1) The model should have sufficient accuracy and
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.
B. Linear Ion Drift Model
Figure 2. HP Labs Memristor Model
A linear ion drift model which was proposed in  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,
Ǥ ݅ሺݐሻ (4)
ݒሺݐሻ ൌ ൬ܴ݊Ǥ
ܴ݂݂ ቀͳ െ
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 .
݂ሺݓሻ ൌ ͳ െ ሺ
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 .
The Biolek window suggests some changes in the window
function give as,
݂ሺݓሻ ൌ ͳ െ ሺ
ݐݏሺ݅ሻ ൌ ൜
ͳ ǡ ݅ Ͳ
Ͳ ǡ ݅ ൏ Ͳ
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 .
݂ሺݓሻ ൌ ݆ሺͳ െ ሾቀ
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 , the proposed model is based on experimental
results. The current and voltage relationship is given as
݅ሺݐሻ ൌ ݓሺݐሻ
ߚ ൫ߙݒሺݐሻ൯ ߯ሾ൫ߛݒሺݐሻ൯ െ 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 ൫ߙݒሺݐሻ൯ 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 ߯ൣ൫ߛݒሺݐሻ൯ െ ͳ൧ resembles an
ideal diode equation. A non linear dependence on voltage
in the state variable equation is assumed in this model,
ൌ ܽǤ ݂ሺݓሻǤ ݒሺݐሻ
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
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  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 ,
ቇ ቈെ ݁ݔ ቆ
ݔ െ ܽ
ǡ ݅ Ͳ
൰ ቈെ ݁ݔ ቆെ
ݔ െ ܽ
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
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
ݕ ൌ െܽሺݔ െ ݄ሻଶ
Now considering the magnitude of the window function to
be '1', the equation for the parabolic window can be
ݕ ൌ ͳ െ ܽ ቀ
where w denotes the state variable while D is the physical
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
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.
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 ሺݔሻ. The proposed
equation can be written as ,
݂ሺݓሻ ൌ ݆ ቆͳ െ ቀ
݂ሺݓሻ ൌ ݆ ቆͳ െ ቀ
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
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
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
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.
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