Statistical distribution of the electric field-driven switching of the Verwey state in Fe₃O₄

Magnetite is an archetypal strongly correlated transition metal oxide, with properties not well described by single-particle band structure. Below 858 K, magnetite, which may be written as Fe³⁺_A(Fe²⁺Fe³⁺)_BO₄, is ferrimagnetically ordered, with the A and B sublattices having oppositely directed magnetizations. The moments of the five unpaired 3d electrons of the tetrahedrally coordinated A-site Fe³⁺ ions are compensated for by those of the octahedrally coordinated B-site Fe³⁺ ions. The net magnetization results from the octahedrally coordinated B-site Fe²⁺ that have four unpaired 3d electrons [1]. Upon cooling, bulk magnetite undergoes a first-order phase transition from a moderately conducting high-temperature state to a more insulating low-temperature state at what is now called the Verwey [2] temperature, T_V ≈ 122 K. The change in electronic properties coincides with a structural transition from a high-temperature cubic inverse spinel to a low-temperature monoclinic unit cell. The nature of the ordered insulating state remains an active topic of current research [3–6]. Experiments indicate the onset of multiferroicity [7] in magnetite below 40 K [8], further highlighting the rich physics in this correlated system.

In recent years, nanostructured electrodes have been used to apply strong electric fields in the plane of magnetite films [9, 10]. Below T_V, a sufficiently large applied voltage triggers the breakdown of the comparatively insulating low-temperature state and a sudden increase in conduction [9, 10]. This is an example of the electric field-driven breakdown of a gapped state in strongly correlated oxides [11–13], similar to the Landau–Zener breakdown in classic semiconductors. The electric field-driven transition in magnetite is consistent with the expectations [13] based on such a mechanism (via geometric scaling [9, 10], lack of intrinsic hysteresis [14] and changes of both contact and bulk resistance at the transition [15, 16]). These previous experiments examined films of various thicknesses, ranging from 30 to 100 nm. No strong thickness dependence was observed in the switching properties, consistent with the applied lateral electric field at the sample surface acting as the driver of the breakdown (although thinner films showed a less pronounced Verwey transition in low-bias resistance versus temperature measurements, consistent with expectations).

Here we report studies of the statistical variations of this electric field-driven transition in Fe₃O₄ as a function of the temperature and the magnetic field perpendicular to the film surface (out-of-plane). We find that there is a statistical distribution of switching voltages, V_SW, that becomes broader and shifts to higher voltages when T is reduced. We discuss these trends in the context of switching kinetics in other systems exhibiting similar trends. The application of a magnetic field perpendicular to the plane of the Fe₃O₄ film alters V_SW, shifting the mean by several mV (several per cent) and changing its shape, within a range of fields comparable to that required to reorient the magnetization out of plane.

The 50 nm Fe₃O₄ (100) thin films used in the present study were grown on (100) oriented MgO single-crystal substrates as described elsewhere [17, 18]. Contact electrodes (2 nm adhesion layer of Ti and 15 nm layer of Au) were patterned by e-beam lithography on the surface of the Fe₃O₄ film. As before [9, 14], V_SW scales linearly with the channel length, L (the electrode spacing), implying an electric field-driven transition. Long channels (L > 100 nm) required large switching voltages that would alter the electrode geometry over numerous switching cycles, distorting the shape of V_SW histograms. To minimize V_SW, electrodes separated by 10–30 nm were patterned using a self-aligned technique [10]. Electrical characterization of the devices was performed using a semiconductor parameter analyzer (HP 4155A). To minimize self-heating when in the conducting state, the voltage was applied as pulses 500 μs in duration with a 5 ms period [14, 15]. The samples were cooled below T_V with no magnetic field applied, and the distribution of V_SW was obtained by executing several thousands of consecutive forward pulsed I–V sweeps in the vicinity of the transition point (typically the 0.2–0.3 V range) at a fixed (to within 50 mK) temperature, and recording the number of switching events at each voltage.

Each voltage value is essentially an independent test to see if switching takes place under the pulse conditions. Hence, some sweeps show one (figure 1(a)) or several (figure 1(b)) switching events. Even if the system is switched to the conducting state at V_SW(1), it may return to the Off state between pulses and then switch to the On state at some higher voltage, V_SW(2), and so on. The V_SW distribution at a particular temperature is built by recording all switching events over several thousands (3000–6000) of I–V cycles and then counting the number of switchings at a certain V_SW, to produce a 'number of counts' versus V_SW histogram. A typical V_SW distribution at 90 K is shown in figure 1(d). The distribution is a single peak, symmetrical around the most probable V_SW value.

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This procedure was repeated at each temperature below T_V (∼110 K for the devices under test; see figure 2(b) inset), down to ∼75 K. At T < 75 K, the high values of V_SW necessarily led to irreversible alteration of the electrode geometry, resulting in asymmetric, distorted V_SW histograms. As T was decreased, I–V cycles with multiple V_SW events (figure 1(b)) were observed more frequently. Thus, the total number of switching events observed varied with T, even with a fixed number of I–V cycles at each temperature. To compare V_SW distributions at different temperatures, the distributions were normalized, plotted as (the number of counts)/(max. number of counts) versus V_SW, where 'max. number of counts' is the number of events at the most probable V_SW and 'the number of counts' is the number of events at a certain V_SW. Figure 2(a) is an example of normalized V_SW distributions in the temperature range 77–105 K. The measured widths of the V_SW distributions are not limited by temperature stability.

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As has been discussed elsewhere [14], the use of pulses is essential to minimize the role of self-heating once the system has been driven into the more conducting state. This self-heating and the short timescale [14] required to raise significantly the local temperature in the channel make it extremely challenging to determine directly whether the initial breakdown takes place through the formation of a conducting filament or through a uniform switching; once a highly conducting path is formed, the whole channel rapidly becomes conducting through self-heating. The filamentary picture is certainly likely, based on other breakdown phenomena in solids, and the statistical variation in V_SW is consistent with the idea of a process involving run-to-run variability associated with local details rather than global material properties, but this is not definitive.

The V_SW distribution at each temperature is characterized by two main parameters: the mean switching value $\bar {V}_{\mathrm {SW}}=(\sum _{i=1}^N V_{{\mathrm {SW}},i})/N$, where N is the total number of switching events, and the width of the distribution, calculated as a standard deviation: $\sigma (V_{\mathrm {SW}})=\sqrt {(\sum _{i=1}^N (V_{{\mathrm {SW}},i}-\bar {V}_{\mathrm {sw}})^2)/{N-1}}$. As expected, the $\bar {V}_{\mathrm {SW}}(T)$ has the same T dependence (see figure 2(b)) as V_SW(T) in single I–V experiments described in previous publications [9, 14]. More interesting is the σ(V_SW) temperature dependence, showing a broadening of the V_SW distribution as the temperature decreases (figure 2(c)). We note a deviation from the monotonic temperature dependence of σ(V_SW) at 100 K, observed in several devices tested. This is a temperature well below T_V = 110 K (see figure 2(b) inset), where several physical parameters (resistance, heat capacity and magnetoresistance (MR)) change abruptly.

This increase in σ(V_SW) as the temperature decreases is rather counter-intuitive. One might expect 'freezing' of temperature fluctuations and a decrease in thermal noise as the temperature decreases and, thus, narrowing of V_SW distributions. The field-driven breakdown of the insulating state is an example of the 'escape-over-barrier' problem, addressed generally by Garg [19]. Below T_V, the (temperature-dependent) effective free energy of the electronic system is at a global minimum value in the insulating state, while the external electric field modifies the free energy landscape, lowering the free energy of another local minimum corresponding to the more conducting state. As the external field is increased beyond some critical value, the minimum corresponding to the more conducting state becomes the global minimum. The nonequilibrium transition to the conducting state then corresponds to some process that crosses the free energy barrier between these minima. At a sufficiently large value of the external field, the free energy has only one minimum, corresponding to the conducting state.

As Garg showed, one may consider thermal activation over the free energy barrier as well as the possibility of quantum escape. This free energy picture predicts a broadening of the transition driving force (V_SW here) distribution as the absolute value of the driving force increases, consistent with our observations. This free energy picture has proven useful in studying other nonequilibrium transitions, such as magnetization reversal in nanoparticles [20] and nanowires [21, 22]. Pinning due to local disorder is one way of finding increasing distribution widths as T → 0, as seen in investigations of field-driven magnetization reversal in nanowires [21, 22]. Unfortunately, quantitative modeling in this framework requires several free parameters and is difficult without a detailed understanding of the underlying mechanism.

A qualitatively similar phenomenology (a distribution of switching thresholds that broadens when T is decreased) is also observed in the current-driven superconducting–normal transition in ultrathin nanowires [24, 25]. In the latter case as in ours, self-heating in the switched state is of critical importance, as is the temperature variation of the local thermal path. Again, quantitative modeling using this self-heating approach would require the introduction of multiple parameters that are difficult to constrain experimentally, as well as detailed thermal modeling of the nanoscale local effective temperature distribution, and is beyond the scope of this paper.

We also examined the dependence of the switching distributions on the applied out-of-plane magnetic field, H. V_SW distributions (3000 cycles each) were collected consecutively at 12 magnetic field values: 0 T (first) →0.2 T → 0.4 T → 0.6 T → 0.8 T → 1 T →  2 T → 3 T → 4 T → 5 T → 6 T → 0 T (last). Figure 3(a) shows the resultant V_SW distributions at 80 K at several selected magnetic fields. As can be seen, magnetic field shifts the V_SW peak to higher V values and narrows the V_SW distributions. To assure ourselves that the observed $\bar {V}_{\mathrm {SW}}$ shift is not from irreversible changes in the device, a control experiment returning to H = 0 T was performed after experiments in all nonzero magnetic fields. The V_SW distributions at H = 0 T initially [H = 0 T (first)] and in the end [H = 0 T (last)] are identical (see figure 3(a)), meaning that the observed changes in V_SW distributions (shift and narrowing) are indeed caused by the applied magnetic field rather than irreversible changes to the device. Figure 3(b) quantifies the dependence of $\bar {V}_{\mathrm {SW}}$ and σ(V_SW). It is clear that both parameters saturate when H is increased beyond 1 T, i.e. further increases of H up to 6 T have no significant effect. Magnetic field of the opposite polarity (not shown) has exactly the same effect on the position and width of V_SW distributions. Note that the shape of the distribution, in particular its asymmetry about the peak value of V_SW, evolves nontrivially with magnetic field, becoming more symmetric in the high field limit.

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We consider whether this magnetic field dependence of the switching originates with some dependence of the bulk resistance or contact resistances, as this would alter the electric field distribution in the channel. Since V_SW scales with the channel length, L [9, 14], i.e. as does the resistance of the channel (R ∼ L), one might expect that an increase in V_SW could result from an increase in the device resistance with applied magnetic field. However, Fe₃O₄ has a negative MR [23, 26, 27]. Figure 4(a) shows an example of the normalized resistance dependence, R/R(H = 0T), on the out-of-plane magnetic field at 80 K. The resistance remains effectively unchanged up to ∼0.4 T and then decreases as |H| increases. Thus, when $\bar {V}_{\mathrm {SW}}$ and σ(V_SW) experience the predominance of changes upon H application (H < 1 T, see figure 3), the resistance of the device either stays constant or decreases. In the H range when R experiences significant changes (see figure 4a), $\bar {V}_{\mathrm {SW}}$ and σ(V_SW) remain essentially unchanged (figure 3). Therefore, the shift of $\bar {V}_{\mathrm {SW}}$ in the presence of H does not originate from the change in the resistance value of the device.

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Another Fe₃O₄ film parameter affected by H is the magnetization of the film. Figure 4 shows the normalized out-of-plane magnetization, M/M_s, as a function of the out of plane H, where M is the magnetization of the film and M_s is the saturated magnetization. These data were consistent with previous measurements on magnetite films [28]. While we do not know the microscopic arrangement of M in the film in the absence of an external H, magnetostatic energy considerations mean that M under that condition lies in the plane of the film. The H range over which M is fully reoriented out of the plane (up to 1 T) matches the H range of changes in the position of $\bar {V}_{\mathrm {SW}}$ and σ(V_SW,T) (figure 3(b)). This suggests (though does not prove) that the switching kinetics parameters $\bar {V}_{\mathrm {SW}}$ and σ(V_SW,T) and therefore the stability of the gapped, low-temperature, insulating state are tied to the magnetization direction of magnetite films.

This observation is intriguing because it is not clear how the nonequilibrium breakdown of the low-temperature state would be coupled to the magnetization. Possible factors include magnetoelastic effects such as magnetostriction [29] (∼parts in 10⁴ per tesla), affecting the tunneling matrix element between B-site iron atoms; and spin–orbit coupling, playing a similar role [30]. There have been reports of significant magnetoelectric and multiferroic effects in magnetite [7, 8], and a recent calculation [30] argues that these originate through the interplay of orbital ordering and on-site spin–orbit interactions of the B-site electrons. In this picture, reorientation of the spin distorts the partially filled minority-spin orbitals occupied on the B-site (formally) Fe²⁺ ions. Such a distortion would be a natural explanation for the observed correlation between M and the kinetics of the electric field-driven breakdown of the ordered state, which directly involves the motions of those charge carriers. It is unclear how this kind of spin–orbit physics would explain the evolution of the V_SW distribution, however. It would also be worth considering whether there is a correlation between the characteristics of the switching distributions reported here and the recently observed glassy relaxor ferroelectric relaxations in bulk magnetite crystals [31].

Additional, detailed experiments as a function of directionality of H, M and crystallographic orientation would be able to test these alternatives. With the existing (100) films, studies of $\bar {V}_{\mathrm {SW}}$ and σ(V_SW,T) as a function of H in the plane as well as perpendicular to the plane would be able to access the tensorial form of the H dependence. Comparison with appropriately directed M versus H data as a function of temperature would be a clear test of whether the observed agreement between H-field scales (in V_SW) and reorientation of M is a coincidence. Further measurements on films grown with different crystallographic orientations would serve as a cross-check. It is important to note, however, that the acquisition of such data is very time-intensive due to the need to acquire many thousands of switching events. In turn, there is an accompanying requirement for extremely good device stability to avoid irreversible changes in the metal configuration over thousands of switching cycles.

We have studied the statistical distribution of the electric field needed for the breakdown of the low-temperature insulating state of Fe₃O₄. The distribution of critical switching voltages moves to higher voltages and broadens when T is reduced. This broadening is consistent with phenomenology in other nonequilibrium experimental systems incorporating disorder and thermal runaway effects. The breakdown distributions are altered by modest external magnetic fields normal to the film, suggesting the need for further experiments to understand the connection between magnetization and the breakdown of the correlated state.

The authors acknowledge valuable conversations with Paul Goldbart and David Pekker. This work was supported by the US Department of Energy grant no. DE-FG02-06ER46337. DN also acknowledges the David and Lucille Packard Foundation and the Research Corporation. RGSS and IVS acknowledge the Science Foundation of Ireland grant number 06/IN.1/I91.