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Dongale et al. Nano Convergence (2016) 3:16
DOI 10.1186/s40580-016-0076-8
RESEARCH
TiO2 based nanostructured memristor
forRRAM andneuromorphic applications:
a simulation approach
T. D. Dongale1*, P. J. Patil1, N. K. Desai1, P. P. Chougule1, S. M. Kumbhar2, P. P. Waifalkar3, P. B. Patil3, R. S. Vhatkar3,
M. V. Takale3, P. K. Gaikwad4 and R. K. Kamat4
Abstract
We report simulation of nanostructured memristor device using piecewise linear and nonlinear window functions
for RRAM and neuromorphic applications. The linear drift model of memristor has been exploited for the simula-
tion purpose with the linear and non-linear window function as the mathematical and scripting basis. The results
evidences that the piecewise linear window function can aptly simulate the memristor characteristics pertaining to
RRAM application. However, the nonlinear window function could exhibit the nonlinear phenomenon in simulation
only at the lower magnitude of control parameter. This has motivated us to propose a new nonlinear window func-
tion for emulating the simulation model of the memristor. Interestingly, the proposed window function is scalable up
to f(x) = 1 and exhibits the nonlinear behavior at higher magnitude of control parameter. Moreover, the simulation
results of proposed nonlinear window function are encouraging and reveals the smooth nonlinear change from LRS
to HRS and vice versa and therefore useful for the neuromorphic applications.
Keywords: Memristor, Window function, RRAM, Neuromorphic applications
© 2016 The Author(s). This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium,
provided 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.
1 Background
Memristor which is poised to establish as the fourth cir-
cuit element in addition to the R, L and C, was theorized
way back in the year 1971 by Chua [1]. Later in the year
2008 the same was validated by the HP research group
[2]. e peculiar characteristics of remembering the data
in terms of low resistance state (LRS) and high resistance
state (HRS) makes the memristor a unique attribute for
many interesting applications not feasible with the con-
ventional circuit elements. Moreover, the passivity and
nonlinearity are some of the important characteristics
of the memristor, which leads to its usage in the appli-
cations of diversified domains such as biomedical, resis-
tive random access memory (RRAM), neural computing,
nonlinear dynamics, neuromorphic computing realm
etc. as reported widely in the literature [3–10]. As these
applications are important so is the accurate modelling
of nonlinear memristor which has been the very basis of
the scientific investigations. Incidentally many research
group including ours are actively working in this direc-
tion as put forth briefly in the following paragraph to set
the background of the present investigation.
Recently, Li etal. reported a new modelling method
with multinomial window function. is method is
derived through the statistical fitting of an experimental
data of a memristor device [11]. Batas et al. have come
out with the behavioral model of magnetic flux-con-
trolled memristor device. e reported model is simu-
lated on integrated circuits emphasis i.e. SPICE platform
[12]. Valsa et al. [13] have investigated the analogue
model of the memristor device which has been duly veri-
fied for various test signals with the results showing good
resemblance with the ones reported in literature. Shin
etal. [14] have put forth a compact circuit model and
hardware emulator for memristor device with its appli-
cations for the arithmetic operations. Kolka et al. [15]
Open Access
*Correspondence: tdd.snst@unishivaji.ac.in
1 Computational Electronics and Nanoscience Research Laboratory,
School of Nanoscience and Biotechnology, Shivaji University,
Kolhapur 416004, India
Full list of author information is available at the end of the article
Page 2 of 7
Dongale et al. Nano Convergence (2016) 3:16
reported the hardware emulator for the mem-systems
based on the memristor, memcapacitor, and memin-
ductor which can be further programmed to realize the
above mentioned trio. Quite relevant to the theme of
present paper are the investigations carried out by Biolek
etal. and Yu etal. on the nonlinear and piecewise linear
window function aspects for modelling the memristor
respectively [16, 17].
Recently, our research group too has reported two new
window functions for the modeling of the memristor
device [18]. e present research paper is an extension of
our previously reported work [3–10, 18]. While our pre-
vious papers report more of the details related to math-
ematical aspects, the present paper is a value addition as
it actually showcases the simulation in light of the RRAM
and neuromorphic applications. Both these applications
are currently in profound demand. RRAM’s seems to
be the only solution in the age of big data while a com-
pletely new paradigm of brain inspired computing is cur-
rently been explored through the neuromorphic domain.
e main achievement of the present manuscript is the
modified nonlinear window function which accurately
models the nonlinearity of memristor device. e rest of
the paper is as follows, after brief introduction in the first
section, second section deals with the overview of piece-
wise linear and nonlinear window functions. e third
section further divulges the simulation details of memris-
tor with above mentioned window function. is section
also deals with the modified nonlinear window function.
At the end results and conclusion has been placed.
2 Overview ofpiecewise linear andnonlinear
window functions
e HP research group modeled the memristor device
based on linear drift model. is model assumes that the
state variable (w) is directly proportional to charges flow-
ing through the device [2]. e structure of HP memris-
tor (Pt/TiO2/Pt memristor) is shown in Fig.1. It has two
prominent regions namely doped low resistance and
undoped high resistance.
e reported literature reveals that drifting of vacan-
cies has been highly nonlinear near the boundary
interfaces. is is attributed to the nanoscale phenom-
ena by which even a small voltage can produce large
electric field across the device. is large electric field
further generates nonlinear drifting of vacancies near
the boundary interfaces [19]. Another problem with lin-
ear drift model of memristor is that, the state variable
‘w’ never reaches to zero physical length which indicates
that the oxygen vacancies are absent in the devices [20].
e boundary problem can be minimized by adopting
window function f(x). In general, the window function
can be multiplied to state equation of memristor which
is given as,
where, w is a state variable, the parameter η indicates
the polarity of memristor e.g. η=1 indicates the expan-
sion of doped region and η=−1 indicates the shrinking
of doped region, μV is a average drift velocity of oxygen
vacancies, RON is a low resistance state of memristor
device or ON state resistance, D corresponds to total
length of the active region, and i is a current through the
device. e function f(x) should have its highest value at
the center of the device (x=0.5) and zero at the bounda-
ries (x=0 and x=1) of memristor device [20]. From the
physics point of view, window functions lower down the
speed of oxygen vacancies near the boundaries which
ultimately leads to the nonlinear behavior.
In the backdrop of the above theoretical propositions,
it is appropriate to select the fitting window functions
most suited to the intended applications to the core of
our research group. With the RRAM and neuromorphic
domain of applications, piecewise linear and nonlinear
window functions have been applied owing to their ben-
efits in terms of parameter adjustment flexibility. Upon
applying, the piecewise linear window function exhib-
its continuously differentiability at LHS bounds, middle
region and RHS bounds. It shows the nonlinear behavior
at lower values of control parameter ‘p’ and linear behav-
ior at higher values of control parameter ‘p’. e nonlinear
window function shows the quasi nonlinear behavior at
higher values of control parameter ‘p’. One of the advan-
tages of these window functions is that the control param-
eter ‘p’ can be adjusted to get required characteristics of
memristor pertaining to RRAM and neuromorphic appli-
cations [18]. Equation2 represents the generalized piece-
wise linear window function such that, [18]
(1)
dw
(
t
)
dt
=ηµv
R
ON
D
i(t)∗f(x
)
(2)
f(x)=
px
mX0for 0≤x≤X0
p
mfor X0≤x≤Y
0
p(1−x)
m(
1−
Y0)
for Y0≤x≤1
Pt
Pt Tio
2
Undoped
Tio
2-x
Doped
D
w(t)
Fig. 1 Structure of memristor reported by HP research group [2, 18]
Page 3 of 7
Dongale et al. Nano Convergence (2016) 3:16
where, 0 < X0 < Y0 < 1 and p and m ∈ R+. e nonlinear
window function has similar characteristics with respect
to piecewise linear window function except, it has non-
linear characteristics at the boundaries. Equation3 repre-
sents the nonlinear window function such that, [18]
where, 0 < X0 < Y0 < 1, Y0=(1−X0) and p ∈ R+. Figure2
represents the piecewise linear and nonlinear window
function with various values of control parameter ‘p’.
3 Simulation ofmemristor device using piecewise
linear andnonlinear window functions
e main intent behind the modeling and simulation is to
help the designers to come out with the apt device char-
acteristics per application. In the present case the main
rationale is to fine tune the memristor attributes through
simulation for two fold purposes viz. fast transition from
LRS to HRS for RRAM applications while slow transi-
tion from LRS to HRS for the neuromorphic domain. e
modeling for the above mentioned attributes has been
obtained by applying the piecewise linear and nonlinear
window functions. After zeroing down on the technique
for modeling the simulation was accomplished.
Figure3 represents the simulated I–V characteristics
of nanostructured memristor device with piecewise lin-
ear window function. e present simulation is carried
out for control parameter p=10, 20, and 30 with state
variable x=0.3, 0.5, and 0.7. In the other words, these
state variables represent the growth location of con-
ductive filament in the memristor device. An illustra-
tion of such mechanism is shown in the Fig.4. e state
(3)
f(x)
x
1
pfor 0≤x≤X0
x
1
p
0for X0≤x≤Y
0
(
x
−
1
)
1
p
for Y0
≤
x
≤
1
variable x= 0.3, 0.5, and 0.7 represents LHS bounds,
middle region and RHS bounds respectively and it can
be visualize from Fig.2. e simulation results suggested
that the current in the device increases as the control
parameter p increases from 10 to 30. e relationship
between current and control parameter with various val-
ues of state variable is shown in the Fig.5. From Figs.3
and 5, it is seen that the memristor device shows RRAM
kind of characteristics at higher magnitude of control
parameter. e results also suggest that abrupt switch-
ing occurs at the higher magnitude of control parameter.
For the lower magnitude of control parameter, current
and pinched hysteresis loop (PHL) become small. e
area under the PHL is also increases as the magnitude of
control parameter increases. e change in the control
parameter can be used for the switching from one state to
another state. In the other words, if one can have power
over the control parameter then switching of the device
can be controlled. is characteristic is very similar to
digital memory and has application in the digital memory
domain. From the results it is clearly evident that mem-
ristor will be work as a promising RRAM building block
at the higher values of control parameter, when it is mod-
eled with piecewise linear window function.
Figure6 represents the simulated I–V characteristics of
nanostructured memristor device with nonlinear window
function. For the present simulation control parameter ‘p’
varies as 1, 3, 5, 10, 20, and 30. For each control param-
eter, I–V characteristics is simulated at state variable
x=0.5 (insignificant change is observed at state variable
x=0.3 and 0.7 with respect to x=0.5). e results indi-
cates that the memristor device shows nonlinear behav-
ior only at the lower magnitude of control parameter
(p=1, 3, and 5) and I–V characteristics does not alter at
higher magnitude of control parameter (p=10, 20, and
00.20.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
State Variable 'x'
f(x)
p = 1
p = 3
p = 5
p = 10
p = 20
p = 30
00.20.4 0.60.8 1
0
0.2
0.4
0.6
0.8
1
State Variable 'x'
f(x)
p = 1
p = 3
p = 5
p = 10
p = 20
p = 30
a
b
Fig. 2 Piecewise linear and nonlinear window functions with various values of control parameter ‘p’ and state variable location
Page 4 of 7
Dongale et al. Nano Convergence (2016) 3:16
30). e area under the curve is higher only at the lower
magnitude of control parameter and becomes approxi-
mately same at higher magnitude of control parameter.
is window function does not reaches to f(x) = 1 (it
becomes f(x)=1 only at p=∞) and is a main limitation
of nonlinear window function. To rectify this limitation,
we are proposing a new window function which can
be scaled up to f(x)=1 at higher magnitude of control
parameter. e proposed nonlinear window function can
be defined as,
Fig. 3 I–V characteristics of nanostructured memristor device with piecewise linear window function. a1–a3 I–V characteristics of memristor
device at x = 0.3, x = 0.5, and x = 0.7 respectively with control parameter p = 10. b1–b3 I–V characteristics of memristor device at x = 0.3, x = 0.5,
and x = 0.7 respectively with control parameter p = 20. c1–c3 I–V characteristics of memristor device at x = 0.3, x = 0.5, and x = 0.7 respectively
with control parameter p = 30
Page 5 of 7
Dongale et al. Nano Convergence (2016) 3:16
(4)
If p <p
0
then,
f(x)=
x1
pfor 0≤x≤X0
x
1
p
0for X0≤x≤Y0
|(x−1)|
1
pfor Y0≤x≤1
Otherwise if p =p0then,
f(x)=
x1−x0x
pfor 0≤x≤X0
x
p0−p
p
0for X0≤x≤Y
0
|
(x
−
1)
|−x0x−1
p
for Y0
≤
x
≤
1
where, 0 < X0< Y0< 1, Y0=(1−X0) and p ∈ R+. Figure7
shows the difference between two nonlinear window
functions with various values of control parameter ‘p’.
e results suggested that the proposed window func-
tion scaled up to f(x)=1 at higher magnitude of con-
trol parameter. e simulation of memristor device with
modified nonlinear window function is shown in the
Fig.8. e results suggested that modified nonlinear win-
dow function is able to simulate the memristor character-
istics at higher magnitude of control parameter. From the
results it is clear that the current in the device increases
as a function of state variable i.e. magnitude of current
increases as value of state variable increases. I–V char-
acteristic shows smooth nonlinear change from LRS to
HRS and vice versa. is is due to the fact that the drift-
ing of oxygen vacancies (state variables of the memristor)
are highly nonlinear when switching occurred. Further-
more, the applied bias, device geometry, and conduction
mechanism also influences the device switching dynam-
ics [2, 19]. is is very similar to analog memory and has
application in the neuromorphic engineering domain.
In nutshell, modified nonlinear window function can be
used for the analog memory applications.
4 Conclusion
e present investigation reports the simulation of TiO2
nanostructured memristor device for RRAM and neu-
romorphic applications. e results strongly indicate
the suitability of piecewise linear window function to
carve the model of the nanostructured memristor device
characteristics for RRAM application which is further
been validated through simulation. Altering the control
Top Electrode
Bottom Electrode
Conductive
Filament
x= 0.7
Top Electrode
Bottom Electrode
Conductive
Filament
x= 0.5
To p Electrode
Bottom Electrode
Conductive
Filament
x= 0.3
Fig. 4 Illustration of growth location of conductive filament in the memristor device. The state variable x = 0.3, 0.5, and 0.7 represents LHS bounds,
middle region and RHS bounds respectively
10 15 20 25 30
0.0
0.2
0.4
0.6
0.8
1.0
Current (A)
Control Parameter 'p'
x = 0.3
x = 0.5
x = 0.7
Fig. 5 The relationship between current and control parameter with
various values of state variable (x). It is clear that current abruptly
increases at higher value of control parameter
Page 6 of 7
Dongale et al. Nano Convergence (2016) 3:16
parameter from one state to another state makes the
piecewise linear window function a best fit for the RRAM
application. e modified nonlinear window function
eliminates the scaling issue and thus accomplishes simu-
lation of the memristor characteristics at higher magni-
tude of control parameters. e results are encouraging
Fig. 6 I–V characteristics of nanostructured memristor device with nonlinear window function. a–f The I–V characteristics of memristor device at
x = 0.5 and control parameter p = 1, 3, 5, 10, 20, and 30 respectively
00.20.40.60.8 1
0
0.2
0.4
0.6
0.8
1
State Variable 'x'
f(x)
p = 1
p = 3
p = 5
p = 10
p = 20
p = 30
00.20.40.60.8 1
0
0.2
0.4
0.6
0.8
1
State Variable 'x'
f(x)
p = 1
p = 3
p = 5
p = 10
p = 20
p = 30
a b
Fig. 7 Difference between two nonlinear window functions with various values of control parameter ‘p’, a nonlinear window function, b modified
nonlinear window function
Page 7 of 7
Dongale et al. Nano Convergence (2016) 3:16
and show strong applicability towards neuromorphic
engineering domains on which our research investiga-
tions are in progress.
Authors’ contributions
TDD, PJP, and SMK designed the mathematical model. TDD, NKD, PPC, PPW,
and PBP developed the MATLAB and Mathematica code. TDD, RSV, and MVT
analyzed the results. RSV, MVT, PKG and RKK provided the advice on and
coordinated the study. TDD and RKK documented the manuscript. All authors
reviewed the manuscript. All authors read and approved the final manuscript.
Author details
1 Computational Electronics and Nanoscience Research Laboratory, School
of Nanoscience and Biotechnology, Shivaji University, Kolhapur 416004, India.
2 Rajarambapu Institute of Technology, Sakharale 415414, India. 3 Department
of Physics, Shivaji University, Kolhapur 416004, India. 4 Embedded System
and VLSI Research Laboratory, Department of Electronics, Shivaji University,
Kolhapur 416004, India.
Acknowledgements
The authors are very much thankful to Dr. S. S. Kumbhar for fruitful discussion
on window functions. The authors are very much thankful to Prof. P. S. Patil for
providing research facilities.
Competing interests
The authors declare that they have no competing interests.
Received: 24 March 2016 Accepted: 13 June 2016
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Fig. 8 a–c Simulation of memristor device with modified nonlinear window function at control parameter p = 30 and state variable x = 0.3, 0.5
and 0.7 respectively