Transverse barrier formation by electrical triggering of a metal-to-insulator transition

Application of an electric stimulus to a material with a metal-insulator transition can trigger a large resistance change. Resistive switching from an insulating into a metallic phase, which typically occurs by the formation of a conducting filament parallel to the current flow, is a highly active research topic. Using the magneto-optical Kerr imaging, we found that the opposite type of resistive switching, from a metal into an insulator, occurs in a reciprocal characteristic spatial pattern: the formation of an insulating barrier perpendicular to the driving current. This barrier formation leads to an unusual N-type negative differential resistance in the current-voltage characteristics. We further demonstrate that electrically inducing a transverse barrier enables a unique approach to voltage-controlled magnetism. By triggering the metal-to-insulator resistive switching in a magnetic material, local on/off control of ferromagnetism is achieved using a global voltage bias applied to the whole device.

Below we list the evidence supporting the electro-thermal origin of the observed switching in LSMO: 1. Reproducible switching is present in every as-made LSMO device without the need of an electroforming. In contrast, electroforming is often necessary to initiate oxygen migration 1 .
2. The switching is volatile, i.e. the device resets in the initial low-resistance state automatically upon turning off the driving voltage/current as expected for the Joule heating mediated process.
Supplementary Fig. 1. a, b, X-ray diffraction of the LSMO film: specular θ-2θ scan in the vicinity of the SrTiO 3 (002) peak (a) and reciprocal space map in the vicinity of the SrTiO 3 (103) peak (b). c, Resistance vs. temperature curves of the LSMO film (grey line) and two 50×100 μm 2 devices (green and red lines) showing similar behavior. . This permanent change of the resistancetemperature dependence could be due to the initiation of oxygen migration in the LSMO device and/or degradation of the electrode/oxide interface. During the acquisition of data presented in the main text and in the Supplementary Information, the application of high voltages/currents that could damage the device was avoided, unless stated otherwise. In this demonstration, the resistive switching measurements were performed 200 K.
4. The switching occurs in a wide temperature range up to Tc ≈ 340 K where the strong I-V nonlinearities disappear, which indicates a close relation between the switching and the MIT (see Fig. 2 b and c in the main text and Supplementary Fig. 2.3).
5. The high resistance state of ~5.5 kΩ attained after the switching remains the same independent of the measurement temperature. This high-resistance state corresponds to the maximum resistance in the equilibrium resistance-temperature dependence as demonstrated in Supplementary Fig. 2 Fig. 2.3. a, Voltagecontrolled I-V curves recorded in a 60-340 K range using a step of 20 K. b, Switching power and voltage corresponding to the onset of the NDR region extracted from the I-V curves in a. through the volatile resistive switching, even when the switching is induced in the entire 50×100 μm 2 device under the application of large voltages/currents (see Figs. 2 and 3 in the main text and Supplementary Fig. 3). If the ionic migration was responsible for the observed switching, it would be impossible that the stoichiometry and crystal structure could spontaneously restore to the pristine state within the 50×100 μm 2 device area when the voltage/current are turned off and no indication of chemical/structural change during the switching could be found in the MIT and ferromagnetic properties. The observed behavior, on the other hand, is consistent with the Joule heating origin of the switching. Warming up the device just past Tc ≈ 340 K is not expected to cause material decomposition, thus neither MIT not ferromagnetic properties are affected by the switching.
8. The switching power steadily increases with decreasing temperature, as expected for the Joule heating mediated process. Supplementary Fig. 2.3a shows voltage-controlled I-V curves recorded in a 60-340 K temperature range. Several I-V curves from this figure are also shown in the main text in Fig. 1c.
To characterize the switching parameters, we extracted the currents and voltages corresponding to the onset of negative differential resistance (NDR), i.e. the point at which dV/dI changes sign. Supplementary Fig. 2.3b shows the switching power and the switching voltage dependence on temperature. While switching power has a monotonic temperature dependence, the switching voltage shows a non-monotonic behavior. This result suggests that Joule heating rather than electric field drives the metal-to-insulator switching in our LSMO devices.
9. The switching is perfectly symmetric with respect to changing the driving voltage/current polarity (i.e. a unipolar switching). As shown in Supplementary Fig. 2.4, the I-V curves recorded using positive and negative driving stimuli perfectly coincide with each other when plotted on the absolute current/voltage scale. This is the expected behavior for the Joule heating mediated process because the electric power dissipated in the device is proportional to the square of driving current/voltage, P = I 2 R = V 2 /R. In this demonstration, the resistive switching measurements were performed at room temperature.
10. The switching is repeatable over a large number of cycles having virtually no cycle-to-cycle variations of low-and high-resistance states, of switching voltages and currents, of the shape of the I-V curves, etc. Supplementary  11. Under constant applied voltage, the high-resistance state can be maintained persistently for prolonged time without any signs of device degradation. In contrast, continual resistance change under constant voltage is often observed in memristors based on oxygen migration 4-8 . Supplementary Fig. 2.6 shows an 8-hour long voltage stress test. The measurements were done using 20 V stress, which is high enough to induce and maintain the high-resistance state at room temperature. While the applied voltage was kept on, we observed no resistance drift over the entire duration of the experiment. The I-V curves before and after the test perfectly coincide with each other indicating that no nonvolatile switching or irreversible damage were induced by holding large stress voltage over a long time. The stress test was done at room temperature.
12. The switching in LSMO shows high endurance and no cycle-to-cycle variability over 5×10 6 highspeed switching cycles. The electrical circuit used in the high-speed measurements is shown in Fig.  2.7a. Because of the large size devices optimized for MOKE imaging (50×100 μm 2 ), we had to use a combination of a function generator and an amplifier to produce large enough voltage/current to induce the switching of the entire device. The performance of the high-voltage amplifier determined the limit of how fast a switching cycle can be performed (15-ms-period waveform) and ultimately set the limit of how many switching cycles (5×10 6 ) can be acquired in a reasonable amount of time. Reducing the device size down to nanoscale dimensions should reduce the switching voltage/current enabling fast speed measurements to determine the ultimate switching time in LSMO and to probe the device endurance past several million cycles. Fig. 2.7b shows the dynamic I-V curves measured in a 50×100 μm 2 LSMO device. The dynamic I-V curves have similar appearance to the dc I-V curves (for example in Supplementary Fig. 2.4a). The larger hysteresis in the dynamic I-V curves is consistent with the Joule heating origin of the switching as thermal equilibrium in a large-size device cannot be established quickly. All the I-V curves recorded between the 1 and 5×10 6 cycles are identical resulting in no apparent dependence of the low-and high-resistance states on cycling ( Supplementary Fig. 2.7c).  The statistical analysis of the low-and high-resistance states over the switching cycles gives Rlow = 1150 ± 16 Ω and Rhigh = 4050 ± 10 Ω. The observed resistance cycle-to-cycle deviations (16 Ω and 10 Ω) are well within the accuracy of the "single-shot" oscilloscope measurements, indicating the absence of the device degradation and extremely high repeatability of the volatile resistive switching in LSMO. While high-endurance switching can be achieved in systems based on ionic migration, the lack of variability in the I-V shape or the low-and high-resistance states over 5×10 6 cycles suggests that the observed volatile switching in LSMO is not related to ionic migration.
13. The switching is independent of the oxygen partial pressure. Supplementary Fig. 2.8 compares currentand voltage controlled I-V curves recorded in air and in high vacuum (~10 -7 Torr). The switching behavior in both cases is identical, which indicates that the oxygen partial pressure does not play a significant role in the volatile resistive switching in LSMO. On the contrary, the nonvolatile resistive switching in LSMO driven by oxygen migration has strong dependence on the environmental oxygen pressure 9 . In Supplementary Fig. 2.7, small deviations in the I-V curves recorded in air and in vacuum most likely are due to a small modification of thermal conditions, which are expected to play a significant role in the Joule heating mediated switching process. In vacuum, the LSMO sample exchanges heat with the sample stage. In air, the sample can exchange heat both with the sample stage and with air. The switching measurements were performed at room temperature.
14. It is possible to induce nonvolatile resistive switching in our devices by first applying a large voltage (i.e. performing electroforming) and then cycling the device in moderate voltages ( Supplementary Fig.  2.9). During the electroforming, an irreversible breakdown occurs at ~45 V. After the breakdown, a hysteretic I-V curve showing a consistent nonvolatile switching can be obtained by cycling the device in ±6 V. This voltage is much lower, about factor of 5, compared to the voltage required to induce the MIT-based volatile switching, which could be an indication that the nonvolatile switching occurs in a small volume within the device, while the volatile switching happens throughout the entire device (as we established using MOKE measurements). The nonvolatile switching occurs between Rlow ~ 230 Ω and Rhigh ~ 440 Ω giving the resistance switching ratio of ∆R/R ~ 90%. We note that both Rhigh and Rlow states after the electroforming are noticeably different from the original resistance of ~660 Ω Supplementary Fig. 2.7. Fast switching cycling test. a, The measurement circuit that was used to apply a triangular 15-msperiod waveform to probe the switching in the LSMO device. b, Overlaid dynamic I-V curves recorded by an oscilloscope for different consecutive switching cycles ranging from cycle 1 to cycle 5×10 6 . No change in the I-V shape can be observed. c, Low-and high-resistance values extracted from the I-V curves as functions of switching cycles. The two resistance states are well separated and remained unaffected by 5×10 6 switching cycles. before the electroforming. In addition, the resistance-temperature dependence before and after the electroforming are dramatically different. The above observations indicate that the electroforming induced chemical and/or structural change inside the LSMO device. Overall, the I-V curves recoded during the nonvolatile switching have different shape and display a noticeably large cycle-to-cycle variability as compared to the volatile MIT-based switching, which highlights the different physical origin between the nonvolatile and volatile resistive switching in LSMO. Inducing a nonvolatile switching in LSMO by applying a high voltage is a qualitatively similar phenomenon as found in VO2 and V2O3, where a small voltages/currents cause a volatile insulator-to-metal switching, but a large stimulus triggers the nonvolatile oxygen migration 10 .
While several of the above properties could be sometimes found in resistive switching systems based on ionic migration, the fact that all of those properties are present at the same time provide strong evidence that the origin of the volatile resistive switching in LSMO devices is the triggering of MIT mediated by Joule heating.
We note that inducing the volatile resistive switching in our LSMO devices required the currents ranging from ~4 mA near room temperature to ~25 mA at 60 K. For a 50×100×0.02 μm 3 device, those current corresponds to the current density of 0.4-2.5×10 6 A/cm 2 . The current densities used in our work are considerably larger compared to the previous work that reported the absence of Joule heating mediated switching in LSMO in 10 4 -10 5 A/cm 2 range 11 . Supplementary Fig. 2.8. Current-(a) and voltagecontrolled (b) resistive switching measurements in high vacuum (~10 -7 Torr, blue line) and in air (red line). Oxygen partial pressure has no significant impact on the volatile MIT-based resistive switching in LSMO. The measurements were done at room temperature.  Supplementary Fig. 2.5), the nonvolatile switching displays noticeable cycle-to-cycle variability, which is common behavior in resistive switching systems based on oxygen migration. c, Comparison of the resistance-temperature dependence before and after the electroforming. Unlike the volatile MIT-based switching (see Supplementary Fig. 2.3), the electroforming (which is necessary to initiate the nonvolatile switching in our devices) dramatically changes the resistance-temperature dependence. The electroforming and nonvolatile switching measurements were done at 200 K. In both cases, we observed the same behavior as described in the main text: the switching from a metal into an insulator occurs by the formation of an insulating barrier that spans through the entire device width in the direction perpendicular to the current flow. The repeatability of the switching behavior indicates that the formation of an insulating barrier is a general property of the metal-to-insulator switching.  3. a, b, Simultaneously recorded I-V curve (center) and MOKE amplitude xy-maps (sides) in two different samples having different device geometry. Device in a is patterned in a 20 nm thick film and has 50×100 μm 2 size. Device in b is patterned in a 50 nm thick film and has 50×50 μm 2 size. The field of view in the MOKE maps is 90×140 μm 2 in a and 75×75 μm 2 in b. In the maps, the current flows horizontally. the All measurements were performed at 100 K.
To check whether the formation of an insulating barrier is not just an anomaly of the LSMO film, which, for example, could be introduced during the device fabrication, we measured the MOKE maps over 100-400 K temperature range without applying voltage (Supplementary Fig. 4). We used the same imaging procedure and the same magnetic field settings as described in Methods section in the main text. Under equilibrium conditions, we observed a spatially uniform transition throughout the device. Importantly, we found no correlation to the formation of an insulating barrier that we observed during resistive switching. This result demonstrate that the formation of an insulating barrier is a special property of the electricallydriven metal-to-insulator transition. The above equations were derived assuming ≪ in order to obtain analytically solvable equations by dropping ( / ) 3 terms. Because of the missing cubic terms, system (5.5) actually does not have an exact solution. Supplementary Fig. 5.2b shows graphically that close to the intercept point, the first equation in (5.5) does not have a real value because the √ 2 − 4 term becomes imaginary. However, the two curves in Supplementary Fig. 5.2b come very close to the interception point. Therefore, we can write a condition for an approximate solution as following System of equations (5.6) contains four unknowns: , ℎ , , and . In order to obtain an approximate solution, we used and 0 as estimations for the insulator and metal regions temperatures, and . As we found in our numerical simulations (see Fig. 3 in the main text), such estimations are justified. In addition, the temperature coefficients in (5.4), ( − 0 ) and ( − ), are always multiplied by other parameters, such as , , . It is possible to "absorb" the modest deviations of ( − 0 ) and ( − ) from ( − 0 ) into other parameters. Therefore, using ≈ and ≈ 0 does fundamentally alter the Supplementary Fig. 5.2. a, Schematic plot of equation (5.3) for three voltages: below and above the threshold (grey dashed lines) and at the threshold (red line). The minimum size of the insulating barrier is highlighted. b, Schematic plot of system of equations (5.5). The intercept of the two curves corresponds to the minimum insulating barrier size and threshold voltage ℎ to induce such a barrier. The shaded area highlights the region where Eq. 1 becomes imaginary. ℎ , on the resistivity ratio, device length, and film thickness. We plot the curves given by the approximate equations (5.7-5.8) (red and blue lines) and by numerically solving equation (5.3) using the ≈ and ≈ 0 temperature estimations (grey dashed lines) and using the condition that the derivative of (5.3) is zero at ℎ (see the discussion on p.6). The material and device parameters were = 340 K, 0 = 100 K, = 2×10 -6 Ω•cm, = 20, = 100 μm, = 20 nm, = 5×10 6 W•K -1 •m -2 , = 3×10 8 W•K -1 •m -2 . These parameters give = 2.2 μm and ℎ = 15.4 V, which is very close to the experimentally observed values. The two approaches give almost the same results, which further supports the validity of the approximation introduced in (5.6). The analytical model predicts that the minimum insulating barrier size can be substantially reduced, potentially down to nanoscale, by selecting a material with large Supplementary Fig. 5.3. Minimum insulating barrier size (top graphs) and threshold voltage (bottom graphs) dependence on resistivity ratio (a), device length (b) and film thickness (c). Continuous red and blue curves were obtained using equations (5.7-5.8). Dashed gray lines were obtained by solving numerically equation (5.3). Material and device parameters were set to the values presented in the text (p. 7), with the exception of the specific parameter for which and ℎ were calculated (i.e. resistivity ratio (a), device length (b) and film thickness (c)). insulator/metal resistivity ratio or by reducing the device dimensions, length and film thickness. We note that because of the simplifications made in our model these results should be regarded as guidelines rather than exact predictions.

Supplementary Information 6
The computational analysis of metal-to-insulator resistive switching was based on a resistor network model as shown schematically in Supplementary Fig. 6a. Each site of the 50×100 grid is represented by a 4-resistor node. The resistance of individual elements inside the nodes depends on temperature as These parameters were chosen to provide a semi-quantitative fit of the experimental ( ) (see Fig. 1a in the main text). Supplementary Fig. 6b shows the ( ) plot of the full resistor network.
I-V curves and resistance maps were calculated in an iterative way. For a given applied voltage, the resistor network is solved to obtain local voltages at each ( , ) site. Then these voltages are used to update local temperatures following the thermal diffusion equation: where = 2.0 and ℎ =0.4. Using the local temperatures, local resistances are updated according to equations (6.1-6.2). The iterative process of solving the resistor network and updating the local temperatures and local resistances is repeated until a steady state is found. Supplementary Fig. 6. a, A schematic of the resistor network used in simulations of the metal-to-insulator resistive switching. Resistor values at each node depend on local temperature given by equations (6.1-6.2). b, Simulated resistance-temperature dependence of the resistor network. c, Simulated voltage-controlled I-V curves in 60 -330 K temperature range.