T_V$, the backgrounds are identical (after Ref. 17). Solid and dashed lines are heat-capacity baselines calculated from theoretical curves (see the text). Note that constant pressure heat capacity $C_p$ is measured, while constant volume $C_V$ is calculated.

Figure 3

The comparison of time spectra from different samples at (a) 50 K and (b) 150 K, proving the similarity of the hyperfine splitting (to a higher extent at $T$ < $T_V$).

Figure 4

Experimental Fe DOS for stoichiometric magnetite [here and in Figs. 5, 6, 7, $g$($E$) is per the Fe atom, unlike in Figs. 8 and 9, where the DOS were normalized to the formula unit]. In the full spectra (a) only the data for temperatures at 50 K (below $T_V$) and 150 K (above $T_V$) are shown, while the DOS divided by $E^2$ to check the quadratic $g$($E$) relation are plotted in panel (b) for all measured temperatures. In the inset of panel (a), a lower-energy part is presented, and the characteristic structures as well as phonon modes, mentioned in Ref. 9, are marked.

Figure 5

Experimental Fe DOS for the Zn-doped sample. Note the difference between low- and high-temperature spectra, mainly below $E$ = 15 meV and around the second peak at 22 meV, the feature present in both samples.

Figure 6

Comparison of experimental partial DOS for both samples at 50 K in (a) full and (b) limited energy ranges. In the inset of panel (a), a lower-energy part is presented using a better resolution.

Figure 7

Same as in Fig. 6 but at 150 K. Note that $g$($E$) for both samples and both temperatures are nearly identical.

Figure 8

Calculated phonon DOS for the magnetite cubic phase for the electron-electron correlation parameter $U$ = 0 eV (squares) and $U$ = 4 eV (line, red online). Note that Fe DOS scales are expanded in comparison with total and oxygen DOS.

Figure 9

Comparison of measured DOS, for stoichiometric magnetite and at $T$ = 50 and 150 K (points + lines), with calculated total DOS for iron (lines) for $U$ = 0 [panel (a)] and $U$ = 4 eV [panel (b)]; in each case, the tetrahedral component is presented separately. In the insets, the important low-energy region is shown.

Figure 10

Experimental heat capacity ($C_p$) curves for $x_{\mathrm{Zn}}$ = 0.028 (stars, blue online)[17] and stoichiometric magnetite sample (triangles) compared to calculated curves ($C_V$). For iron vibration, the measured DOS were used. For the stoichiometric sample (hollow black bullets), oxygen vibrations were described by calculated DOS with $U$ = 4 eV at $T$ < $T_V$ and with $U$ = 0 for $T$ > $T_V$. For the Zn-doped sample, either the same procedure was applied (#1, bulk blue squares), or $U$ = 0 oxygen DOS was used (#2, open blue squares) Note that the low-temperature upturn for type-II samples is not reproduced in any case. Finally, the Schottky-like two-level system model with an energy separation of ∼170 K and equal degeneracies was used in addition to experimental Fe DOS for $x_{\mathrm{Zn}}$ = 0.03 and oxygen vibrations for $U$ = 0 (red line). Note also that the large heat-capacity peak for stoichiometric magnetite was only marked.

Figure 11

Measured DOS for both samples as well as the results from Ref. 19 in $g$($E$)${}^{*}{E}^2$ for increasing the energy $E$ representation to amplify the changes in the second peak. Unlike in previous results, this peak also seems to be related to the Verwey transition.

Figure 12

Fe DOS for the stoichiometric magnetite bulk sample as well as for the stoichiometric thin film[19] at low and high temperatures and in $g/E^2$ vs $E$ coordinates. The gap starting below $E$ = 10 meV is apparent for both samples, although the spectrum for the bulk crystal is shifted to higher energies. Bullets show ${\Delta }_5$ and $X_3$ phonon modes as calculated in Ref. 9.