Transient dynamics of NbO_x threshold switches explained by Poole-Frenkel based thermal feedback mechanism

Niobium oxide (NbO_x) based two-terminal devices can display current-controlled negative differential resistance (CC-NDR) when operated using a current source or, equivalently, threshold switching when operated using a voltage source, making them potential candidates as selector devices in crossbar memory arrays^1–9 and as building blocks for neuromorphic systems.^10–16 While CC-NDR or threshold switching in NbO_x has been known for decades,¹⁷ its physical mechanism is still under debate. Various mechanisms have been proposed, including thermally driven insulator-metal transition (IMT),^2,9,11,18,19 electrically driven IMT,^7,20 and thermal feedback (also known as “thermal runaway”) based on Poole-Frenkel conduction which is essentially the modulation of temperature-dependent electrical conductivity by device self-heating.^14,21–26

Historically, we have discovered some evidence, i.e., model fitting to quasi-static and transient electrical characteristics, which is in support of the high-temperature (∼1000 K) IMT mechanism in NbO_x threshold switches.¹⁸ However, we have recently discovered stronger evidence that supports the lower temperature (∼400 K) Poole-Frenkel based thermal feedback mechanism, including in-operando temperature mapping and transmission X-ray spectromicroscopy,²⁴ and model fitting to quasi-static electrical characteristics^23,25 which is considerably better than what we obtained for the IMT mechanism (see Sec. 3 in the supplementary material for details). While the IMT mechanism can coexist with the thermal feedback mechanism during the large temperature excursions associated with threshold switching,²⁴ our work provides evidence that the lower-temperature thermal feedback mechanism triggers such a threshold switching.

The above observations, together with other recent reports,^21,22 indicate that thermal feedback is likely the correct mechanism for threshold switching in NbO_x. However, while the IMT mechanism has been shown to reproduce the transient electrical characteristics during threshold switching in NbO_x,^11,18 the ability for the thermal feedback mechanism to model such transient characteristics remains unexplored.

Here, we show the that thermal feedback mechanism based on Poole-Frenkel conduction can explain both the quasi-static and the transient electrical characteristics experimentally observed for NbO_x threshold switches, providing strong support for the validity of this mechanism. Moreover, using our NbO_x devices as an example, we present a clear picture of the transient dynamics during threshold switching for devices where thermal feedback is important. This leads to a better understanding of the transient characteristics of thermal feedback threshold switches, which is important for their application in both memory arrays and neuromorphic computing.

The NbO_x devices used in this study are crossbar-type (2 μm × 2 μm in size) fabricated on top of freely suspended Si₃N₄ membrane windows. The material stack from the bottom electrode (BE) to the top electrode (TE) is Pt (15 nm)/NbO_x (15 nm)/Ti (2 nm)/Pt (25 nm), where NbO_x has an estimated stoichiometry of x = 2.2–2.3.²⁵ The fabrication process and the device structure are identical to those described elsewhere,²⁵ except for an additional 30 nm thick sputtered Si₃N₄ passivation layer. Electrical measurements were performed using an Agilent 4156C semiconductor analyzer, a Keithley 3402 pulse generator, and an Agilent MSO7104A oscilloscope. In all cases, signals were applied to TE, with BE grounded. A Temptronic TP03010 thermo-chuck was used to control the temperature during electrical measurements. A forming process using current sweeps was used to condition the as-fabricated devices. Note that this forming process is gradual and spatially uniform (see Sec. 1 in the supplementary material for details), and so, the NbO_x in formed devices is likely to remain spatially uniform.

Figure 1 displays the quasi-static IV characteristics of the NbO_x devices at different ambient temperatures, showing good agreement between experimental data and simulation results using the Poole-Frenkel based thermal feedback mechanism. A complete description of the quasi-static simulation can be found in Ref. 25. Briefly, the electrical conductivity within the active material (NbO_x) is modeled by modified 3D Poole-Frenkel conduction²³ described by

where $F$ is the electric field, $εi$ is the high frequency dielectric constant, $σp$ is the prefactor, and $Ea$ is the activation energy. During device operation, the temperature is approximated to be spatially uniform within the active NbO_x region, and so, the heat transfer at the quasi-static limit can be approximated by the following equation:

where $T$ is the temperature within the active NbO_x region, $Tamb$ is the ambient temperature, and $Rth$ is the thermal resistance. $Vdev$ is the voltage across NbO_x rather than the applied voltage $(Vapp)$⁠, which is related through $Vapp=Vdev+IRseries_DC$⁠. These equations are solved self-consistently to generate the simulated quasi-static IV curves. $Ea$ and $σp$ are experimentally extracted from Arrhenius plots by measuring the temperature dependence of low-field electrical conductivity, as is shown in the inset of Fig. 1, and are used as input parameters for the simulation. $Rth$⁠, $Rseries_DC$⁠, and $εi$ are allowed as fitting parameters, with their extracted values listed in Fig. 1. Note that due to the temperature dependence of resistivity of Pt electrode bars, $Rseries_DC$ at $Tamb=70 °C$ is slightly larger than at $Tamb=20 °C$⁠. Also note that the maximum temperature during device operation (corresponding to the maximum sweeping current of $I=2 mA$⁠) is calculated to be $∼300 °C$⁠, which is well below the IMT temperature of NbO_x (⁠$∼800 °C$⁠).²⁷ 

For transient measurements, we use the measurement setup shown in the inset of Fig. 2. Current is monitored by measuring the voltage across a 50 $Ω$ load resistor using an oscilloscope. This load resistor, together with the 50 $Ω$ output impedance of the pulse generator, gives a total series resistance during the transient measurement (⁠$Rseries_AC)$ that is 100 $Ω$ larger than the quasi-static case (⁠$Rseries_AC=Rseries_DC+100 Ω)$⁠. The input pulse $[Vappt]$ has a very short (∼2 ns) rise/fall time compared with the pulse width (200 $μs$⁠), so that it can be approximated as a step function.

Figure 2 shows the transient electrical characteristics of the NbO_x devices at different ambient temperatures and pulse voltages, showing good agreement between experimental data and simulation results using the Poole-Frenkel based thermal feedback mechanism. Details about the transient simulation can be found in Sec. 2 of the supplementary material. Importantly, we use the same equations as in the quasi-static simulation except for (2), which is replaced by the more general heat transfer equation

The parameters determined from quasi-static measurements and fitting are used as input for the transient simulation. Note that the $Rth$ extracted from fitting quasi-static IV curves is used here to calibrate the thermal conductivity of Si₃N₄ in the finite element model, so that the $Rth$ calculated from the finite element model exactly matches the $Rth$ extracted from fitting quasi-static IV curves (see Sec. 2 in the supplementary material for details). It is important that the quasi-static and the transient simulation use the same set of parameters, i.e., the model can simultaneously and self-consistently explain both the quasi-static and the transient electrical characteristics of NbO_x threshold switches. This strongly supports that thermal feedback based on Poole-Frenkel conduction is the correct physics in our NbO_x threshold switches.

One possible complication about modeling transient characteristics of self-heating-modulated devices (such as NbO_x threshold switches) is that there are two RC transients at play, i.e., an electrical RC transient and a thermal RC transient. Both can affect the transient electrical response of the device, which may prevent a straightforward and unambiguous interpretation of the results. However, if one of the two RC transients is significantly slower than the other, then the switching transient will be dominated by the slower one. In such a case, it would be reasonable to assume that the faster transient can be neglected when modeling the switching transient.

The membrane-suspended device structure used in this study is chosen for its slow thermal RC transient so that the thermal RC transient is slow enough compared to the electrical RC transient. Figure 3 shows experimental current waveform exhibiting a fast (∼50 ns) transient around t = 0, which is not captured in the simulation result. Since the simulation neglects the electrical RC transient, this fast transient should be the electrical RC transient, and it is indeed negligible compared to the switching transient (∼100 $μs$⁠). Note that while we have focused on the case where the condition “electrical RC transient is negligible” is met, the validity of the conclusion for the Poole-Frenkel based thermal feedback mechanism is clearly not limited by this condition.

In real applications, the electrical RC transient might not be necessarily faster than the thermal RC transient. In such cases, the electrical RC transient would not be negligible. However, when the electrical RC transient is slower than the thermal RC transient, the speed of the voltage change across the device would be slower than the speed of re-establishing steady state heat transfer in response to the voltage change. This means that heat transfer would be approximately at a steady state at all times. In other words, the device would behave as if operated using a quasi-static voltage sweep, which would be a trivial case for understanding of switching dynamics. Therefore, in the following discussion, we will focus on the case where the electrical RC transient is faster than the thermal RC transient. Also, when the speed of electrical RC transient and thermal RC transient happens to be comparable, their interaction could result in complicated switching dynamics, which may be addressed in future publications.

Figure 4(a) shows the simulated time evolution of temperature $T$⁠, electric field $F$⁠, and electrical conductivity $σ(F,T)$ in the active NbO_x region during the switching transient. Importantly, the electrical conductivity shows an abrupt jump at t = 0 and then increases gradually after that before saturating. This two-step increase in electrical conductivity indicates a two-step switching process. In the first step (t = 0), the electric field turns on almost instantaneously (due to the fast electrical RC transient), while the temperature cannot change instantaneously (due to the slow thermal RC transient), and so, the abrupt jump in electrical conductivity at t = 0 is driven by the electric field [through the electric field related terms in (1)]. In the second step (t > 0), the temperature increases gradually, while the electric field decreases gradually, and so, the gradual increase followed by saturation in electrical conductivity at t > 0 is driven by Joule heating and counteracted by the decrease in the electric field. The decrease in the electric field is not surprising because as the electrical conductivity of NbO_x increases, the resistance of NbO_x decreases (while the series resistance remains unchanged), and so, the voltage drop on NbO_x decreases (while the voltage drop on the series resistance increases since the total applied voltage is fixed). Figure 4(b) further illustrates the importance of series resistance. Clearly, without the series resistance, temperature and current increase uncontrollably, eventually damaging the device. When the series resistance is present, temperature and current increase in a more controlled way, eventually reaching a steady state.

Figure 5 illustrates the connection between quasi-static and transient electrical characteristics of the NbO_x devices. The green curve is a hypothetical device where the thermal insulation is set to zero so that its temperature is clamped to the ambient temperature. Therefore, point “A” should align horizontally with point “C,” with the electric field turned on but no temperature rise yet. On the other hand, the transient should eventually settle to a steady state, and so, point “B” should align horizontally with the asymptote (t = $∞$⁠) of the transient waveform. Therefore, thermal feedback during switching transient can be visualized as going vertically from point “A” to point “B” on the quasi-static IV plot. Clearly, as $Vapp$ moves into the subthreshold region, points “A” and “B” become closer, and so, thermal feedback has much smaller effect in the subthreshold region compared to the superthreshold region. For selector applications, this means that thermal feedback is contributive to the on/off ratio (typically defined as the current at some read voltage $Vread$ divided by the current at $12Vread$⁠). Therefore, if the NbO_x selectors are operated at a frequency so high that thermal feedback cannot progress to a large extent, not only the current drive but also the on/off ratio will be significantly compromised.

While this discussion on the transient dynamics of thermal feedback has used our NbO_x devices as an example, it is actually applicable to any device where thermal feedback is important. Thermal feedback would be important when the following two conditions are met: (i) the temperature rise due to Joule heating is significant and (ii) the electrical conductance of the device increases rapidly with temperature. For example, Goodwill et al.²⁸ have shown that electrical conductivity in TaO_x can also be described by (1). Funck et al.²² and Slesazeck et al.²¹ have proposed expressions for NbO_x electrical conductivity which are different from (1) and also increase rapidly with temperature. Also, in all these cases, the temperature rise due to Joule heating is shown to be significant. Therefore, our picture of thermal feedback during switching transient is applicable to these cases as well.

In conclusion, our experiments and analysis show that the thermal feedback mechanism assuming Poole-Frenkel conduction can explain both the quasi-static and the transient electrical characteristics of NbO_x threshold switches, supporting the validity of this mechanism. In addition, a clear picture of the transient dynamics of the thermal feedback during the threshold switching is presented, providing useful insights required to model nonlinear devices where thermal feedback is important.

See supplementary material for details on the forming process, the transient simulation, comparison with our previous work on the IMT mechanism, and some comments on activation energy, electrical capacitance, temperature dependence of $Rth$⁠, and electronic instability.

We gratefully thank Dr. R. Stanley Williams for useful suggestions. This work was supported by the member companies of Stanford Non-Volatile Memory Technology Research Initiative (NMTRI). Z. Wang was additionally supported by the Stanford Graduate Fellowship. Device fabrication was performed at the Stanford Nanofabrication Facility that was supported by National Science Foundation through the NNIN under Grant No. ECS-9731293.