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1st ieee-publication


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1st ieee-publication

  1. 1. Trigonometric Window Functions For Memristive Device Modeling Joy Chowdhury PG Student, School of Electronics Engineering KIIT University, Bhubaneswar 1 J. K. Das, 2 N. K. Rout 1,2 Associate Professor School of Electronics Engineering KIIT University, Bhubaneswar, Abstract—After the fourth passive circuit element(Memristor) came to existence in 2008 its tremendous potential to be used as a replacement for MOSFETS made its mathematical modeling very imperative. Various device models for the memristor had been proposed earlier such as the linear ion drift model, non linear ion drift model, tunnel barrier model, and a recently proposed TEAM model. In case of linear ion drift model a window function is needed to restrict the state variable within device bounds. In this paper we propose a modified window function which accounts for greater non linearity than the existing windows such as the Jogelkar, Biolek or Prodromakis windows. Index Terms— Memristors, Memrisristive Systems, Model, non linearity, Window function. I. INTRODUCTION The basic concepts of physics mention about three two terminal passive circuit elements : resistor, capacitor, and inductor. In 2008, May some scientists at Hewlett-Packard Laboratories claimed the invention of the fourth passive element- memristor [1,2] in a paper in 'Nature' whose existence was first postulated by Leon Chua in the 1970s . In that paper, Strukov et al proposed a model that gave a simple explanation for several puzzling phenomenon in nano-scale devices which could be solved with knowledge of basic algebra and calculus. Electric current 'i' is defined as the time derivative of electric charge, ݅ ൌ ௗ௤ ௗ௧ (1) Faraday's law defines voltage 'v' as the time derivative of magnetic flux 'ij' ‫ݒ‬ ൌ ௗఝ ௗ௧ (2) There are four fundamental circuit variables are R = dv/di rate of change of voltage with current C = dq/dv rate of change of charge with voltage L = d /di rate of change of magnetic flux with current Figure 1. Four basic circuit elements[2] Using symmetry arguments Chua reasoned the existence of a forth fundamental element , which he called the memristor (short for memory resistor) M , which was defined by a functional relation between charge and magnetic flux. ൌ ௗఝ ௗ௤ (3) In 2008, Stanley Williams , et al., at Hewlett Packard Lab reported the first fabricated memristor. According to the mathematical relations governing the model, the memristor's electrical resistance is not constant rather it depends on the history of current that had previously flowed through the device, i.e., its present resistance depends on how much electric charge has flowed in what direction through it in the past[3]. The device remembers its history, that is, when the electric power supply is turned off, the memristor remembers its most recent resistance until it is turned on again. II. PREVIOUSLY PROPOSED MEMRISTIVE DEVICE MODELS A. An Effective Memristive Device Model should have following features: 2015 Fifth International Conference on Advanced Computing Communication Technologies 2327-0659/15 $31.00 © 2015 IEEE DOI 10.1109/ACCT.2015.25 157 2015 Fifth International Conference on Advanced Computing Communication Technologies 2327-0659/15 $31.00 © 2015 IEEE DOI 10.1109/ACCT.2015.25 157
  2. 2. 1) The model should have sufficient accuracy and computational efficiency. 2) The model should be simple, intuitive and closed form. 3) Preferably, the model should be generalized so that it can be tuned to suit different Memristive devices[15]. B. Linear Ion Drift Model Figure 2. HP Labs Memristor Model A linear ion drift model which was proposed in [5] for memristive devices assumes that a device having physical width 'D' consists of two regions as shown in figure 2. One region comprises of high dopant concentration having width 'w' (acting as the state variable of the Memristive system). Originally the high concentration of dopants is denoted by oxygen vacancies of TiO2, given as TiO2-x. The second region has a width of 'D-w' which is actually the oxide region. The region having a higher concentration of the dopants has a higher conductance compared to the normal oxide region as shown in figure 2. Several assumptions are taken such as : ohmic conductance, linear ion drift in uniform field, equal average ion mobility. The derivative of the state variable and voltage can be given as, ௗ௪ ௗ௧ ൌ ߤ௩ ோ௢௡ ஽ Ǥ ݅ሺ‫ݐ‬ሻ (4) ‫ݒ‬ሺ‫ݐ‬ሻ ൌ ൬ܴ‫݊݋‬Ǥ ௪ሺ௧ሻ ஽ ൅ ܴ‫݂݂݋‬ ቀͳ െ ௪ሺ௧ሻ ஽ ቁ൰ (5) where Ron is the resistance when w(t)=D , while Roff is the resistance when w(t)=0. C. Window Function In order to limit the state variable within physical device bounds , equation (4) is multiplied by a function which nullifies the derivative and forces equation (4) to zero when 'w' is at a bound. Ideal Rectangular window function is one of the possible approaches. In order to add non-linear ion drift phenomenon (like decrease in drift speed of the ions when closer to the bounds) a different window function has to be considered like the one given by Jogelkar in [9]. ݂ሺ‫ݓ‬ሻ ൌ ͳ െ ሺ ଶ௪ ஽ െ ͳሻଶ௣ (6) where p is a positive integer. When the value of p is very large the Jogelkar window resembles a rectangular window which has reduced non-linear drift phenomenon. The Jogelkar window function in equation (4) suffers from a significant discrepancy while modeling practical devices, as the derivative of 'w' is forced to zero the internal state of the device is not able to change when 'w' reaches one of the device bounds. This modeling in accuracy is corrected using another window function suggested by Biolek in [10]. The Biolek window suggests some changes in the window function give as, ݂ሺ‫ݓ‬ሻ ൌ ͳ െ ሺ ௪ ஽ െ ‫݌ݐݏ‬ሺെ݅ሻሻଶ௣ (7) ‫݌ݐݏ‬ሺ݅ሻ ൌ ൜ ͳ ǡ ݅ ൐ Ͳ Ͳ ǡ ݅ ൏ Ͳ (8) where i is the current in the memristor device. Now the Biolek Window has a limitation which is that it does not have a scaling factor and thus it cannot be adjusted to have the maximum value of the window greater or lesser than one. This limitation is overcome using a minor modification - adding a multiplicative scaling factor to the window function as suggested by Prodromakis in [11]. ݂ሺ‫ݓ‬ሻ ൌ ݆ሺͳ െ ሾቀ ௪ ஽ െ ͲǤͷቁ ଶ ൅ ͲǤ͹ͷሿ௣ ሻ (9) where j is a control parameter determining the maximum value of f(w). D. Non Linear Ion Drift Model Experiments indicate that the behavior of the fabricated memristive devices differ significantly from the Linear ion drift model , exhibiting high non-linearity. This non-linearity in the I-V characteristics is desirable for the design of logic circuits. Hence, more appropriate device models have been proposed for the memristor. In [12], the proposed model is based on experimental results. The current and voltage relationship is given as ݅ሺ‫ݐ‬ሻ ൌ ‫ݓ‬ሺ‫ݐ‬ሻ௡ ߚ •‹Š൫ߙ‫ݒ‬ሺ‫ݐ‬ሻ൯ ൅ ߯ሾ‡š’൫ߛ‫ݒ‬ሺ‫ݐ‬ሻ൯ െ ͳሿ (10) where Į , ȕ , Ȗ , Ȥ are experimental fitting parameters, while n is a parameter which determines the effect of state variable on current. In this model, the state variable is considered a normalized parameter limited to the interval [0,1], and asymmetric switching behavior is assumed. In the ON state of the device, the state variable w is close to one and the first expression of equation (10) dominates the current, where ߚ •‹Š൫ߙ‫ݒ‬ሺ‫ݐ‬ሻ൯ describes the tunneling phenomenon. When in OFF state , the state variable w is close to zero and the current is dominated by the second expression in equation (10), where ߯ൣ‡š’൫ߛ‫ݒ‬ሺ‫ݐ‬ሻ൯ െ ͳ൧ resembles an ideal diode equation[16]. A non linear dependence on voltage in the state variable equation is assumed in this model, ௗ௪ ௗ௧ ൌ ܽǤ ݂ሺ‫ݓ‬ሻǤ ‫ݒ‬ሺ‫ݐ‬ሻ௠ (11) where a and m are constants, m is an odd integer, while f(w) is a window function. The same I-V relationship with a more complex state drift derivative has also been described[14]. 158158
  3. 3. E. Simmons Tunnel Barrier Model The linear and non-linear ion drift models emphasizes on representing the two regions of the oxide - doped and undoped as two series connected resistors. In [13] a more accurate physical model had been proposed which assumes nonlinear and asymmetric switching behavior which is due to the exponential nature of the ionized dopants movements, namely, the state variable changes. Unlike linear drift this model represents a resistor in series with an electron tunnel barrier, as shown in figure 3. Here, the state variable x denotes the Simmons tunnel barrier width. The derivative of x can be interpreted as the drift velocity of the oxygen vacancies, and is given as , ݀‫ݔ‬ሺ‫ݐ‬ሻ ݀‫ݐ‬ ൌ ‫ە‬ ۖ ‫۔‬ ۖ ‫ۓ‬ܿ௢௙௙ •‹Š ቆ ݅ ݅௢௙௙ ቇ ‡š’ ቈെ ݁‫݌ݔ‬ ቆ ‫ݔ‬ െ ܽ௢௙௙ ‫ݓ‬௖ െ ȁ݅ȁ ܾ ቇ െ ‫ݔ‬ ‫ݓ‬௖ ቉ ǡ ݅ ൐ Ͳ ܿ௢௡ •‹Š ൬ ݅ ݅௢௡ ൰ ‡š’ ቈെ ݁‫݌ݔ‬ ቆെ ‫ݔ‬ െ ܽ௢௡ ‫ݓ‬௖ െ ȁ݅ȁ ܾ ቇ െ ‫ݔ‬ ‫ݓ‬௖ ቉ ǡ ݅ ൏ Ͳ where ܿ௢௙௙ ǡ ܿ௢௡ ǡ ݅௢௡ ǡ ݅௢௙௙ ǡ ܽ௢௡ ǡ ܽ௢௙௙ǡ ‫ݓ‬௖ and b are all curve fitting parameters. III. PROPOSED MODIFIED WINDOW FUNCTIONS In case of the previously proposed window functions which were to be used with the linear ion drift model several modifications had been done to introduce the non linearity in the ion drift. But it was seen from simulation results of these window functions that the amount of non linearity resulting from these windows is less than that is depicted by practically fabricated devices. As a result to account for this extra non linearity we had to introduce different models such as the non linear ion drift model, Simmons Tunnel barrier models which had complex mathematical equations and computational time required was more. Also we had to make a lot of assumptions relating to the device parameters. A. Parabolic Window Function So here we propose a modified window function which allows us to use the simple linear ion drift model with considerable amount of non linearity introduced as a result of the modified window function. Careful study of the Jogelkar window reveals the fact that the window function is inspired from the equation of the parabola which is of the form; ‫ݕ‬ ൌ ܽሺ‫ݔ‬ െ ݄ሻଶ ൅ ݇ (10) where (h,k) represents the vertex of the parabola. When the value of 'a' is less than zero the parabola is inverted resembling a window function. The resulting equation can be represented as, ‫ݕ‬ ൌ െܽሺ‫ݔ‬ െ ݄ሻଶ ൅ ݇ (11) Now considering the magnitude of the window function to be '1', the equation for the parabolic window can be presented as ‫ݕ‬ ൌ ͳ െ ܽ ቀ ௪ ஽ െ ͲǤͷቁ ଶ௣ (12) where w denotes the state variable while D is the physical device dimensions. The window function was simulated for different values of the scaling factor 'a' and it can be seen that the amount of non linearity depicted in this window function is improved as compared to the Jogelkar window when the value of 'a' is 4. The simulation results are shown in figure . Figure 3. Simulation result of Parabolic Window Function B. Non Linear Window Function Analysis of the Prodromakis window suggests that the amount of non linearity can be further improved by changing the positive integer 'p' in the window function to a varying non linear function which will account for the damping effect in the ion drift near the device boundaries. Trigonometric functions can be substituted in place of the exponent integer. The first modified equation was taken with ͳ ൅ •‹ ሺ‫ݔ‬ሻ as a substitute for p. The resulting window function is given as ݂ሺ‫ݓ‬ሻ ൌ ݆ ቆͳ െ ൤ቀ ௪ ஽ െ ͲǤͷቁ ଶ ൅ ͲǤ͹ͷ൨ ሺଵାୱ୧୬ሺ௣ሻሻ ቇ (13) where 'j' is a control parameter which determines the maximum value of f(w). From the comparison of the simulation results of this window shown in figure with that of the Prodromakis window it can be seen that a greater degree of non linearity can be achieved with the use of this window. 159159
  4. 4. Figure 4. Simulation Result of Sine Window The second modification can be done by replacing the ͳ ൅ •‹ ሺ‫ݔ‬ሻ term in equation(13) by a hyperbolic trigonometric function such as –ƒŠሺ‫ݔ‬ሻ. The proposed equation can be written as , ݂ሺ‫ݓ‬ሻ ൌ ݆ ቆͳ െ ൤ቀ ௪ ஽ െ ͲǤͷቁ ଶ ൅ ͲǤ͹ͷ൨ ୲ୟ୬୦ ሺ௣ሻ ቇ (14) ݂ሺ‫ݓ‬ሻ ൌ ݆ ቆͳ െ ൤ቀ ௪ ஽ െ ͲǤͷቁ ଶ ൅ ͲǤ͹ͷ൨ ୡ୭୲୦ ሺ௣ሻ ቇ (15) The simulation result of the hyperbolic windows are shown in figure which indicate that better non linearity is achieved near the device bounds than the Prodromakis window due to the use of hyperbolic function. From the result of the window function it can be seen that the plotted curve represents the drift of the ions. Near the device boundaries the damping phenomenon is very smoothly exhibited by the ions and the ion drift gradually drops to zero unlike the steep change that occurs in case of Prodromakis window. The tan hyperbolic window exhibits the better non linearity than the cot hyperbolic function as it has considerable resemblance to the Prodromakis window for higher values of 'p'. The cot hyperbolic window can even produce a window magnitude of reasonable high value as compared to the tan hyperbolic window. IV. CONCLUSION The striking feature of a memristor is that irrespective of its past state, or resistance, it freezes that state until another voltage is applied to change it. There is no power requirement for maintaining the past state . This feature makes the memristor stand apart from a dynamic RAM cell, which requires regular charging at the nodes to maintain its state. The consequence is that memristors can substitute for massive banks of power-consuming memory. Figure 5. Simulation results of window functions of Prodromakis versus Tan Hyperbolic window Figure 6. Simulation Result of Prodromakis window versus Cot Hyperbolic Window From the study of the various previously proposed window functions and the modified windows proposed in this paper it can be inferred that to exploit the simplicity of the linear ion drift model a window function is needed to introduce the non linear ion drift phenomenon for accurate modeling of Memristive systems without getting into the complexity of other models like the Simmons tunnel Barrier Model, Team Model, etc. The modified windows proposed in this paper provide a better non linearity to characterize the memristor device than some of the already proposed windows as verified from the simulation results of the window functions. 160160
  5. 5. REFERENCE [1] Chua, L.O., ʊMemristor- the missing circuit element , IEEE Trans. Circuit Theory, 1971, vol. CT-18, no. 5, pp. 507-519. [2] Chua, L.O. and Kang, S.M.. ʊMemristive devices and systems , Proceedings of the IEEE, 1976, vol. 64, no. 2, pp. 209-223. [3] Frank Y. Wang, Memristor for introductory physics, arXiv:0808.0286vl [physics.class-ph] 4Aug2008. [4] S.Thakoor, A. Moopenn, T. Daud, and A.P. Thakoor, Solid-state thin-film memistor for electronic neural networks, Journal of Applied Physics, vol. 67, March 15, 1990, pp. 3132-3135 [5] Dimitri Strukov, Gregory Snider , Duncan Stewart , and Stanley Williams, The missing memristor found, Nature. [6] A. G. Radwan , M. Affan Zidan and K. N. Salama, On the Mathematical Modeling of Memristors, International Conference on Microelectronics 2010. [7] Chris Yakopcic , A Memristor Device Model, IEEE Electron Device Letters, Vol.32, No.10, Oct 2011. [8] Y. N. Joglekar and S. J. Wolf, “The Elusive memristor: Properties of basic electrical circuits,” Eur. J. Phys., vol. 30, no. 4, pp. 661– 675, Jul. 2009. [9] Z. Biolek, D. Biolek, and V. Biolkova, “SPICE model of memristor with nonlinear dopant drift,” Radioengineering, vol. 18, no. 2, pp. 210–214, Jun. 2009. [10] T. Prodromakis, B. P. Peh, C. Papavassiliou, and C. Toumazou, “A versatile memristor model with non-linear dopant kinetics,” IEEE Trans.Electron Devices, vol. 58, no. 9, pp. 3099–3105, Sep. 2011. [11] E. Lehtonen and M. Laiho, “CNN using memristors for neighborhood connections,” in Proc. Int. Workshop Cell. Nanoscale Netw. Their Appl., Feb. 2010, pp. 1–4. [12] J. G. Simmons, “Generalized formula for the electric tunnel effect between similar electrodes separated by a thin insulating film,” J. Appl. Phys., vol. 34, no. 6, pp. 1793–1803, Jan. 1963. [13] E. Lehtonen, J. Poikonen, M. Laiho, and W. Lu, “Time- dependency of the threshold voltage in memristive devices,” in Proc. IEEE Int. Symp. Circuits Syst., May 2011, pp. 2245–2248. [14] Shahar Kvatinsky, Eby G. Friedman, Avinoam Kolodny, TEAM: ThreEshold Adaptive Memristor Model, IEEE Transactions on Circuits and Systems-I, vol.60, no.1, jan-2013. [15] M. D. Pickett, D. B. Strukov, J. L. Borghetti, J. J. Yang, G. S. Snider, D. R. Stewart, and R. S. Williams, “Switching dynamics in titanium dioxide memristive devices,” J. Appl. Phys., vol. 106, no. 7, pp. 1–6, Oct. 2009. 161161