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A Unified Capacitive-Coupled Memristive Model for the Non-Pinched Current–Voltage Hysteresis Loop Bai Sun, Yuanzheng Chen, Ming Xiao, Guangdong Zhou, Shubham Ranjan, Wentao Hou, Xiaoli Zhu, Yong Zhao, Simon A.T. Redfern, and Y. Norman Zhou Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.9b02683 • Publication Date (Web): 21 Aug 2019 Downloaded from pubs.acs.org on August 22, 2019
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A Unified Capacitive-Coupled Memristive Model for the Non-Pinched Current–Voltage Hysteresis Loop Bai Sun,†,‡ Yuanzheng Chen,*,‡ Ming Xiao, † Guangdong Zhou,§ Shubham Ranjan,‖ Wentao Hou,† Xiaoli Zhu,‖ Yong Zhao,‡ Simon A.T. Redfern,¶ and Y. Norman Zhou*,† † Department
of Mechanics and Mechatronics Engineering, Centre for Advanced Materials Joining,
Waterloo Institute for Nanotechnology, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada ‡ School
of Physical Science and Technology, Key Laboratory of Advanced Technology of Materials
(Ministry of Education of China), Southwest Jiaotong University, Chengdu, Sichuan 610031, China §
Scholl of Artificial Intelligence, Southwest University, Chongqing 400715, China
‖
Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
¶ Department
of Earth Sciences, University of Cambridge, Downing Street, Cambridge, CB2 3EQ, United
Kingdom
Supporting Information ABSTRACT: The concept of the memristor, a resistor with memory, was proposed by Chua in 1971 as the fourth basic element of electric circuitry. Despite a significant amount
of
effort
devoted
to
the
understanding of memristor theory, our understanding of the non-pinched currentvoltage (I–V) hysteresis loop in memristors remains incomplete. Here we propose a physical model of a memristor, with a capacitor connected in parallel, which explains how the non-pinched I-V hysteresis behaviour originates from the capacitive-coupled memristive effect. Our model replicates eight types of characteristic nonlinear I-V behavior, which explains all observed non-pinched I-V curves seen in experiments. Furthermore, a reversible transition from a non-pinched I–V hysteresis loop to an ideal pinched I–V hysteresis loop is found, which explains the experimental data obtained in C15H11O6-based devices when subjected to an external stimulus (e.g, voltage, moisture, or temperature). Our results provide the vital physics models and materials insights for elucidating the origins of non-pinched I-V hysteresis loops ascribed to capacitive-coupled memristive behavior. KEYWORDS: capacitive-coupled memristive, unified model, non-pinched, current–voltage curve, hysteresis loop
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The memristor has attracted significant interest since it was first theoretically proposed by Chua and claimed to be realized physically by Strukov.1,2 It was claimed that its appearance would revolutionize the electronics and expand the field of electronic science research through the coordination and integration of the basic the concept of the memristor, a resistor with memory, was proposed as the fourth basic element of electric circuitry.3-5 Beyond that, the memristor has many extremely promising potential applications, including uses as resistive random access memory (RRAM),6,7 analogues for biological synapses,8,9 bio-inspired computing systems,10-12 and integrated neural networks.13 As a novel circuit component, the memristor exhibits many attractive physical characteristics and multifunctional properties.14-17 Despite a significant amount of effort devoted to the understanding of memristor theory, our understanding of the components of electrical circuits (resistors, inductors, capacitors, memristors) is not enough. From a circuit perspective, the memristor, a resistor with memory, can be defined by the statedependent Ohm’s law:1
𝑖 = G(𝑥, 𝑣) ∙ 𝑣 ;
d𝑥 d𝑡
= 𝑓(𝑥,𝑣)
(1)
where the G(x, v) represents the conductance of the memristor cell, x is the state variable, i is current, v is voltage, and t is time. This equation indicates that a memristor has a characteristic fingerprint: it must exhibit a pinched hysteresis loop in the current–voltage (I-V) curve when driven by a bipolar periodic signal.18-21 In other words, the I-V curve must retain a zero-crossing behavior. However, a non-pinched or “non-zero-crossing” I-V hysteresis loop has been recently reported in some electrochemical systems.22-31 Subsequently, the hysteresis loop of the memristor, when subjected to a periodic stimulus, can be categorized into zero-crossing (type I memristor) or nonzero-crossing (type II memristor). Although these non-pinched I-V hysteresis behaviors are considered by many to originate from the contribution of additional capacitive effects,22-31 its origin and nature remain elusive due to the rather incomplete theory of these components and the lack of understanding of the influence of coupling among fundamental circuit elements.
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Figure 1. Capacitive-coupled memristive behavior. (a) Relevant mathematical relationships. (b) Circuit schematic diagram for a memristor with connected a capacitor in parallel. (c) The coupling relationship between memristor and capacitor.
Facing this problem, we have performed a theoretical analysis and simulation to investigate the origins of non-pinched I-V behavior. It is well known that there are four basic variables in a circuit (current i, voltage v, charge q and magnetic flux φ). They can be described mathematically as: (Figure 1a):2
𝑣=
𝑑𝜑 𝑑𝑡 ;
𝑑𝑞
𝑑𝜑
(2)
𝐶 = 𝑑𝑣 ; 𝑀 = 𝑑𝑞
where the t is time, C is capacitance and M is memristance. From equation (2), the relationship between C and M can be further obtained as follows: 1𝑑𝜑
(3)
𝑀 = 𝐶 𝑑𝑣
𝑑𝜑
Equation (3) implies that the capacitance can couple with memristance (CM = 𝑑𝑣 ). To depict this correlation, a model of an ideal memristor connected to a capacitor in parallel has been constructed (Figure 1b). It can be described according to: (4)
𝑣1 = 𝑣2 𝑑𝑣2
𝑑𝐶
𝑖1 = 𝑖2 = 𝑖𝑀 + 𝑖𝐶 = 𝐶 𝑑𝑡 +𝑣2 𝑑𝑡
(5)
where the current i1 = i2 is equal to the total of the memristor‒current (𝑖𝑀) plus the capacitor‒current (𝑖𝑐). Following on from equations (4) and (5), and adopting the ideal I-V curves of a memristor (M1, M2, M3, M4) and capacitor with positive capacitance effect C1 and negative 3 ACS Paragon Plus Environment
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capacitance effect C2,32 we further simulate the I-V curves of the parallel memristor-capacitor model. Based on the direction of I-V response and the switching direction (Figure S1, S2 and details given in Supplementary note 1), the I-V curves of this model exhibits, in total, eight types of nonpinched characteristic behaviour, namely C1M1, C1M2, C1M3, C1M4, C2M1, C2M2, C2M3, C2M4, as shown in Figure 2. Among, these nonlinear non-pinched I-V hysteresis behaviors, C1M1, C1M2, C1M4 have be found in some experimental reports.22-31 Although the C1M3 type I-V curve has not yet been reported, as far as we known, the similar C2M4 type I-V curve has already been described by circuit simulation.33 To the best of our knowledge, the results in Figure 2 contain all reported non-pinched I-V hysteretic behaviors thus far known, strongly supporting the validity of our simulation.
Figure 2. The schematic diagram of eight non-pinched I-V hysteresis curves types (CxMy; x = 1, 2; y = 1, 2, 3, 4) duo to the capacitive-coupled memristor effect between the capacitive state (Cx) and the memristive state (My).
In addition to these known types, our model also predicts novel types such as C2M1, C2M2, C2M3 and some rare non-pinched I-V hysteresis curves which have two crossing-points such as the 4 ACS Paragon Plus Environment
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C1M3 and C2M4 types. In the case of C2M1, C2M2, C2M3, C2M4, the non-pinched I-V hysteresis behaviors correspond to the ideal negative capacitance effect C2. This concept has been proposed previously in some special materials,34-36 which could possibly be related to the recently reported coexistence of the negative differential resistive effect and the memristive effect.37 In Figure 2 we show, for the first time, all possible non-pinched I-V types for a memristor in parallel with a capacitor. This model clearly indicates the fact the non-pinched I-V curves are the result of the 𝑑𝑉2
𝑑𝐶
memristive-effect ( 𝑑𝑡 ) coupled to the capacitive-effect ( 𝑑𝑡 ). The origin and nature of non-pinched IV hysteresis behaviors are attributed to an inherent coupling in the capacitive-memristive effect (Figure 1c). According to the equation (3), C is inversely proportional to M, that is, a larger memristive effect corresponds to a smaller capacitive effect, and vice versa. This suggests that we should expect an ideal pinched I-V hysteresis curve from a non-pinched I-V hysteresis curve by decreasing the capacitive effect such that it disappears, or by hindering the coupling of the capacitive and memristive effects. Some stimulus would presumably be required to achieve such a transformation from a capacitive-coupled memristive system to ideal memristive behavior. To test this hypothesis, we fabricated a C15H11O6 based-device (Figure S3 and experimental details given in Supplementary note 2) to search for this transition in behavior. We firstly measured the I-V curve of the Ag/C15H11O6/FTO device from -2.0 V to 2.0 V and observed obvious nonpinched I-V hysteresis characteristics at ambient conditions (Figure 3a). This non-pinched I-V hysteresis curve corresponds to a typical capacitive-coupled memristive circuit of C1M4 type. When applying a larger voltage it is interesting to note that the I-V hysteresis curve changes from nonpinched (non-zero-crossing) to pinched (zero-crossing), as we had anticipated, and as is shown in Figure 3b and Figure S4. In addition, we note that C15H11O6 can react with moisture (Figure S5),38 therefore we moistened the C15H11O6 layer. When the moisture content increases beyond a certain value (for detailed discussion see Figure S6 and Supplementary note 2), it is found that the capacitive effect completely disappears and the device’s I-V characteristics transform to a pinched 5 ACS Paragon Plus Environment
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I–V behavior (Figure 3c). Moreover, by increasing the temperature from 25 oC to 250 oC, the nonpinched I–V at low temperature changes into a pinched I–V loop at high temperature (Figure 3d and Figure S7). Interestingly, when these external stimulations are removed or decreased to a special value, the capacitive effect appears again and then the device can go back to the capacitive-coupled memristive behavior, showing a reversible transition (details discussed in Supplementary note 3, 4, 5). All these phenomena reflect that the capacitive-coupled memristive effect can be controlled and regulated, supporting our hypothesis of the possibility of transformation between capacitivecoupled memristive and pure memristive behavior via appropriate stimulus.
Figure 3. The transition between the capacitive-coupled memristive and pure memristive effect. (a) The initial I-V curve. (b) Applying larger voltage. (c) Adding moisture. (d) Increasing temperature.
To understand the transformation between the capacitive-coupled memristive and ideal memristive behavior, we have studied its mechanism in Ag/C15H11O6/FTO devices. As is well known, the C15H11O6 molecule has functional groups such as hydroxyl and oxygen. In particular, the oxygen in the organic film is prone to react readily with oxidizable electrodes at the organic/metal interface. The oxygen atom (O) inside the C15H11O6 film can be ionized to O2- by gaining electron (e–) under sufficiently high electric field:39 6 ACS Paragon Plus Environment
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O + 2e– → O2-
(6)
At the same time, the Ag atoms in the top electrode can be ionized to Ag+ and e– under an applied voltage:40,41 Ag → Ag+ + e–
(7)
Therefore, the capacitive effect occurs when an external voltage is applied to the top and bottom electrodes of the device (Figure 4a).
Figure 4. Schematic diagram of physical mechanism. (a) Capacitive-coupled memristive mechanism. (b) Mechanism of pinched I-V under larger voltage. (c) Mechanism of moisture induced a pinched I-V curve.
In the initial state, the small amount of O in active materials is ionized into O2- under an external electric field (E) (Equation 6), and a small amount of the Ag in the top electrode is ionized into Ag+ (Equation 7). At the same time, the material commonly contains oxygen vacancies (Vo2+), thus the Vo2+ and Ag+ move towards the FTO electrode along the direction of applied field E, while O2– and e– move towards the Ag electrode against the direction of E. As Ag+ and Vo2+ accumulate near the bottom electrode and O2– and e– accumulate near the top electrode, to a certain extent, and an internal electric field (Ei), opposed to E, will be generated. When Ei is equal to E, the migration of 7 ACS Paragon Plus Environment
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ions will cease. At this point, the effective dielectric thickness (de) of the capacitor is reduced, which is equivalent to reaching a low resistance state (LRS) for the device. As E decreases gradually, the Vo2+ and Ag+ return to the top electrode Ag along the direction of the Ei, while the O2and e– return to the bottom electrode in the opposite direction to Ei. In this process, some of the positive and negative ions will be neutralized, resulting in increases in the effective dielectric thickness of the capacitor to deˊ (deˊ ˃ de), which is equivalent to reaching the high resistance state (HRS). It is clear that the device has obvious capacitive effect during the ion transport process, which may be attributed to the non-pinched I-V hysteresis curve. With increasing the applied voltage (here up to 5.0 V), the following major changes will occur within the device: i) more Ag+ and e– will be produced under higher applied voltage; ii) a larger number of O atoms will be ionized to O2+; iii) more Vo2+ vacancies will be generated immediately after that. These phenomena are helpful to facilitate the Ag+ and Vo2+ conduction pathways inside the C15H11O6 layer (Figure 4b), leading to a pinched I-V curve. Similarly, when increasing the temperature, the ionic activity in the active material is also increased due to reinforcing thermal shock, which makes it easier to ionise atoms that may be transported more rapidly. As a result, it is easy to form ion conduction paths, resulting in a I-V curve with pinched behavior. For the case of moisture increase on the sample, water molecules (H2O) decompose in an electric field. At the interface of Ag/C15H11O6, H2O becomes more likely to generate OH− according to:42 O2 + 2H2O+ 4e− ⇆ 4OH−
(8)
The chemical reaction at the Ag/C15H11O6 interface is depicted in Figure 4c. Simultaneously, watersplitting can occur at the interface of C15H11O6/FTO. By consuming Vo2+, the disruption of the Vo2+ path is accelerated by the additional H2O, and the reaction equation is:42 H2O + O + Vo2+ ⇆ 2OH−
(9)
The concentration of OH− is mainly determined by equations (8) and (9). However, considering that a large number of e– is accumulated at the interfaces of Ag/C15H11O6, Equation (8) should be more 8 ACS Paragon Plus Environment
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significant than Equation (9). Once the reaction given in Equation (8) occurs, a half-cell reaction for gas phase H2O absorbed at the interface of Ag/C15H11O6 may also occur:43 2H2O ⇆ O2 + 4H+ + 4e−
(10)
As a result, a large amount of OH− is produced in active materials, which assists in the formation of conductive paths, resulting in a pinched I-V curve. Comparison with experimental observations reveals that the non-pinched hysteresis loop, a C1M4 type I-V curve is given by a C15H11O6 memristor which shows a capacitor-coupled memristive effect. When subjected to an external stimulus such as voltage, moisture, or temperature, this can transform to pinched hysteresis loop, verifying our model. We have reason to expect that this transformation also occurs in other type non-pinched I-V curves, by reference to our proposed model. In summary, we have proposed a physical model of a memristor in terms of a capacitor connected in parallel, which shows all possible non-pinched I-V hysteresis loop in a memristor and reveals that its origin and nature are due to the presence of a capacitor in parallel with a capacitivecoupled memristive effect. Subsequently, we have investigated the C1M4 type I-V curve of the memristor C15H11O6 which shows a capacitor-coupled memristive effect. When subjected to an external stimulus (e.g, voltage, moisture, and temperature), this can transform to pinched hysteresis loop behaviuor. Through this experiment, we find that conversion between capacitive-coupled memristive and pure memristive behavior can be happen through appropriate external stimulation. This work expands circuit theory between capacitor and memristor, and depicts the capacitivememristive coupled I-V behaviors, pushing the development of memristor.
ASSOCIATED CONTENT Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION Corresponding Author *E-mail:
[email protected] (Y. Chen);
[email protected] (Y. N. Zhou). Author Contributions B. Sun and Y. Chen contributed equally for this work. M. Xiao and G. Zhou made important discussion. Y. N. Zhou and B. Sun conceived and designed the experiments. Y. Chen and B. Sun written the manuscript. Simon A.T. Redfern made finally revision. All authors have discussed related results and approval to the final version of the manuscript. Notes The authors declare no competing financial interest.
ACKNOWLEDGMENTS We acknowledged the discussion with Xiaolei Feng, Hanyu Liu, and Zhongfang Chen. This work was supported by the National Natural Science Foundation of China (No. 11604270, 11704050), the Fundamental Research Funds for the Central Universities (2682017CX052, 2018GF08), the Natural Sciences and Engineering Research Council (NSERC) and Canada Research Chairs (CRC) Programs.
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(36) Khan, A. I.; Chatterjee, K.; Wang, B.; Drapcho, S.; You, L.; Serrao, C.; Bakaul, S. R.; Ramesh, R.; Salahuddin, S. Nature Materials 2015, 14, 182. (37) Zhou, G.; Duan, S.; Li, P.; Sun, B.; Wu, B.; Yao, Y.; Yang, X.; Han, J.; Wu, J.; Wang, G.; Liao, L.; Lin, C.; Hu, W.; Xu, C.; Liu, D.; Chen, T.; Chen, L.; Zhou, A. and Song, Q. Adv. Electron. Mater. 2018, 4, 1700567. (38) Chien, C.-Y.; Hsu, B.-D. Solar Energy 2013, 98, 203-211. (39) Xu, Z.; Jin, K.; Gu, L.; Jin, Y.; Ge, C.; Wang, C.; Guo, H.; Lu, H.; Zhao, R.; Yang, G. Small 2012, 8, 1279. (40) Tappertzhofen, S.; Valov, I.; Tsuruoka, T.; Hasegawa, T.; Waser, R. and Aono, M. ACS Nano 2013, 7, 6396. (41) Zhou, G.; Ren, Z.; Wang, L.; Wu, J.; Sun, B.; Zhou, A.; Zhang, G.; Zheng, S.; Duan, S.; Song, Q. Nano Energy 2019, 63, 103793. (42) Torija, M. A.; Sharma, M.; Gazquez, J.; Varela, M.; He, C.; Schmitt, J.; Borchers, J. A.; Laver, M.; Khatib, S. E.; Leighton, C. Adv. Mater. 2011, 23, 2711-2715. (43) Merkle, R.; Maier, J. Angew. Chem. Int. Ed. 2008, 47, 3874-3894.
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TOC Graphic
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Figure 1 297x86mm (150 x 150 DPI)
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Figure 2 298x154mm (150 x 150 DPI)
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Figure 3 246x142mm (150 x 150 DPI)
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Figure 4 259x181mm (150 x 150 DPI)
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