An improved analytical model for the statistics of SET emergence point in HfO2 memristive device Cite as: AIP Advances 9, 025118 (2019); https://doi.org/10.1063/1.5085685 Submitted: 13 December 2018 . Accepted: 14 February 2019 . Published Online: 22 February 2019 Dong Xiang, Rulin Zhang, Yu Li, Cong Ye, Enrique Miranda, Jordi Suñé, and Shibing Long ARTICLES YOU MAY BE INTERESTED IN Sub-nanosecond pulse programming and device design strategy for analog resistive switching in HfOx-based resistive random access memory Applied Physics Letters 114, 112102 (2019); https://doi.org/10.1063/1.5078782 Perspective: A review on memristive hardware for neuromorphic computation Journal of Applied Physics 124, 151903 (2018); https://doi.org/10.1063/1.5037835 Comprehensive numerical modeling of filamentary RRAM devices including voltage ramp- rate and cycle-to-cycle variations Journal of Applied Physics 124, 174502 (2018); https://doi.org/10.1063/1.5042789 AIP Advances 9, 025118 (2019); https://doi.org/10.1063/1.5085685 9, 025118 © 2019 Author(s). AIP Advances ARTICLE scitation.org/journal/adv An improved analytical model for the statistics of SET emergence point in HfO2 memristive device Cite as: AIP Advances 9, 025118 (2019); doi: 10.1063/1.5085685 Submitted: 13 December 2018 • Accepted: 14 February 2019 • Published Online: 22 February 2019 Dong Xiang,1 Rulin Zhang,2 Yu Li,3 Cong Ye,2,a) Enrique Miranda,4 Jordi Suñé,4 and Shibing Long5,a) AFFILIATIONS 1 School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, China 2 Hubei Key Laboratory of Applied Mathematics, Faculty of Physics and Electronic Science, Hubei University, Wuhan 430062, China 3 Key Laboratory of Microelectronics Devices & Integration Technology, Institute of Microelectronics of Chinese Academy of Sciences, Beijing 100029, China 4 Departamentd’Enginyeria Electrònica, Universitat Autònoma de Barcelona, Bellaterra 08193, Spain 5 School of Microelectronics, University of Science and Technology of China, Hefei 230027, China a) Corresponding authors: yecong@issp.ac.cn (Cong Ye); shibinglong@ustc.edu.cn (Shibing Long) ABSTRACT In this work, an improved analytical model for the SET switching statistics of HfO2 memristive device is developed from the cell-based percolation model. The statistical results of the SET emergence point related to the beginning stage during SET process are systematically discussed. Moreover, the deviation from Weibull model in high percentiles region is found to orig- inate from the uneven distribution of defect density. Our improved model exhibits excellent consistency with experimental results in Cu/HfO2 /Pt device. Besides, we explain the relationship between the parameters of the model and SET resistance. The underlying mechanism of SET process for HfO2 memristive device is fully illuminated. © 2019 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5085685 Memristor, also called resistance switching device, is a (CF) in the resistive switching (RS) layer. Moreover, the under- two-terminal device whose resistance values can be modu- lying mechanisms of SET/RESET switching processes have lated by the external stimulation, and has attracted exten- not been fully illuminated. Hence, understanding the RS sive interest as promising candidates for non-volatile memo- mechanism is the key aspect to control the parameter ran- ries, reconfigurable switches, and bio-inspired neuromorphic domness. By means of statistical analysis with Weibull model, computing.1–4 As a promising device applied in non-volatile the statistical distributions of resistive parameters can be memory field, memristor has shown highly desirable prop- investigated,6 which would help to reveal the undetected erties including low power, fast switching speed, excellent behavior of CF and disclose the RS mechanism.7–12 retention and scalability.2 Especially, HfO2 material, which is We have investigated the dynamic CF evolution in SET highly compatible with complementary metal oxide semicon- process of memristive device.6,13,14 Different from the usually ductor (CMOS) technology, is applied in memristor, show- reported abrupt SET switching, some RRAM devices, for exam- ing narrow resistance distributions, reliable endurance, and ple, the Cu/HfO2 /Pt device studied in this work, show com- robust thermal stability.5 However, the wide fluctuation of plicated SET process with multiple steps. At a low voltage, the resistive switching parameters is a major obstacle for prac- resistance keeps comparatively stable because the external tical application. The randomness is mainly induced by the electrical stimulations are not strong enough to induce the CF uncontrolled formation and rupture of conductive filament growth in the RS layer. With the increase of the applied volt- AIP Advances 9, 025118 (2019); doi: 10.1063/1.5085685 9, 025118-1 © Author(s) 2019 AIP Advances ARTICLE scitation.org/journal/adv age, the SET emergence point occurs, indicating the beginning dynamic evolution of CF can be regarded as the cell’s evolu- of CF growth. The SET emergence point is a bridge between tion in the most constrictive part, i.e., the bottleneck region.20 the initial high resistance state and the SET process, and it Here, we deduce that the bottleneck region contains a certain obviously affects the self-accelerating SET process. Generally, number of cells, which have the same switching probability the CF shows obvious fluctuation at a high voltage and leads irrespective of their spatial position. The cells are assumed to to a final abrupt formation of the connected CF, which is cor- be divided into n slices, describing the length of the bottleneck responding to the SET point. As the SET switching includes region. Each slice contains N cells, which indicates the width different stages similar to those in RESET switching,14 it’s nec- of the region. According to the geometric model, the SET essary to reveal the property of the SET emergence point dynamic process for Cu/HfO2 /Pt device can be conveniently through experiments especially the statistic ones and also considered on the variation in those n × N cells. The thickness deeply understand its physical mechanism. In our previous of every cell is a0 . Moreover, it is supposed that these insu- work, cell-based analytical percolation model to study the lating cells share the same conductive probability λc . Impor- SET switching process has been built, and Weibull distribu- tantly, the emergence point in SET process corresponding to tions of the SET parameters have been demonstrated on the the abrupt rise of current (Fig. 1(b)) is defined as the criti- basis of the Weibull model.6,13,14 However, if we apply the cal point, which indicates that at least one cell switches into Weibull model to investigate the SET emergence point, the conductive state. Thus, the SET emergence probability can be tail bits phenomenon are apt to appear in high percentiles represented as region. Although it has been reported that the defects in RS FSET = 1 − (1 − λc )nN , (1) layer are inclined to cluster when the defect density is high Where FSET is the cumulative probability. Therefore, Weibull enough,15–19 the Weibull model cannot solve the tail bits phe- distribution function W = Ln[−Ln(1 − F)] can be replaced by nomenon for SET process in high percentiles region. It is nec- WSET = Ln −Ln(1 − λc )nN , f g essary to introduce a more exact or improved model to analyze (2) the statistical experiment data, which can explore the inherent RS mechanism of HfO2 memristive device more precisely. In previous Weibull models, one tends to assume that defects In this work, as concerned as SET emergence point, are evenly distributed in the whole region with an average an improved clustering statistic model with the influence of defect density (D0 ) and the density function ( f(D)) of uniform defect density is taken into account to analyze the statisti- defect density (D) is a “delta” function f(D) = δ(D − D0 ).10 cal characteristic of the experimental data for Cu/HfO2 /Pt According to the Poisson distribution function, the probability device. The relationship between SET resistance (RSET ) and RS λc can be represented as parameters (Weibull slope, scale factor and clustering factor) λc = 1 − exp(−D0 ), (3) is discussed to understand the underlying switching mech- anism. The improved modeling method provide an excellent In the general case of emergence point, conductive probability method for analyzing resistive parameters and the deviation in λc is comparatively low (λc  1), thus the Weibull distribution high percentiles region, which provides guidance for under- function W SET can be approximated expressed as follows: standing the RS parameters fluctuation for HfO2 memristive WSET  ln(nND0 ), (4) device. In order to investigate the CF evolution and explain the Furthermore, when we use Weibull distribution to describe statistical characteristics of RS parameters in SET process for the SET switching distribution, it can be defined as W(x) = Cu/HfO2 /Pt device, the cell-based model shown in Fig. 1(a)   x β Ln x63% , so the Weibull distribution of V SET and ISET can also has been commonly proposed.8–10 The formation/rupture of be innovatively expressed as: CF mainly occurs in a small cell-described region, and the ! VSET WVSET = βVSET Ln , (5) VSET,63% ! ISET WISET = βISET Ln , (6) ISET,63% where βVSET (or βISET ) and V SET,63% (or ISET,63% ) are the Weibull slope and scale factor of V SET (or ISET ) distribution, respec- tively. It is worth noting that these expressions for improved analytical model in HfO2 memristive device have never been adopted to describe the statistics of SET emergence point previously. On the basis of traditional Weibull model, we assume the defects are evenly distributed in the bottleneck region. How- FIG. 1. (a) Schematic of n × N cells in the cell-based region. (b) I−V characteristics ever, when the sweep voltage is relatively high, the defects of SET (blue line) and RESET (red line) in Cu/HfO2 /Pt device. The blue and red existing in the bottleneck region tend to cluster and the defect triangles indicate the emergence point of SET and RESET process, respectively. density increases, hence the distribution of defect density Inset of (b) shows the magnified plot of low voltage region in SET process. becomes uneven.17,19,21 As a result, the Gamma distribution AIP Advances 9, 025118 (2019); doi: 10.1063/1.5085685 9, 025118-2 © Author(s) 2019 AIP Advances ARTICLE scitation.org/journal/adv is adopted to describe density function of defect instead of On the basis of the above serials formula corrections and delta distribution. According to the Poisson-gamma mixture analysis, Fig. 2(a) displays the results of the Weibull model and model,22 the probability λ0c can be expressed as: the improved clustering model under different values of clus- ! −αSET tering factor α. It is worth noting that at low percentiles region D0 with relatively low V SET , Weibull model and clustering model λ0c = 1 − 1 + , (7) αSET have the same tendency and nearly overlap. The driven force where αSET is represented as the clustering factor, reflecting of electric field is not strong enough to drive the defects to the impact of uneven defect density on the SET process.16 induce significant clustering effects,16,20 and the defects are At the same time, we should notice that when αSET → ∞, still apt to have relatively uniform distribution upon the bot- Eq. (7) can be simplified to Eq. (3), which means the Pois- tleneck region. As a result, when αSET → ∞, the improved son model has been changed to delta model and the defect clustering model tends to coincide with the Weibull model, in the bottleneck region becomes uniform distribution. When which means the clustering effect can be ignored. However, we substitute Eq. (7) into Eq. (2), we can get at high percentiles region with high V SET , the deviation of both models can be clearly observed, which can be assumed ! −nNαSET   D0 to the clustering phenomenon of the defects. Fig. 2 (b) W0SET = Ln −Ln(1 − λ0c )nN = Ln −Ln 1 + f g   αSET  , (8) displays the results of the Weibull model and the improved clustering model under different values of Weibull slope β.  In RS layer, the defect density (D0 ) is relatively low, thus, when The V SET,63% = 0.5 means the separation point of both mod- D0  1, Eq. (8) becomes els. Especially, when the value of Weibull slope β changes, " !# the cross point is fixed. Fig. 2(c) displays the Weibull model nND0 W0SET = Ln(αSET ) + Ln Ln 1 + . (9) and clustering model under different values of scale factor αSET V SET,63% . It can be observed that the curves of both mod- Eq. (9) expresses our improved clustering model with cluster- els are consistent in low percentiles region, but the curves ing factor (αSET ) for the SET emergence point statistics. By of improved clustering model bend downward in high per- combining Eqs. (4), (5), (6) and (9), we can get the complete centiles region. To explore the separation point of two models, expression of the distribution of V SET and ISET : the crossing point of fitting lines with Ln[−Ln(1 − F)] = 0 can be defined as the separation of two models, which are displayed ! βV 1 VSET in Fig. 2(d).When the SET voltage is less than V SET,63% , indi-  SET   W0VSET = Ln(αVSET ) + Ln Ln*1 + + , (10)  , αVSET VSET,63% cating that the distribution of defects is uniform, so it can be - seen that the results extracted from Weibull model and clus- tering model show consistency well. When V SET is higher than 1 ISET βISET +   W0ISET = Ln(αISET ) + Ln Ln*1 + ( ) V SET,63% , the defects emerge to clusterand the results of both  , αISET ISET,63% , (11) - models show much derivation, which suggest that the Weibull FIG. 2. Plots of previous Weibull model(dashed lines) and our improved analytical model(solid lines) under different values of(a) clustering factor α, (b) Weibull slope β and (c) scale factor V SET,63% . (d) Results extracted from improved analytical model and Weibull model. The intersection points indicate the value of V SET,63% . AIP Advances 9, 025118 (2019); doi: 10.1063/1.5085685 9, 025118-3 © Author(s) 2019 AIP Advances ARTICLE scitation.org/journal/adv 1   m model needs to be improved and our analytical model is (τT0 γm (m + 1)) m+1 tgap m+1 necessary. VSET,63% = 1 ∝ Ln(RSET ). (14) Just as we have reported, the initial conductance strongly (nN) (m+1)ρ affects the current evolution in each SET cycle,23 we use the Using the same method, we can get: SET resistance (RSET = V SET /ISET ) just at the SET emergence point, where a current jump can be observed (as shown in βISET = (m + 1)ρ, (15) Fig. 1(a)) for the analysis of the VSET statistics. Normally, RSET 1 ISET,63% ∝ . (16) is related to the CF size and morphology at the set emergence Ln(RSET ) point. In order to explore the influence factor of the distribu- In order to investigate the statistics of the SET emergence tion parameters (Weibull slope, scale factor, clustering factor), point, resistance screening method is adopted, so that the we develop a deterministic model to relate the SET parame- results would be expected to exhibit resistance-dependent ters’ statistics with SET resistance. In the DC voltage sweeping characteristics. Figs. 3(a) and (b) show the distributions of V SET mode (VSM), which is used for the practical measurement of and ISET in 4000 cycles of Cu/HfO2 /Pt device. The dots, dot- the device,24 a ramped voltage with V(t) = γt is applied. Con- ted lines and solid lines represent the experiment data, fit- sidering the dynamic evolution of defects, the defect density ting results of Weibull model and improved clustering model, D0 can be described by the power law8 respectively. From Fig. 3, we can see both the fitting lines  ρ  tSET 1 (dotted lines and solid lines) reveal good consistency with D0 =   dt  , (12) experimental data in low percentiles regions. However, as the  0 τT(t)  SET voltage increases to high percentile regions, the Weibull V −m where τT(t) = τT0 ( tgap ) is the characteristic time for defect model begins to separate from the experiment results while generation8 with τT0 and m being constants, and ρ is anexpo- the improved clustering model still shows excellent consis- nent that provides flexibility to the model. According to our tency with the experiments. From our improved clustering previous study,9,13 a tunnel gap (the bottleneck region which model, this phenomenon can be explained as the influence of consist of n × N cells, as shown in Fig. 1(a)) associated with the uneven defect density in high SET voltage region, as discussed first quantized subband forms at the constriction bottleneck. above. If we use the Weibull distribution to describe the set Here, we define its length as tgap . According to the quantum emergence point of our experiment, we can observe the “tail point contact (QPC) model, which has been used to success- bit” effect in high percentiles region.22 The reason lies in that fully describe the I−V curves ofmemristive the new generation of uneven defects would form cluster- q device in both the ing phenomena in high percentiles region and the traditional 2 LRS and HRS,25,26 we can get tgap = ~ Ln(G0 Nch RSET ), π m* φB Weibull model deviates from the experiment. ∗ where ~ is the reduced Planck constant, m is the effective Analogous to the method in Weibull model, values of three electron mass, φ B is the height of the tunneling barrier in the relevant parameters related to V SET and ISET distributions are conducting channels, G0 = 2e2 /h is the quantum of conduc- extracted from experimental results, as shown in Fig. 4. It can tance, the reduced Planck constant and Nch is the number of be seen that when the RSET increases, the Weibull slopes and opened conducting channels.27 By introducing the result of clustering factors keep constant. However, the scale factors the QPC model, the physical nature such as effective elec- exhibit different dependence with the RSET . It can be seen tron mass, barrier height, etc, which has been deeply stud- with the increase of RSET , V SET,63% increases while ISET,63% ied in previous work,28–30 is combined into our analytical decreases linearly. Actually, the clustering factor α is a con- model. Combining the above equation and Eqs. (12) and (9), stant independent of RSET , which is related to the properties of the Weibull slope and scale factor of V SET distribution can be materials and structure for the memristive device.24,31 Mean- expressed as: while, Eqs. (13) and (15) explain well that the Weibull slopes βVSET = (m + 1)ρ, (13) constant is independent on RSET . Eqs. (14) and (16) show a FIG. 3. Statistical results of (a) V SET and (b) ISET in 4000 cycles of Cu/HfO2 /Pt device. AIP Advances 9, 025118 (2019); doi: 10.1063/1.5085685 9, 025118-4 © Author(s) 2019 AIP Advances ARTICLE scitation.org/journal/adv FIG. 4. Distribution parameters extracted from statistical results of (a) V SET and (b) ISET distributions. clear relation between ISET,63% (or V SET,63% ) and RSET , which 3 F. Pan, C. Chen, Z. S. Wang, and J. Yang, Prog Nat. Sci. Mater. 20, 1 agrees well with the experimental results shown in Fig. 4. (2010). 4 Furthermore, we find that the parameter V SET,63% (or ISET,63% ) C. Ye, J. J. Wu, G. He, J. Zhang, T. F. Deng, P. He, and H. Wang, J. Mater. Sci. can be controlled by adjusting the value of RSET , which pro- Technol. 32, 1 (2016). 5 vides guidance for understanding the switching mechanism A. Padovani, J. Woo, H. Hwang, and L. 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