Solid-State Electronics 185 (2021) 108083 Contents lists available at ScienceDirect Solid State Electronics journal homepage: www.elsevier.com/locate/sse A simple, robust, and accurate compact model for a wide variety of complementary resistive switching devices M. Saludes-Tapia a, *, M.B. Gonzalez b, F. Campabadal b, J. Suñé a, E. Miranda a a Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona, 08193 Cerdanyola del Valles, Spain b Institut de Microelectrònica de Barcelona, IMB-CNM, CSIC, 08193 Cerdanyola del Valles, Spain A R T I C L E I N F O A B S T R A C T Keywords: Complementary Resistive Switching (CRS) using memristive devices has been intensively investigated in the last Complementary Resistive Switching decade. The objective of CRS is to generate low and high resistance windows in the I-V characteristic of the Memristor selector device with the aim of reducing the sneak-path conduction problem in crossbar arrays. Though a wide Memdiode variety of compact models for CRS have been proposed, the one presented here stands out for its simplicity, Snapback Snapforward robustness, and accuracy. The flexibility of the memdiode model is demonstrated through a series of fitting exercises using experimental data found in the literature. The model script for the LTSpice XVII simulator is also provided. 1. Introduction constitute the CRS structure. In this regard, the model of each memristor needs to include the crucial features required for the realistic simulation Complementary Resistive Switching (CRS) takes place when two of the curves, such as the snapback (SB) and snapforward (SF) effects memristive devices are anti-serially connected forming a single func­ (see Fig. 2). These effects correspond to the vanishing and formation of a tional structure. This simple device arrangement has been proposed as a gap along the atomic filamentary structure, respectively, and are way of reducing the sneak-path conduction problem in crossbar arrays essential for correctly addressing the device behavior. used for information storage and neuromorphic computing [1]. As After the completion of the filament formation process (SET) in a illustrated in Figs. 1 and 2, the combined action of the two resistive memristive structure, the voltage at the constriction suddenly drops (SB switches, with their respective low (LRS) and high (HRS) resistance effect) following the circuit load line dictated by the series resistance Ri states, leads to the alternate appearance of ON and OFF conducting (green solid line in Fig. 2). This resistance can be internal, external, or windows in the I-V characteristic. Remarkably, these windows can both. Beyond this point, the filament laterally expands or, alternatively, exhibit a wide variety of behaviors both in shape (abrupt/progressive) accumulates defects at a constant voltage called the transition voltage Vt and magnitude (small/large) that are not only a consequence of the (vertical line in Fig. 2). When in LRS, the curve reaches the RESET point selected materials (metals and dielectric) but also the result of the at negative bias and drops (SF effect) following again the load line of the particular features of the generated filamentary structures (oxygen va­ device. The red solid line in Fig. 2 corresponds to the same curve but cancy- or metal ion-based pathways, lateral size, internal series resis­ taking into account the additional potential drop across the series tance, etc.). In this work, a simple, robust, and accurate compact model resistance. Depending on the magnitude of the SB and SF effects capable of reproducing different CRS behaviors reported in the literature occurring in each memristive device, the CRS I-V curve can exhibit is presented and tested. The CRS conduction characteristics are modeled abrupt or gradual transitions as those illustrated in Fig. 1. In order to connecting two opposite-biased quasi-static memdiodes [2–4] and the demonstrate the flexibility of our model to cope with these situations system is simulated in LTSpice XVII from Linear Technologies. A quasi- (Section II), a number of experimental curves found in the literature static approach is followed here because neither programming steps were analyzed and simulated (Section III). Interestingly, the proposed nor frequency effects are within the scope of this paper. Although this model can also be used to deconstruct the CRS experimental curve into may sound obvious, a key issue for the accurate modeling of the CRS I-V its separate constituents when the corresponding information is not characteristic is the detailed modeling of the individual memristors that available. * Corresponding author. E-mail address: mercedes.saludes@e-campus.uab.cat (M. Saludes-Tapia). https://doi.org/10.1016/j.sse.2021.108083 Available online 31 May 2021 0038-1101/© 2021 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). M. Saludes-Tapia et al. Solid State Electronics 185 (2021) 108083 Fig. 4. Experimental and model results for two Ti/HfO2/Hf-based structures Fig. 1. Simulated I-V characteristics with gradual (green line) and abrupt (red anti-serially connected [5]. line) transitions between the ON and OFF states. The curves were simulated with the model reported in this work. Fig. 2. Simulated I-V characteristics for a single memristor with (green line) Fig. 5. Experimental and model results for a SiO2/GeSe-based CRS structure and without (red line) SB and SF corrections (V-IRi). Notice that the Vr=-Vt in [1]. Notice that the I-V characteristic is not symmetric. the corrected curve. ISB denotes the triggering current for the SET process. used for I0 . (1) resembles the I-V relationship for two opposite-biased diodes with shared series resistance (see Fig. 3(b)). Eq.(1) replaces the Lambert function and the Hermite-Padé approximation considered in [4]. The second equation relates the memory state λ to the voltage drop across the constriction VC = V − (RS (λ) + Ri )I through the recursive hysteresis operator [2]: { [ ← ]} λ(VC ) = min Γ− (VC ), max λ(VC ), Γ+ (VC ) where Γ+ and Γ− are the so-called ridge functions, which physically (←) represent the ion/vacancy movement. λ VC is the memory value a timestep before (hysteretic behavior). The model contains other pa­ rameters for the fine-tuning of the simulated curves. In particular, the Fig. 3. Schematic circuit for (a) the CRS structure consisting in two memristors gam parameter, not considered before [4], is used to fit the reset region anti-serially connected; (b) the internal implementation of the QMM for a single of the devices. Eqns. (1) and (2) are implemented in LTSpice using an device (see Fig. 10 in the appendix). The model contains a two-port circuit for equivalent circuital approach with behavioral components and sources the I-V characteristic and a memory circuit for tracking the hysteresis effect. (see the script in the Appendix). A very important feature of the model is The memory circuit controls the diode parameters. that the SET event can be triggered not only by a set voltage VS but also by a threshold current ISB (see Fig 2). This leads to a variety of behaviors 2. The memdiode model for CRS devices suitable for capturing the details of the experimental curves. In this re­ gard, it is worth mentioning the possibility of generating gradual or The Quasi-static Memdiode Model (QMM) is considered here for abrupt ON transitions in the CRS curves as well as departures from the simulating each device in the CRS structure [2–4]. As illustrated in Fig. 3 abrupt OFF transitions. The key issue behind the observation of abrupt (a), two anti-serially connected devices of this kind define the CRS ON transitions is a noticeable SB effect in the individual devices. structure. The I-V characteristic for each memdiode reads: I(V) = I0 (λ)sinh{α(λ)[V − (RS (λ) + Ri )I ] } 3. Experimental curves and simulations where I0 (λ) = Iomin + (Iomax − Iomin )λ is the current amplitude factor. In order to assess the versatility of the QMM for CRS devices, we Iomin and Iomax are calibration parameters. V is the voltage across the present simulations for a number of experimental I-V characteristics terminals, RS a variable series resistance, and α a fitting parameter. Both reported by other authors. The model parameters are indicated in the α and RS , if required, can be described by relationships similar to that table next to each curve. Figure 4 shows experimental and simulation 2 M. Saludes-Tapia et al. Solid State Electronics 185 (2021) 108083 Fig. 6. Experimental and model results for a symmetric Tantalum oxide-based Fig. 9. Experimental and model results for TiN/TiOxNy/TiO2-x/Pt memristors CRS structure [6]. forming a CRS structure [9]. (gam). Figure 8 shows results for the I-V curve of carbon nanotubes (CNT)-based devices [8]. The parameter values used in this simulation are very different from those used before because the switching voltages are higher than those observed in the other materials. Finally, Fig. 9, which corresponds to a combination of TiN/TiOxNy/TiO2-x/Pt devices [9], exhibits both gradual and large ON and OFF transitions in the CRS conduction characteristic. They are well captured by the proposed model. It is worth mentioning that, though in all the cases we attempted to use in the simulations parameter values close to their corresponding experimental ones, it was sometimes difficult to identify the right values. The current behavior is affected by a combination of parameters and the only information available is the final CRS curve. In our simulations, and for simplicity, we have considered initial states λ=0 and λ=1 for each memdiode. Experimentally, both memdiodes are initially in the HRS state and reach the complementary behavior in the following cycles. The Fig. 7. Experimental and model results for two-sided graphene oxide doped simulated curves are shown for the stationary loop. silicon oxide-based memristors forming a CRS structure [7]. 4. Conclusions Complementary resistive switching is a key element to control the crosstalk effects in memristor-based crossbar arrays. In this paper, we have reported and discussed a compact model (the quasi-static Fig. 8. Experimental and model results for a-C(amorphous-Carbon)/ CNT- based memristors forming a CRS structure [8]. results for combined TiN/HfO2/Hf devices [5]. In this plot, the SET and RESET transitions are abrupt and have associated SB and SF effects, respectively. Notice the small departures of the experimental curve at the end of the OFF transitions and how the model is able to capture the detail. Figure 5 illustrates completely abrupt CRS transitions for two Pt/ SiO2/GeSe/Cu devices [1]. Figure 6 shows a similar behavior for two bilayer Pd/Ta2O5-x/TaOy/Pd combined devices but in log-linear axis [6]. Figure 7 shows experimental and simulation results for graphene/ Zr-based structures [7]. As reported by these authors, the two-sided graphene oxide (GO) structures exhibit an intrinsic current restriction ability and uniform switching (the model can also be applied in this case). Notice the details of the OFF transitions achieved by means of the Fig. 10. LTSpice script for the QMM used in RS. The label QMM designs the effective voltage reduction parameter in the reset equation of the model memdiode. To simulate CRS, two devices of this kind are connected in series. 3 M. Saludes-Tapia et al. Solid State Electronics 185 (2021) 108083 [7] Chang K-C, et al. Physical and chemical mechanisms in oxide-based resistance random access memory. Nanoscale Res. Lett. Dec. 2015;10(1):120. https://doi.org/ 10.1186/s11671-015-0740-7. [8] Yang Chai et al., “Resistive switching of carbon-based RRAM with CNT electrodes for ultra-dense memory,” in 2010 International Electron Devices Meeting, 2010, pp. 9.3.1-9.3.4. DOI: 10.1109/IEDM.2010.5703328. [9] Panda D, Simanjuntak FM, Tseng T-Y. Temperature induced complementary switching in titanium oxide resistive random access memory. AIP Adv. Jul. 2016;6 (7):075314. https://doi.org/10.1063/1.4959799. Mercedes Saludes Tapia received the bacheloŕs degree in physical chemistry from the University of Barcelona and the M. S degree in advanced nanoscience and nanotechnology from the Autonomous University of Barcelona, where is currently Fig. 11. Model CRS curve obtained with the script shown in Fig. 10. The inset pursuing the PhD degree in computational nanoelectronics. of the figure shows the schematic used for the simulation. Her current research interests include electrical charac­ terization and compact modeling of resistive switching devices. memdiode model) suitable for the simulation of a wide variety of experimental data. The model for each individual device consists in two equations, one for the electron transport and a second equation for the memory state of the device. When both structures are combined, the CRS behavior emerges. To the best of our knowledge, no other published model can cope with the variety of curves reported here. Mireia Bargallo Gonzalez received the degree in physics from the University of Barcelona, Barcelona, Spain, and the Ph.D. degree on the topic of stress analysis and defect characteriza­ Declaration of Competing Interest tion techniques of semiconductor materials and devices, from Katholieke Universiteit Leuven, Leuven, Belgium, in 2011. She The authors declare that they have no known competing financial pursued her Ph.D thesis with the Interuniversity Microelec­ tronics Center (imec), Leuven, Belgium. interests or personal relationships that could have appeared to influence In 2011, she joined the Institut de Microelectrònica de the work reported in this paper. Barcelona (IMB-CNM, CSIC). Her current research interests include electrical characterization, modeling and applications of resistive switching devices. Acknowledgements This work was supported by the Spanish Ministry of Science, Inno­ Francesca Campabadal received the Ph.D. degree in physics vation, and Universities through projects TEC2017-84321-C4-1-R, from the Universitat Autònoma de Barcelona, Bellaterra, Spain, TEC2017-84321-C4-4-R and WAKeMeUP, co-funded by grants from in 1986. She joined the Institut de Microelectrònica de Barce­ the Spanish Ministry of Science and Innovation and the Electronic lona, Consejo Superior de Investigaciones Científicas, Barce­ lona, Spain, in 1987, where she is currently a Research Components and Systems for European Leadership-European Union Professor. Her current research interests include the deposition (ECSEL-EU) Joint Undertaking. of high-k dielectric layers, their electrical characteristics, and the resistive switching phenomena in RRAM devices. Appendix The code shown in Fig. 10 for LTSpice XVII corresponds to a single memristive structure. When connected in series two of these devices as illustrated in the inset of Fig. 11, the CRS behavior shows up. The script contains four sections: i) model parameter definitions, ii) memory Jordi Suñé is a full professor of Electronics at the Universitat equation using the recursive operator, iii) I-V characteristic with series Autonoma de Barcelona (UAB). He is the coordinator of the NANOCOMP research group, dedicated to the modeling and resistance effects, and iv) definition of the auxiliary functions. The simulation of electron devices with a multi-scale approach. His model can be simplified further but was written this way for the sake of main contributions are in the area of gate oxide reliability for clarity. The output H corresponds to the memory state of the device and CMOS technology. In 2008, he received the IBM Faculty award for a long-lasting collaboration with IBM Microelectronics in can be eliminated as a pin. this field. Since 2008, he has worked in the area of memristive devices and their application to neuromorphic circuits. In 2010, he received the ICREA ACADEMIA award and, in 2012 and References 2013, he was awarded the Chinese Academy of Sciences Pro­ fessorship for Senior International Scientists, for a collaboration with IMECAS (Beijing, [1] Linn E, Rosezin R, Kügeler C, Waser R. Complementary resistive switches for passive China). He has (co)authored more than 400 papers (h-index = 44) in international journals nanocrossbar memories. Nat. Mater. May 2010;9(5):403–6. https://doi.org/ and relevant conferences, including 14 IEDM papers, several invited papers, and five tu­ 10.1038/nmat2748. torials on oxide reliability at the IEEE-IRPS. Recently, he launched a new research group/ [2] Miranda E. Compact Model for the Major and Minor Hysteretic I-V Loops in network (neuromimeTICs.org) dedicated to the application of neuromorphic electronics to Nonlinear Memristive Devices. IEEE Trans. Nanotechnol. Sep. 2015;14(5):787–9. artificial intelligence and to dissemination activities. https://doi.org/10.1109/TNANO.2015.2455235. [3] Patterson GA, Sune J, Miranda E. “Voltage-Driven Hysteresis Model for Resistive Switching: SPICE Modeling and Circuit Applications”, IEEE Trans. Comput. Des. Integr. Circuits Syst. 2017;36(12):2044–51. https://doi.org/10.1109/ TCAD.2017.2756561. [4] Miranda E, Frohlich K. Compact Modeling of Complementary Resistive Switching Devices Using Memdiodes. IEEE Trans. Electron Devices 2019;66(6):2831. https:// doi.org/10.1109/TED.2019.2913322. [5] Wouters DJ, et al. Analysis of Complementary RRAM Switching. IEEE Electron Device Lett. Aug. 2012;33(8):1186–8. https://doi.org/10.1109/LED.2012.2198789. [6] Yang Y, Sheridan P, Lu W. Complementary resistive switching in tantalum oxide- based resistive memory devices. Appl. Phys. Lett. May 2012;100(20):203112. https://doi.org/10.1063/1.4719198. 4 M. Saludes-Tapia et al. Solid State Electronics 185 (2021) 108083 Enrique Miranda is Professor at the Universitat Autonoma de Barcelona (UAB), Spain. He has a PhD in Electronics Engi­ neering from the UAB (1999) and a PhD in Physics from the Universidad de Buenos Aires, Argentina (2001). He received numerous scholarships and awards including: RAMON y CAJAL (UAB), DAAD (Technical University Hamburg-Harburg), MATSUMAE (Tokyo Institute of Technology, Japan), TAN CHIN TUAN (Nanyang Technological University, Singapore), WALTON award from Science Foundation Ireland (Tyndall National Institute), Distinguished Visitor Award (Royal Acad­ emy of Engineering, UK), CESAR MILSTEIN (CNEA, Argentina), Visiting Professorships from the Abdus Salam International Centre for Theoretical Physics, Slovak Academy of Sciences, Politecnico di Torino, Leverhulme Trust (University College London, UK), and Nokia Foundation (University of Turku, Finland). He serves as member of the Distinguished Lecturer program of the Electron Devices Society (EDS-IEEE) since 2001 and as Associate Editor of Microelectronics Reliability since 2003. He has authored and co-authored around 250 peer-review journal papers. 5