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. 2017 Sep 11;7:11204. doi: 10.1038/s41598-017-11165-5

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© The Author(s) 2017

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Simulation results obtained from various operation schemes. (a) Heat map of the SET probability for different gap thickness t _gap and gap voltage V. Plots of different I − V characteristics under (b–d) VSM, (g) CSM and (h,i) GVR operation schemes. (c) and (d) I − V characteristics corresponding to different values of I _cc and different stress times in the compliances state (i.e. at I = I _cc), respectively. The insets of (c,d) are statistical results of conductance. (e) Fluctuations of current under VSM. The inset exhibits the simulation result of I − V characteristics (black line) and its trendline (red line). (f) Simulated tested I − t curve in Cu/HfO₂/M structure. In the simulation, a constant voltage stress of 1 mV is used, close to that used in the experiment. Inset of (h) The schematic of the circuit diagram in the GVR mode. (i) The plot of conductance jump corresponding to (h). Inset (i1) The evolution of the gap thickness when G < G ₀ (the CF gap is not yet completely closed). Inset (i2) The magnified plot of conductance when G > G ₀, and the conductance changes show discrete quantum effects.