The present paper reports two new window functions viz. piecewise linear window function and nonlinear window function for modelling of the nanostructured memristor device. The piecewise linear window function can be used for modelling of symmetric pinched hysteresis loop in I-V plane (for digital memory applications) and the nonlinear window function can be used for modelling of nonlinear pinched hysteresis loop in I-V plane (for analog memory applications). Flexibility in the parameter selection is the main attractive feature of these window functions

Delekar, S.D.

Dongale, T.D.

Gaikwad, P.K.

Kamat, R.K.

More, K.V.

Mullani, S.B.

Patil, K.P.

Patil, P.J.

English

SocioEconomic Challenges Journal (Sumy State University)

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ 
Vol. 7 No 3, 03012(4pp) (2015) Том 7 № 3, 03012(4cc) (2015) 
 
 
2077-6772/2015/7(3)03012(4) 03012-1  2015 Sumy State University 
Piecewise Linear and Nonlinear Window Functions for  
Modelling of Nanostructured Memristor Device 
 
T.D. Dongale1,*, P.J. Patil1, K.P. Patil1, S.B. Mullani2, K.V. More2, S.D. Delekar2, 
P.K. Gaikwad3, R.K. Kamat3,† 
 
1 Computational Electronics and Nanoscience Research Laboratory,  
School of Nanoscience and Biotechnology, Shivaji University, Kolhapur-416004, India 
2 Nanoscience Research Laboratory, Department of Chemistry, Shivaji University, Kolhapur-416004, India 
3 Embedded System and VLSI Research Laboratory,  
Department of Electronics, Shivaji University, Kolhapur-416004, India 
 
(Received 09 February 2015; revised manuscript received 14 October 2015; published online 20 October 2015) 
 
The present paper reports two new window functions viz. piecewise linear window function and nonlin-
ear window function for modelling of the nanostructured memristor device. The piecewise linear window 
function can be used for modelling of symmetric pinched hysteresis loop in I-V plane (for digital memory 
applications) and the nonlinear window function can be used for modelling of nonlinear pinched hysteresis 
loop in I-V plane (for analog memory applications). Flexibility in the parameter selection is the main at-
tractive feature of these window functions. 
 
Keywords: Memristor, Simulation, Window function. 
 
 PACS numbers: 07.05.Tp, 73.40.Rw, 68.60.Wm. 
 
 
                                                                
* tdd.snst@unishivaji.ac.in 
† rkk_eln@unishivaji.ac.in 
1. INTRODUCTION 
 
The memristor is a fourth fundamental circuit com-
ponent and it has potential application in the area of 
Resistive Random Access Memory (RRAM), biomedical 
applications, new kind of electronic circuits and many 
more [1-5]. The pinched hysteresis loop in I-V plane is 
one of the fingerprint characteristics of memristor. This 
loop is analogues with the Low Resistance State (LRS) 
and High Resistance State (HRS) of RRAM based de-
vices. The LRS and HRS loop can be symmetrical and 
asymmetrical (nonlinear) in the nature. The symmet-
rical loop can be used for digital memory application 
due to its instantaneous switching from LRS to HRS 
and vice versa. The asymmetrical loop can be used for 
analog memory application such as neuromorphic ap-
plication where switching of memristor can be consid-
ered as smooth, highly nonlinear switching from LRS to 
HRS and vice versa. Recently Dongale et al. developed 
nanostructured TiO2 thin film memristor with low 
symmetric voltage switching ( 0.68 V) [3]. Similarly 
Kundozerova, et al. developed anodic oxides based 
memristor in which I-V curve is asymmetric [6]. 
The literature survey reveals that, most of the re-
searchers applied the nonlinear window function or 
derived nonlinear dependence of the state variable for 
the simulation of nonlinear effects of memristor device 
[7-10]. The explicit relationship between memristance 
(M) and charge (q) cannot be obtained using above 
model and theories. If we would like to simulate the 
memistor with above window functions, then we re-
quire some numerical methods for solving the differen-
tial equations [11, 12]. To deal with this problem, we 
are proposing two window functions viz. Piecewise 
Linear Window Function and Nonlinear Window Func-
tion. 
The rest of the paper portrayed as follows, the first 
section deals with the introduction and outline of the 
problem statement. The second section discusses back-
ground theory of memristor device and simulation of 
memristor device using linear drift model. The third 
section briefly describes the proposed window functions 
and at the end conclusion is reported. 
 
2. BACKGROUND THEORY OF MEMRISTOR 
DEVICE 
 
The memristor is popularly known as fourth fun-
damental and passive circuit component [1]. The 
memristor device was first of all predicted by Prof. L. 
Chua in his seminal research paper [1] in 1971. After 
forty years, in 2008, HP research group reported the 
first physical prototype of memristor device [2]. The 
pinched hysteresis Current-Voltage (I-V) loop and non-
linear curve between Charge (q)-Magnetic Flux (φ) are 
the fingerprint characteristics of nanostructured 
memristor device. The current controlled and voltage 
controlled are the two types of memristor device.  
The HP memristor model considered the drift of oxy-
gen vacancies as a state variable in the Pt / TiO2 / Pt 
structure. The typical structure of HP memristor is de-
picted in the Fig. 1. 
 
PtPt
Tio2
Undoped
Tio2-x
Doped
D
w(t)
 
Fig. 1 – Structure of memristor reported by HP Lab [2] 
 T.D. DONGALE, P.J. PATIL, ET AL. J. NANO- ELECTRON. PHYS. 7, 03012 (2015) 
 
 
03012-2 
The Valency Change Mechanism (VCM) is the basis 
of memristor. The existing literature reveals that the 
most of the memristor is modeled around the ideal HP 
memristor model in which drift of oxygen vacancies 
considered as a state variable, which is similar to 
Chua’s state variable ‘w’ [1, 2]. The reported HP 
memristor consists of Pt / TiO2 / Pt structure in which 
TiO2 – x oxygen rich doped conductive layer plays an 
important role to produce memristor like characteris-
tics. The ‘D’ is a thickness of active sandwich structure 
and ‘w’ is the thickness of doped region. The HP 
memristor is based on linear drift model and consid-
ered the linear ionic drift. If average drift velocity of 
oxygen vacancies μv, then memristor current and volt-
age relation can be represented using following math-
ematical equations (1) [2]: 
 
 
   
 ( ) 1ON OFF
R w t w t
V t R i t
D D
    
         
     
, (1) 
 
Where state variable ‘w’ can be represented as, 
 
 
 
 ON
dw t u R
i t
dt D
  (2) 
 
Integrating equation (2) w.r. to ‘t’ we get, 
 
  ON
u R
w q t
D
  (3) 
Inserting equation (3) into (1) we get memristance 
M(q), 
If RON  ROFF, 
 
  
 
2
1 ONOFF
u R q t
M q R
D
    
 
 (4) 
 
The simulation of linear drift model is represented 
in the Fig. 2. The experimental results suggested that 
the drifts of vacancies are highly nonlinear near the 
boundary interfaces and it is generally known as non-
linear dopent drift [13-16]. It is due to fact that, a small 
voltage across the nanoscale memristor produces large 
electric field across the device. This large electric field 
produces nonlinear drifting of vacancies near the 
boundary interfaces [13-16]. Another problem with 
linear drift model of memristor is that the state varia-
ble ‘w’ never reaches to zero physical length which 
indicates oxygen vacancies are absent in the devices 
[17]. The boundary problem can be minimized using 
window function f(x). In general, the nonlinear dopent 
drift can be obtained by simply multiplying the state 
equation of memristor (equation 2) with the window 
function f(x) [1, 8]. 
 
 
 
   ON
dw t u R
i t f x
dt D

 
  
 
 (5) 
 
The function f(x) should have its highest value at 
the center of the device (x  0.5) and zero at boundaries 
of memristor device [17]. 
a
dc
b
 
 
Fig. 2 – Simulation results of linear drift model of nanostructured memristor device (a) plot of flux vs charge. (b) plot of current vs 
voltage. (c) plot of relationship between current and voltage vs width (state variable). (d) plot of log (current (I)) vs log (voltage (V)) 
 PIECEWISE LINEAR AND NONLINEAR WINDOW FUNCTIONS… J. NANO- ELECTRON. PHYS. 7, 03012 (2015) 
 
 
03012-3 
3. PROPOSED WINDOW FUNCTIONS 
 
3.1 Piecewise Linear Window Function 
 
The proposed window function is piecewise continu-
ously differentiable at three regions viz. LHS bounds, 
middle region and RHS bounds and shows the nonline-
ar behaviour at lower values of control parameter ‘p’ 
and linear behaviour at higher values of control pa-
rameter ‘p’. The equation (6) represents the generalized 
piecewise linear window function such as, 
 
 
0
px
mX
  for 0  x  X0 
f(x)   
p
m
   for X0  x  Y0    (6) 
 
 
 0
1
1
p x
m Y


 for Y0 ≤ x  1 
 
Where, 0  X0  Y0  1 and (p and m ∈ R+). Consider 
a suitable case when LHS bounds lies between 
0  x  1/3 and RHS bounds lies between 2/3 ≤ x  1 and 
constant for middle region such that, 
 
         
3
20
px
  for 0 ≤ x ≤ 1/3 
f(x)  
20
p
  for 1/3 ≤ x  2/3    (7) 
 
 3 1
20
p x
 for 2/3 ≤ x ≤ 1 
 
Where, x(t)  w/D is the normalized form of the 
state variable, and the control parameter ‘p’ is a posi-
tive integer and m  20. Fig. 3 represents the proposed 
piecewise linear window function for various values of 
control parameter ‘p’. 
 
p
 
 
Fig. 3 – Plot of piecewise linear window function for modelling 
of nonlinearities of memristor device for various values of 
control parameter ‘p’ 
 
The Fig. 3 clearly indicates that, as the value of con-
trol parameter ‘p’ increases from 1 to 20, LHS and RHS 
bounds of window function shows the quasi nonlinear 
behaviour of dopent kinetics at the boundaries and 
middle region represents the linear behaviour of 
memristor’s dopent kinetics. This indicates proposed 
model accurately considered the linearity of dopent 
drift at the middle region of the device and quasi non-
linear behaviour at the boundaries. The proposed win-
dow function has more flexibility than other nonlinear 
window functions. The proposed window function simu-
lates the nonlinearity at the boundaries only for the 
lower value of control parameter ‘p’. This drawback is 
rectified by using nonlinear window function. One such 
approach is depicts in the next section. One can adjust 
the values of X0, Y0, m and control parameter ‘p’ for 
modelling of the memristor based digital memories. 
 
3.2 Nonlinear Window Function 
 
The proposed piecewise linear window function 
simulate the quasi nonlinear behaviour of dopent kinet-
ics at the boundaries and linear behaviour in the mid-
dle region. A good window function must possess full 
nonlinear behaviour at the boundaries and linear be-
haviour in the middle region. It is found that the 
piecewise linear window function shows the nonlinear 
behaviour only at lower value of control parameter ‘p’. 
This problem can be minimized by adopting the smooth 
nonlinear window function. The proposed nonlinear 
window function is similar to piecewise linear window 
function except, it has nonlinear characteristics at the 
boundaries. The equation (8) represents the proposed 
nonlinear window function such that, 
 
 
1
px     for 0  x  X0 
f(x)  
1
0
px   for X0  x  Y0       (8)
  
1
1 px    for Y0 ≤ x  1 
 
Where, 0 ˂ X0 ˂ Y0 ˂ 1, Y0  (1 – X0) and (p ∈ R+). 
Consider a suitable case when LHS bounds lies be-
tween 0 ≤ x ≤ 1/5 and RHS bounds lies between 
4/5 ≤ x ≤ 1 and constant for middle region such that, 
 
   
1
px   for 0 ≤ x ≤ 1/5 
f(x)    
1
5 p   for 1/5 ≤ x ≤ 4/5    (9) 
   
1
1 px   for 4/5 ≤ x ≤ 1 
 
Where, x(t)  w/D is the normalized form of the 
state variable and the control parameter ‘p’ is a positive 
integer and m  20. The nonlinear dopent drift can be 
obtained by simply multiplying the state equation of 
memristor with the window function f(x). The Fig. 4 
represents the proposed nonlinear window function for 
various values of control parameter ‘p’. 
Fig. 4 suggested that, as the value of control param-
eter ‘p’ increases from 1 to 20, the LHS and RHS 
bounds of window function show the nonlinear behav-
iour of dopent kinetics at the boundaries, and middle 
region represents the linear region of memristor’s 
dopent kinetics. Moreover, it accurately fitted the non-
linearity for the higher values of control parameter ‘p’.  
 
 
 
 
 T.D. DONGALE, P.J. PATIL, ET AL. J. NANO- ELECTRON. PHYS. 7, 03012 (2015) 
 
 
03012-4 
 
 
Fig. 4 – Plot of nonlinear window function for modelling of 
nonlinearities of memristor device for various values of control 
parameter ‘p’ 
 
This indicates that the model accurately considered 
the linearity of dopent drift at the middle region of the 
device and nonlinear behaviour at the boundaries. The 
proposed window function has more flexibility than 
other nonlinear window functions and proposed piece-
wise linear window function. The proposed nonlinear 
window function can be used for modelling of the 
memristor based analog memories. 
 
4. CONCLUSION 
 
The present paper discuses two new window func-
tions for modelling of memristor based analog and digi-
tal memories. The piecewise linear window function 
accurately models the linearity of dopent drift at the 
middle region of the device and quasi nonlinear behav-
iour at the boundaries. The nonlinear window function 
models the linearity of dopent drift at the middle region 
of the device and nonlinear behaviour at the bounda-
ries. The main feature of these models is the flexibility 
of its operation and one can adjust the parameters of 
window function for modelling of memristor based 
memory devices. 
 
 
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Piecewise Linear and Nonlinear Window Functions for Modelling of Nanostructured Memristor Device

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Piecewise Linear and Nonlinear Window Functions for Modelling of Nanostructured Memristor Device

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