At the 2018 Royal Society of Chemistry Faraday Discussion on memristive phenomena, R. S. Williams challenged Ella Gale's demonstration of computation with 'spiking memristors', arguing that the observed current spikes of her solution-processed Al/TiO₂/Al devices are the classical transient response of a leaky capacitor (an RC circuit), not memristance. Gale's extended response marshalled published and unpublished device data arguing that the devices are non-ideal memristors with parasitic capacitance (an MC circuit): they show pinched hysteresis loops at higher voltages, their low-voltage open hysteresis loops grow with electrode width, a memristor theory fits the spike waveforms, and the spikes underlie the frequency effect, a defining memristor fingerprint. The response explicitly left the question open: the quantitative memristor+capacitance model with geometry-dependent parameters was never published, the microscopic switching mechanism was not identified, and the closing alternative it states for reported memristor current transients in general — that some are memristive in form, that the frequency effect is due to the capacitance found in non-ideal memristors, or that many, possibly all, 'memristors' are just leaky capacitors — was left undecided. The question of whether spiking memristors are true memristors, memristor–capacitor composites, or leaky capacitors remains open.

Background

The memristor ('memory resistor'), conjectured by Leon Chua in 1971 as a fourth fundamental two-terminal circuit element, is a resistor whose value depends on the past history of charge or flux through it. Real-world memristive devices are usually identified by a pinched hysteresis current–voltage loop (the loop passes through the origin), whose lobe area shrinks as the driving frequency increases — the 'frequency effect' — together with other 'fingerprints' formalized by Adhikari, Sah, Kim and Chua (2013). Interest surged after Strukov, Snider, Stewart and Williams at HP demonstrated a TiO₂ nanoscale device (2008) whose resistance is switched by drift of oxygen vacancies (missing oxygen atoms that act as mobile dopants), the basis of much resistive random-access memory (ReRAM) research. Ella Gale's group fabricated a distinct, macroscopic variant: sol-gel solution-processed amorphous TiO₂ films about 44 nm thick, spin-coated from titanium hydroxide precursor onto sputtered aluminium electrodes of 1–6 mm width on flexible plastic, i.e. Al/TiO₂/Al crosspoint devices. Under small d.c. voltage steps these devices show decaying current spikes (their 'd.c. response'), which Gale's group used as the computational primitive for neuromorphic schemes — habituation, sequential logic gates, and a sequence-sensitive perceptron (Faraday Discussions 213, 2019, 521–551). The aluminium electrodes proved essential (gold-electrode devices did not switch), attributed to the native Al₂O₃ layer acting as a source/sink of oxygen ions. Gale analysed the devices with her 'memory-conservation theory' of memristance, a model in which memristance arises from redistribution of a conserved quantity of mobile dopant between doped and undoped regions of the oxide. At the Faraday Discussion, Williams countered that the spike in Fig. 1 of Gale's paper is a classical capacitor transient: from the roughly 5 s decay time and about 2.7 nA asymptotic current he estimated a 37 MΩ parallel resistance and about 135 nF capacitance, concluding one could reproduce the behaviour 'with no memristors involved' — the devices would be leaky capacitors, not memristors; he also noted a second, somewhat longer time scale in the response, which he attributed to electrons escaping from charge traps in the sol-gel oxide. Gale's extended printed response (arXiv:1812.05976) argued the devices are imperfect memristors with some capacitance: some devices are indeed capacitor-like (3 of 64), but most show pinched hysteresis at higher voltage; at low voltage the loops are open ('jelly-bean' curves that do not cross zero, indicating capacitance) yet are not pure-capacitor circles; the memory-conservation theory (plus measured contact resistance) predicts the loop shape and asymmetry but underestimates the measured hysteresis by a roughly constant ratio, with the missing contribution growing with electrode width, which Gale attributed to a geometry-dependent device capacitance; the spike waveforms are fit better by the memristor theory than by exponentials; and stepped-voltage experiments show the spikes generate the shrinking-hysteresis frequency effect, tying them to a defining memristor fingerprint. What the response does not contain is the decisive quantitative device model: the capacitance term, its parameter values, and its functional dependence on electrode geometry were left as planned future work in an unpublished-elsewhere 2013 preprint (arXiv:1312.4422), the microscopic mechanism (oxygen-vacancy ionic drift versus phase change versus filamentary fuse–antifuse switching, with Williams additionally proposing charge-trap detrapping for the slower component) remained unsettled even in Gale's own 2014 review of TiO₂ memristors, and the broader attribution problem — many groups have published transient currents in memristive devices — was posed but not answered. Related definitional debates continued in the literature (e.g., Abraham 2018, arguing against the memristor as a fundamental circuit element), and later transient-relaxation mechanisms were established only for other material systems (e.g., diffusive Ag-filament memristors, Wang et al., Nature Materials 2017), not for sol-gel TiO₂.

Problem Statement

Decide, quantitatively, whether the d.c. current spikes and low-voltage open hysteresis loops of sol-gel-processed Al/TiO₂/Al memristors arise from memristance, from parasitic capacitance, or from both — i.e., whether these 'spiking memristors' are true (non-ideal) memristors best modelled as memristor–capacitor (MC) circuits, as Gale argued, or merely leaky capacitors (RC circuits), as Williams' Faraday Discussion question asserted. Each of these verdicts is an admissible answer: the objective is to establish which description the device evidence supports, not to confirm the MC description in particular.

The quantitative core is the device-model comparison the exchange itself frames. On one side is Williams' leaky-capacitor equivalent circuit for the spike in Fig. 1 of Gale's paper (roughly 5 s 1/e decay time, about 2.7 nA asymptotic current, about 37 MΩ effective parallel resistance, about 135 nF capacitance, together with the second, longer time scale he attributed to electrons escaping from charge traps in the sol-gel oxide). On the other is the memory-conservation memristor theory extended by a capacitance whose parameter values and functional dependence on device geometry (electrode width/area) are measured — the extension Gale announced ('I plan to add the capacitance into the model in that paper before publication'; 'This is something I plan to add to the paper') but never published. The comparison is against these devices' measured responses: the low-voltage open V–I loops with their electrode-size-dependent hysteresis and the d.c. spike I–t waveforms, which are to be accounted for by whichever description is found correct, in a way that discriminates between MC and pure-RC accounts of the same data (e.g., through amplitude, ramp-rate, and frequency dependence). The evidence may come from analysis or re-analysis of the published device data (the electrode-width hysteresis series and the spike traces), from new measurements, or from both. Explicitly, that comparison is to supply the capacitance term, its parameter values, and their dependence on electrode geometry (electrode width/area) whenever the evidence supports a memristor-plus-capacitance description; those quantities are moot if the pure-RC (leaky-capacitor) description is found correct.

One further question is left open in the source but is not a condition of answering it: what microscopic process underlies the switching and relaxation in these devices — oxygen-vacancy ionic drift, interfacial Al₂O₃-mediated phase/chemistry effects, filamentary fuse–antifuse processes, detrapping of electrons from charge traps in the sol-gel oxide, or another mechanism — and what roles the aluminium electrodes and the amorphous oxide play. Gale records that 'figuring out the mechanism behind all of this was rather difficult and time consuming (and still not settled in the field)', and she concludes that the devices are non-ideal memristors without settling it, so the RC-versus-MC question can be answered without a mechanism determination.

The verdict does bear on the attribution alternative Gale leaves open for previously observed memristor current transients in general, and an answer should state which of the enumerated possibilities it supports: that some previously observed 'current transients' are memristive in form, that the frequency effect is due to the capacitance found in non-ideal memristors, or that Williams is correct and many, possibly all, reported 'memristors' are just leaky capacitors. The possibilities are not mutually exclusive, and stating which of them the verdict supports is not an independent field-wide deliverable: it is enough to say which of them the evidence adduced is consistent with, at the scope that evidence covers. Her framing observation is that the term 'current transient' only means the current is short-lived and does not itself define the cause (capacitance or memristance), and that other groups have reported similar spikes in their devices.

References

  1. Ella M. Gale, My, and others', spiking memristors are true memristors: a response to R.S. Williams' question at the New Memory Paradigms: Memristive Phenomena and Neuromorphic Applications Faraday Discussion, arXiv:1812.05976 (2018), https://arxiv.org/abs/1812.05976
  2. Ella M. Gale, Neuromorphic computation with spiking memristors: habituation, experimental instantiation of logic gates and a novel sequence-sensitive perceptron model, Faraday Discussions 213, 521–551 (2019), doi:10.1039/c8fd00111a, https://doi.org/10.1039/c8fd00111a
  3. Alexandra I. Berg, Stefano Brivio, Simon Brown, Geoffrey Burr, Sweety Deswal, Jonas Deuermeier, Ella Gale, Hyunsang Hwang, Daniele Ielmini, Giacomo Indiveri, Anthony J. Kenyon, Asal Kiazadeh, Itir Köymen, Michael Kozicki, Yang Li, Daniel Mannion, Themis Prodromakis, Carlo Ricciardi, Sebastian Siegel, Maximilian Speckbacher, Ilia Valov, Wei Wang, R. Stanley Williams, Dirk Wouters, Yuchao Yang, Synaptic and neuromorphic functions: general discussion, Faraday Discussions 213, 553–578 (2019), doi:10.1039/c8fd90065e, https://doi.org/10.1039/c8fd90065e, peer-reviewed version: https://backend.orbit.dtu.dk/ws/files/163016499/Session_4_c8fd90065e_3_.pdf
  4. R. Stanley Williams, Summary of the Faraday Discussion on New memory paradigms: memristive phenomena and neuromorphic applications, Faraday Discussions 213, 579–587 (2019), doi:10.1039/c8fd90061b, https://doi.org/10.1039/c8fd90061b
  5. Ella M. Gale, Benjamin de Lacy Costello, Andrew Adamatzky, Which Memristor Theory is Best for Relating Devices Properties to Memristive Function?, arXiv:1312.4422 (2013), https://arxiv.org/abs/1312.4422
  6. Ella Gale, Ben de Lacy Costello, Victor Erokhin, Andrew Adamatzky, The short-term memory (d.c. response) of the memristor demonstrates the causes of the memristor frequency effect, 2014 IEEE International Symposium on Circuits and Systems (ISCAS), 2812–2815 (2014), doi:10.1109/ISCAS.2014.6865758, https://doi.org/10.1109/ISCAS.2014.6865758
  7. Ella Gale, TiO₂-based memristors and ReRAM: materials, mechanisms and models (a review), Semiconductor Science and Technology 29, 104004 (2014), doi:10.1088/0268-1242/29/10/104004, https://doi.org/10.1088/0268-1242/29/10/104004
  8. Ella Gale, David Pearson, Steve Kitson, Andrew Adamatzky, Ben de Lacy Costello, The effect of changing electrode metal on solution-processed flexible titanium dioxide memristors, Materials Chemistry and Physics 162, 20–30 (2015), doi:10.1016/j.matchemphys.2015.03.037, https://doi.org/10.1016/j.matchemphys.2015.03.037
  9. S. P. Adhikari, M. P. Sah, H. Kim, L. O. Chua, Three Fingerprints of Memristor, IEEE Transactions on Circuits and Systems I: Regular Papers 60, 3008–3021 (2013), doi:10.1109/TCSI.2013.2256171, https://doi.org/10.1109/TCSI.2013.2256171
  10. Leon O. Chua, Memristor — the missing circuit element, IEEE Transactions on Circuit Theory 18, 507–519 (1971), doi:10.1109/TCT.1971.1083337, https://doi.org/10.1109/TCT.1971.1083337
  11. D. B. Strukov, G. S. Snider, D. R. Stewart, R. S. Williams, The missing memristor found, Nature 453, 80–83 (2008), doi:10.1038/nature06932, https://doi.org/10.1038/nature06932
  12. Isaac Abraham, The case for rejecting the memristor as a fundamental circuit element, Scientific Reports 8, 10972 (2018), doi:10.1038/s41598-018-29394-7, https://doi.org/10.1038/s41598-018-29394-7
  13. Zhongrui Wang, Saumil Joshi, Sergey E. Savel'ev, Hao Jiang, Rivu Midya, Peng Lin, Miao Hu, Ning Ge, John Paul Strachan, Zhiyong Li, Qing Wu, Mark Barnell, Geng-Lin Li, Huolin L. Xin, R. Stanley Williams, Qiangfei Xia, J. Joshua Yang, Memristors with diffusive dynamics as synaptic emulators for neuromorphic computing, Nature Materials 16, 101–108 (2017), doi:10.1038/nmat4756, https://doi.org/10.1038/nmat4756