3D simulation of conductive nanofilaments in multilayer h-BN memristors via a circuit breaker approach

rsc.li/materials-horizons Materials Horizons rsc.li/materials-horizons ISSN 2051-6347 COMMUNICATION Blaise L. Tardy, Orlando J. Rojas et al. Biofabrication of multifunctional nanocellulosic 3D structures: a facile and customizable route Volume 5 Number 3 May 2018 Pages 311-580 Materials Horizons This is an Accepted Manuscript, which has been through the Royal Society of Chemistry peer review process and has been accepted for publication. Accepted Manuscripts are published online shortly after acceptance, before technical editing, formatting and proof reading. Using this free service, authors can make their results available to the community, in citable form, before we publish the edited article. We will replace this Accepted Manuscript with the edited and formatted Advance Article as soon as it is available. You can find more information about Accepted Manuscripts in the Information for Authors. Please note that technical editing may introduce minor changes to the text and/or graphics, which may alter content. The journal’s standard Terms & Conditions and the Ethical guidelines still apply. In no event shall the Royal Society of Chemistry be held responsible for any errors or omissions in this Accepted Manuscript or any consequences arising from the use of any information it contains. Accepted Manuscript View Article Online View Journal This article can be cited before page numbers have been issued, to do this please use: D. Maldonado, A. Cantudo, F. M. Gomez Campos, Y. Yuan, Y. Shen, W. Zheng, M. Lanza and J. B. Roldan, Mater. Horiz., 2024, DOI: 10.1039/D3MH01834B. In this manuscript we present a new 3D memristor simulator based on circuit breakers (CBs). Previous simulators are 2D; therefore, it means a step forward in the description of memristor operation and the resistive switching processes. The CBs can be switched between different resistance values depending on the voltage between their terminals or on the CB temperature. By means of these mechanisms, that reflect the physics and chemistry involved in the operation of memristors, we are able to reproduce experimental data obtained in h-BN memristors and describe the conductive nanofilament formation and rupture that make the device operate. Moreover, we can reproduce reset processes where the current versus voltage curve presents several steps (partial nanofilament rupture). We can also describe defect regions in the dielectric (our simulation domain), allowing the study of pristine dielectrics and the corresponding resistive switching operation. The particularities of 2D materials (the case for the dielectric of our devices, hexagonal boron nitride) have been considered in the simulator and they helped to understand the operation and experimental measurements of our devices. Page 1 of 27 Materials Horizons Materials Horizons Accepted Manuscript Open Access Article. Published on 02 December 2023. Downloaded on 12/2/2023 1:33:39 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online DOI: 10.1039/D3MH01834B 1 Title: 3D simulation of conductive nanofilaments in multilayer h-BN memristors via a circuit breaker approach Authors: D. Maldonado1, A. Cantudo1, F. M. Gómez-Campos1, Yue Yuan2, Yaqing Shen2, Wenwen Zheng2, M. Lanza2, J.B. Roldán1 Address: 1Departamento de Electrónica y Tecnología de Computadores. Universidad de Granada. Facultad de Ciencias. Avd. Fuentenueva s/n, 18071 Granada, Spain. 2Materials Science and Engineering Program, Physical Sciences and Engineering Division, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia. Authors to whom correspondence should be addressed: [email protected], [email protected] Page 2 of 27Materials Horizons Materials Horizons Accepted Manuscript Open Access Article. Published on 02 December 2023. Downloaded on 12/2/2023 1:33:39 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online DOI: 10.1039/D3MH01834B 2 Abstract: A 3D simulation of conductive nanofilaments (CNF) in multilayer hexagonalBN memristors is performed. To do so, a simulation tool based on circuit breakers is developed including for the first time a 3D resistive network. The circuit breakers employed can be modeled with two, three and four resistance states; in addition, a series resistance and a module to account for quantum effects, by means of the quantum point contact model, are also included. Finally, to describe real dielectric situations, regions with a high defect density are modeled with a great variety of geometrical shapes to consider their influence in the resistive switching (RS) process. The simulator has been tuned with measurements of h-BN memristive devices, fabricated with chemical-vapour-deposition grown h-BN layers, that were electrically and physically characterized. We show the formation of CNFs that produce filamentary charge conduction in our devices. Moreover, the simulation tool is employed to describe partial filament rupture in reset processes and show the low dependence of the set voltage on the device area, that is seen experimentally. Index Terms — Memristor, resistive switching, 2D materials, simulation, circuit breaker, variability, defects Page 3 of 27 Materials Horizons Materials Horizons Accepted Manuscript Open Access Article. Published on 02 December 2023. Downloaded on 12/2/2023 1:33:39 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online DOI: 10.1039/D3MH01834B 3 I.-Introduction Memristive devices are being intensively studied in the Academia and industry in the last decade [Chua1976, Corinto2015, Lanza2022]. These devices show a great potential both for standalone and embedded nonvolatile memory applications [Pan2014, Lanza2019, Ielmini2015, Lee2015, Spiga2020]; in fact, they have already been incorporated in different industrial products [Lanza2022]. Their features make them fit the market needs (in general, as storage-class memory) and they are CMOS fabrication technology compatible (some memristive devices have a 4F2 footprint, where “F” is the minimum technology half-pitch) [Spiga2020]. Although memristor-based non-volatile memory applications are the most commercially advanced [Yang2020, Chou2020, Chou2018], these devices play an important role in other fields such as neuromorphic computing [Yu2011, Ambrogio2018, Merolla2014, Alibart2013, Zhu2023, Sebastian2020, Roldan2022, Prezioso2015, Zidan2018, Hui2021]. The neuromorphic engineering approach [Mead1989] allows the acceleration of matrix-vector multiplication (a key operation in Artificial Intelligence (AI) algorithms) that can be implemented through memristive device crossbar arrays [Lanza2022, Sebastian2020]. This approach can get over some of the hurdles of von Neumann’s bottleneck, that are linked to the constant data movement between the memory and the processor. Memristive devices, in this neuromorphic computing context, mimic biological synapses to permit the fabrication of hardware neural networks [Yu2011, Ambrogio2018, Roldan2022, Merolla2014, Alibart2013, Prezioso2015, Zhu2023, Dalgaty2021, Zidan2018]. In this respect, due to the inherent Page 4 of 27Materials Horizons Materials Horizons Accepted Manuscript Open Access Article. Published on 02 December 2023. Downloaded on 12/2/2023 1:33:39 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online DOI: 10.1039/D3MH01834B 4 redundancy of neural circuits, the device requirements do not need to be as strict as in non-volatile memory applications since AI methodologies allow a greater margin of variability and endurance [Sebastian2020]. Resistive switching memristors are fabricated with a thin layer of dielectric sandwiched between metal electrodes; their electrical and thermal features are closely linked to the materials employed. Different authors have described switching and charge conduction making use of the dynamics of metallic ions, oxygen vacancies and other defects, whose concentration evolves with time in the device active part [Aldana2020a, Menzel2017, Dirkmann2018, Bocquet2014, Aldana2020b, Menzel2015, Funck2021]. In particular, for filamentary conduction, the formation and destruction of CNFs is a stochastic process that leads to cycle-to-cycle (C2C) variability [Perez2019, Mikhaylov2021, Ielmini2015, Roldan2023, Lee2015]. This inherent variability (in addition to device-to-device (D2D) variability [Perez2019]) has to be minimized for memory applications; however, it could be beneficial in some cases for deep neural network training to avoid overfitting [Romero-Zaliz2021]. Variability is key for hardware cryptography (an entropy source that allows the fabrication of physical unclonable functions and random number generators) [Carboni2019, Pazos2023, Wei2016, Chen2015b, Lanza2021]. C2C variability is linked to CNFs morphological changes in each RS cycle, where the CF is created (set process) and ruptured (reset process) successively [Aldana2020a, Menzel2017, Dirkmann2018, Menzel2015, Funck2021]. The device C2C and D2D variability, and switching dynamics can be tackled from different simulation and modeling approaches such as: kinetic Monte Carlo (kMC) simulation [Vandelli2015, Aldana2020a, Dirkmann2018, Aldana2020b, Page 5 of 27 Materials Horizons Materials Horizons Accepted Manuscript Open Access Article. Published on 02 December 2023. Downloaded on 12/2/2023 1:33:39 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online DOI: 10.1039/D3MH01834B 5 Guy2015]; advanced statistical modeling [Roldan2019, Alonso2021] and compact modeling (for circuit simulation and design) [Huang2013, González-Cordero2017, Chen2015a, Corinto2015, Guan2012, Huang2017, Jiang2016, Roldan2021]. A different approach, although complementary, is based on RRAM simulation by means of circuit breakers (CB) [Lee2015, Chang2009, Lee2011, Chae2008, Brivio2017, Maldonado2022, Roldan2022b]. These CB-based simulators are bidimensional; nevertheless, a 3D approach is needed if CNFs (in case of filamentary operation) are to be described correctly, see in Ref. [Aldana2018] a study on the appropriateness of a 3D description in comparison with a 2D approach based on a kMC simulation tool. CB-based simulation poses a numerical technique in between kMC and compact modeling for circuit simulation in what is refereed to complexity, although it allows a reasonable description of variability and current versus voltage curves. In this work we present a 3D CB-based simulator that can describe the CNF evolution (that facilitates RS operation) and the charge transport in the filamentary operation regime. Apart from common features for these CBbased simulation tools [Lee2015, Chang2009, Lee2011, Chae2008, Brivio2017], we include quantum effects implemented through the quantum point contact model, the use of circuit breakers with four conductivity stages and a device series resistance. We also consider 3D regions of different shapes within the simulator domain to model dielectric zones with high defect density formed at the fabrication stage that evolve as the RS unfolds. This latter feature is hardly ever taken into consideration in simulation tools. Page 6 of 27Materials Horizons Materials Horizons Accepted Manuscript Open Access Article. Published on 02 December 2023. Downloaded on 12/2/2023 1:33:39 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online DOI: 10.1039/D3MH01834B 6 We have tuned our simulator making use of experimental measurements from hexagonal boron-nitride memristive devices [Lanza2021b] that we have fabricated. The devices characterized here have been studied previously [Roldan2022, Acal2023], physical and electrical experimental characterization was employed. Devices with some similarities in the layer stack were also analyzed from different viewpoints [Pan2017]. All these works were purely experimental. In this work we present an analysis where a strong simulation approach was introduced. The 3D modeling employed was implemented by means of a new CB-based simulator that allows to assess different physical effects on the RS operation and the role of high defect density regions in the dielectric on device variability and reliability. It is important to draw attention to the fact that there exist different physical characterization techniques to visualize conductive nanofilaments [Li2017, Knot2022, Knot2023]. They are based on the use of Conductive Atomic Force Microscopy (C-AFM) and Transmission Electron Microscopy (TEM). These techniques allow outstanding analyses that give us information about the CNF composition, size, shape, charge transport features, etc. This information can be used in the development of models and simulation tools. They are also important for the model and simulator calibration, in addition to electrical measurements. Once the simulators are tuned, they can complement C-AFM and TEM by providing an exact map of the temperature and the electric field in the simulation domain (usually the dielectric, although it could include the electrodes), the progress in the CNF formation, the influence of high defect density regions on RS, quantum effects, and charge transport processes. Page 7 of 27 Materials Horizons Materials Horizons Accepted Manuscript Open Access Article. Published on 02 December 2023. Downloaded on 12/2/2023 1:33:39 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online DOI: 10.1039/D3MH01834B 7 The measurements have been correctly fitted and different operational particularities have been explained in full. In particular, in section II, we introduce the fabrication and measurement details; the simulator description is presented in section III and the results and discussion is given in section IV. Finally, the main conclusions are drawn in section V. II.-Device fabrication and measurement setup The memristive devices used in our study have been introduced previously [Roldan2022]. The electrodes are made of a bilayer of 40 nm Au/10 nm Ti thick (E-beam evaporation is employed). The bottom electrode is deposited on a Si wafer, with 300nm SiO2 on top the Si. Then, a h-BN multilayer (18 layers approximately, see Figures 1a and 1b) film was placed on top of the bottom electrode by wet transfer from a Cu foil where it was grown by chemical vapor deposition. In Figure 1c RS I-V curves are shown. They are obtained with a B1500A Keysight semiconductor parameter analyzer and a probe station (Karl Suss); Ramped Voltage Stress (RVS) is used for the measurement of long RS series with consecutive set and reset cycles. RS operation is filamentary [Roldan2022]. In Figures 1d and 1e we plot the set and reset parameters, the Low and High Resistance States (LRS/HRS) are plotted in Figure 1f. It is clear that the resistance ratio (RHRS/RLRS) allows non-volatile memory applications (see the cumulative distribution functions of RHRS and RLRS in the inset in Figure 1f, a reasonable variability is obtained). Page 8 of 27Materials Horizons Materials Horizons Accepted Manuscript Open Access Article. Published on 02 December 2023. Downloaded on 12/2/2023 1:33:39 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online DOI: 10.1039/D3MH01834B 14 group of simulated curves is obtained by changing the thickness of the geometry shown in Figure 2e, and the probability linked to the CBs within this geometry in the low resistance state at the beginning of the simulation (see Figures 3e and 3f where the group of simulated I-V curves is shown). In spite of the approximations performed in the modeling implemented in the simulator, the curve fitting is reasonable (Figure 3d). In this respect the cycle-to-cycle variability can be described with our simulation tool. In Figure 4a we have plotted a simulated set current versus voltage. Different points have been marked along the curve in order to follow the CNF evolution. The low resistance value circuit breakers (Ron) are shown in red in Figures 4b-e (assuming two resistance values CBs); these panels correspond to the simulation points shown in symbols in Figure 4a. Notice how the CNF is formed as the set process unfolds till (in Figure 4e) it shorts the electrodes and constitutes a fully-formed conduction path. The latter points (2-4) correspond to the sudden current rise that is seen both in simulated and experimental curves close to the set point, where a positive feedback process linked to the CNF formation is triggered [Aldana2020a]. Page 15 of 27 Materials Horizons Materials Horizons Accepted Manuscript Open Access Article. Published on 02 December 2023. Downloaded on 12/2/2023 1:33:39 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online DOI: 10.1039/D3MH01834B 15 Figure 4. a Simulated current versus voltage in a set process. See different points along the I-V that corresponds to different CNF formation stages (the CNFs are plotted in panels be) An 18×18×18 network was employed to obtain this simulation using CBs with two resistance levels: Roff = 65.5 × 106 Ω, Ron = 0.24 Ω. The model parameters in relation with Figure 2 are: Voff = 0.11 V, Von = 0.25 V. A 10 scale factor was employed between horizontal (lower) and vertical (higher) resistances. f Simulated and experimental current versus voltage in a set process. A clear stepped curve is seen due to the partial CNF rupture along the reset process. Some of the experimental curves measured present a stepped-like shape (see Figure 4f) due to a CNF rupture in several stages. Our simulator can Page 16 of 27Materials Horizons Materials Horizons Accepted Manuscript Open Access Article. Published on 02 December 2023. Downloaded on 12/2/2023 1:33:39 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online DOI: 10.1039/D3MH01834B 16 reproduce this behavior. In this respect, a single CNF could be broken in steps, or two CNFs (formed in a previous set event) can go through a reset process and get broken at different times. Experimental and simulated curves are shown in Figure 4f to illustrate this effect. The simulator can also be used to study large area devices. For instance, 100 nm x 100 nm area devices are simulated maintaining the number of CBs per nanometer that corresponds to the description of the h-BN layers of the dielectric described above. In this respect, millions of CBs are taken into consideration (see Figures 5a and 5b for plots of a large area device simulation CB network). In this case, big matrix processing acceleration techniques have been implemented. In Figure 5c we have shown arbitrary I-V curves simulated for devices with different areas. In this case no regions of high defect density were assumed. The probability of finding CBs in the low resistance value at the beginning of the simulation (1% in these examples) was the same in all cases; notice that a different random distribution is generated at the start of each simulation. We have scaled the I-V curves by a factor Areasmallest_area_simulated/Areaactual_device_area in order to fairly compare the curves taking into consideration the purely resistive network we have employed to model the device. It is seen that, although the current curves are close together, the set voltage varies in each of the device areas employed. This parameter depends on the initial random distribution of low resistance CBs. We have plotted the set voltage versus device area in Figure 5d, it can be seen that as the area increases the set voltage decreases. This effect is linked to the higher probability (for the higher area devices) of finding a pre-formed subpath with the random initial Page 17 of 27 Materials Horizons Materials Horizons Accepted Manuscript Open Access Article. Published on 02 December 2023. Downloaded on 12/2/2023 1:33:39 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online DOI: 10.1039/D3MH01834B 17 CB configuration to let the CNF be created. As can be seen, as the area increases, the set voltage reduction saturates, as it is expected for devices where charge conduction is based on filamentary switching (Figure 5d). Figure 5. a Resistance network corresponding to a great area (100 nm x 100 nm) device. b Resistance network corresponding to a great area (100 nm x 100 nm) device (zoomed-in view). c Several simulated set I-V curves for different device areas (the current values are scaled with respect to the lowest area shown in the plot, i.e. (8.33 nm x 8.33 nm)). d Set voltage versus device side length (assuming a device square area obtained as (side length)2). Page 18 of 27Materials Horizons Materials Horizons Accepted Manuscript Open Access Article. Published on 02 December 2023. Downloaded on 12/2/2023 1:33:39 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online DOI: 10.1039/D3MH01834B 18 V.-CONCLUSIONS A 3D simulation tool based on CBs is developed for the first time to describe RS in multilayer h-BN memristors. It is based on CBs that can be modeled with two, three and four resistance states; in addition, a series resistance and a module to account for quantum effects. The simulator has been tuned with measurements of h-BN memristive devices. The influence of the model parameters has been shown in the simulator tuning process. We also show the CNF formation that accounts for filamentary charge conduction in our devices, explaining the current abrupt change when the set event takes place. In doing so, the particularities of the material have been taken into consideration. Moreover, the simulation tool is employed to describe partial filament rupture in reset processes. Finally, the dependence of the set voltage with the device area is described by means of simulations with a massive number of CBs. Page 19 of 27 Materials Horizons Materials Horizons Accepted Manuscript Open Access Article. Published on 02 December 2023. Downloaded on 12/2/2023 1:33:39 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online DOI: 10.1039/D3MH01834B 19 VI. - ACKNOWLEDGMENTS We acknowledge project PID2022-139586NB-44 funded by MCIN/AEI/10.13039/501100011033 and by European Union NextGenerationEU/PRTR. F.M.G.-C. thanks project PP2022.PP-13 funded by “Ayudas del Plan Propio UGR 2022”. M.L. acknowledges generous support from the King Abdullah University of Science and Technology. Data available on request from the authors. The data that support the findings of this study are available from the corresponding author upon reasonable request. Conflict of Interest There are no conflicts of interest to declare. Page 20 of 27Materials Horizons Materials Horizons Accepted Manuscript Open Access Article. Published on 02 December 2023. Downloaded on 12/2/2023 1:33:39 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. 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